NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 71C46A87-4768-404C-BAFD-7A0E093C3C4E final-5af47 #1 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 B64A3240-E65D-4313-B277-39B071DFCFDD final-5af47 #1 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 AEE9F521-D588-4C91-A321-CD26E79E2E2E final-5af47 #1 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 37873CE0-8B3D-4FA8-84FB-B13DB3339FF2 final-5af47 #1 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 3617C809-FFAA-45EA-B4D7-35FFDA1DDA4F final-5af47 #1 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 8547256A-D770-4CE5-88E6-0B42BE966B14 final-5af47 #1 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 943750C9-D196-4383-B878-FF95198EC4A2 final-5af47 #1 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 1882520B-1D98-4A96-B4A8-FBCFAD38571B final-5af47 #1 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 9D3608FB-DEF7-4BDF-8946-0D3A953BC455 final-5af47 #1 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 FD1909D2-45F8-4CE5-8152-48E3FC69BE39 final-5af47 #1 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 1BB569F7-1226-40F0-A711-9D8984FDDC8F final-5af47 #1 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 592495DC-2312-4D9C-B47A-38EB410A8720 final-5af47 #1 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 CF28CD77-09A8-49B7-979C-EF773BDF769A final-5af47 #1 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 94CED21A-C5F2-4106-949B-F5A75D25D881 final-5af47 #1 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 B3D58967-0115-46AD-989A-3F13FDFB31F3 final-5af47 #1 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 765F448B-0C31-4454-BB4B-4622113286C2 final-5af47 #1 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 9839C553-943C-48FC-97F3-DC9CB010E2FC final-5af47 #1 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 9ABF5829-F595-477B-8B3B-829B89F41F5B final-5af47 #1 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 956799F0-AE6A-4BFE-A4A7-C1ED51AA4193 final-5af47 #2 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 2056B418-8878-451A-BFAF-156F0D389A6C final-5af47 #2 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 9021F7F7-B820-475D-BC16-1CAA73B67FE0 final-5af47 #2 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 BFCC5B4A-E4D3-400A-9F8A-50D04A90FB4C final-5af47 #2 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 2D83E31D-AAB9-4E0A-8108-08B293B8BB77 final-5af47 #2 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 266722C3-D2CD-418F-9CE6-2E5478BABE81 final-5af47 #2 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 4459E3C8-5E4D-4D4C-98A2-AFB8F0EE8A1F final-5af47 #2 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 751D68EF-3B0C-4464-800A-C0ACF6D690DF final-5af47 #2 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 0163045E-16D3-4BED-94F5-47A96B55826B final-5af47 #2 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 6665CBD9-7DD0-41FA-A3E4-8C4B572E9E1F final-5af47 #2 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 10752E87-D9A6-4073-8737-20F33B100D7E final-5af47 #2 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 4E7057F9-6758-42D7-B856-1958448170BB final-5af47 #2 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 0B660DBF-AED0-436B-B5DA-47D703959D6D final-5af47 #2 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 9F6F5918-191F-4D86-B380-98D3E60064CF final-5af47 #2 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 0BF18DB0-F294-4CE0-89CD-D1C03AF6AFB1 final-5af47 #2 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 7233BE39-70BB-4824-883E-8F2E4903A0E7 final-5af47 #2 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 E6057108-4F5F-48E6-8BA0-B0409496EBD4 final-5af47 #2 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 A376A963-1239-4A35-845D-7C9130E2A49F final-5af47 #2 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 3F94047C-ECA8-47FE-B7A8-F7B551F88333 final-5af47 #3 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 2CCC57D6-FADE-4B43-9A3A-779014191ED0 final-5af47 #3 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 9F5D0D53-7144-4B97-98F7-4B35ABB7856B final-5af47 #3 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 E4961181-B8B9-4D26-85D5-F8FEE534BF80 final-5af47 #3 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 1B485606-7013-4798-A304-8078FCF30B03 final-5af47 #3 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 CF642B42-2715-4568-B353-1C32E11D2E5E final-5af47 #3 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 4E96A387-0217-423E-9EFD-DED0D5A02F87 final-5af47 #3 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 2C3C5F13-776C-46DE-AB73-EAFA668288AF final-5af47 #3 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 5254EC7F-8119-49B9-B7C8-501ABA2205E5 final-5af47 #3 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 C45D4B86-0187-44B4-9499-D207C1AA1210 final-5af47 #3 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 6AE73E8F-67D6-420F-9B9C-C27EA9B39036 final-5af47 #3 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 C8B8E881-BF37-434F-9C9A-C52170668F5E final-5af47 #3 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 47464929-90C1-409B-889D-7E28DB6239ED final-5af47 #3 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 3B1F9CAF-CC90-4445-8A5B-3B975BF1C410 final-5af47 #3 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 AF286A1C-52E6-4B78-ACC0-4F92174EA533 final-5af47 #3 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 2B618FAB-7D9D-4B7F-9715-42BBBD517260 final-5af47 #3 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 C4E80B1A-F479-44F5-9161-2C7C3EC4DC87 final-5af47 #3 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 7B13A997-789D-476C-BAE8-67937C6AF043 final-5af47 #3 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 C3B1EAEA-D499-4F96-B0BF-ACD4A5D7F9C9 final-5af47 #4 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 3187D2FC-6132-4A31-9789-59D72AF9980B final-5af47 #4 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 3A1DACD1-DCCC-429B-8852-087A6F7886AE final-5af47 #4 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 88AA6C6D-DE1E-4990-AD1E-2B0F26AFDD5F final-5af47 #4 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 6D18E31C-17EA-4A2D-B8B5-020664A7CF91 final-5af47 #4 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 C0BA7716-7A4C-48CB-BF64-F10A1D657D37 final-5af47 #4 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 9ECB419E-098B-473A-914A-7A715F5A94B8 final-5af47 #4 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 0BA186A4-F70E-4B2B-8891-0D8BC9DF719F final-5af47 #4 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 FD81EBAC-F9C2-4E7E-B75D-1854683B616D final-5af47 #4 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 51310CA8-4B0F-46A2-8F22-F30A24E24DF0 final-5af47 #4 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 17B709BF-258F-420C-B8C2-E663684B64CE final-5af47 #4 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 3DDD0312-877B-47EF-A765-42743B817FEB final-5af47 #4 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 1D63565A-BB20-4ACC-8ED8-6263B76D69C9 final-5af47 #4 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 ED6A89BD-B05E-48EC-8D2A-51C9CBC0456B final-5af47 #4 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 9A858EC5-BBF8-417B-820C-556759BF8CF0 final-5af47 #4 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 A6675E34-702E-4118-AE62-AFD2E6F83038 final-5af47 #4 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 B3892771-1590-44DD-A1A5-C682ED18E028 final-5af47 #4 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 31C41264-5F15-4857-B42A-2B3A5277EB51 final-5af47 #4 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 2E25B1ED-1F6C-4CE8-BDD3-F45DCA38B629 final-5af47 #5 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 17831359-113A-488C-93EF-E798B6996756 final-5af47 #5 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 24A87477-BFBD-491F-8A66-665AF9A8B306 final-5af47 #5 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 782FC980-FD15-420B-9C92-53FE6B8CD6A0 final-5af47 #5 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 C29EA3ED-65EA-407A-BBF9-BE6ECC8E3AA8 final-5af47 #5 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 63819DE9-A3A7-4519-96EA-9A091DC6BEA5 final-5af47 #5 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 E81672F1-F9E9-455A-A2A4-1F477FB4F3FF final-5af47 #5 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 ED146C67-EE34-44C6-BD1C-692650BC46C9 final-5af47 #5 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 72E33966-0635-4BE9-9282-12498E3FD286 final-5af47 #5 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 1E5A2A9B-E8EC-41F9-AA00-53A3479B8A42 final-5af47 #5 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 69D5EE4E-2FAA-4A2B-B4C3-61332C273EB9 final-5af47 #5 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 EFB52FB3-5E11-4F48-9945-11A9F7B41498 final-5af47 #5 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 9A9503E4-2E0E-4F8C-8DA0-64EF9E55D2C8 final-5af47 #5 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 4D604C7D-BF1C-4521-8671-8A6A33D6D868 final-5af47 #5 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 D06B855E-4104-4499-9AFF-53AD3287F772 final-5af47 #5 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 BB7DCD31-64C3-4E01-ABE4-7E5120D270AC final-5af47 #5 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 823E756E-3B4F-4DA1-B2DB-8B282C47C439 final-5af47 #5 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 8CF146F1-8928-4B76-994F-567E43931AB5 final-5af47 #5 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 C0C32314-9924-4077-BB14-F7FC99175AFE final-5af47 #6 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 69175839-07D6-4E17-86FD-A97CE150E888 final-5af47 #6 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 969B5C6D-2367-4EE9-A954-27DA5BE9C744 final-5af47 #6 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 A32986F9-A6F4-44D4-AE50-AC4E902EE2D4 final-5af47 #6 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 835F19CD-C0DD-4DC1-A921-6AA8EE989742 final-5af47 #6 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 037ED3A8-397B-4DDF-BFBF-0AC2F6B02628 final-5af47 #6 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 0503DA0E-15CB-407D-941B-0E2EF7AD7613 final-5af47 #6 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 19ECCB69-9E35-40CF-9EE9-97D1C460A98A final-5af47 #6 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 A6A1DEAB-94E3-445D-BCFD-AF00254332F5 final-5af47 #6 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 F2BF67E1-D590-4755-B1AB-68E447591B89 final-5af47 #6 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 3E898E39-AA94-4270-B0E2-8A6A0D2EC759 final-5af47 #6 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 143D6816-0468-47C8-9157-8D73BC2FC44C final-5af47 #6 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 D5DE206A-C8C3-4116-8565-CAC526782FFC final-5af47 #6 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 B8923078-71A9-4B5F-AEBD-7D6392E479F2 final-5af47 #6 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 4C8CDF8F-F4B2-4E03-B2A6-3B58C1323B77 final-5af47 #6 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 5B5EB357-AF10-4754-B2AF-1F5DB2F3A7BF final-5af47 #6 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 91C773D3-1469-4067-9CE6-8FA54F51C304 final-5af47 #6 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 689F3C13-846C-40DF-BF3D-B63B7BD9849E final-5af47 #6 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 68FB277E-09CC-4965-9F70-225A241D8CAB final-5af47 #7 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 8C1EC9E7-9C05-476E-95E3-C10B6D4BAE98 final-5af47 #7 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 A6B5BCC2-9463-4328-8D95-6324DF59683E final-5af47 #7 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 D0E41ECE-8E92-4E12-8D47-24FB7FD4D0A4 final-5af47 #7 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 91EBBD3D-E9AD-4CA5-ABC3-8C0B9CADA974 final-5af47 #7 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 CEE6146C-D39D-4F2B-9963-512209F003BC final-5af47 #7 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 D7AA8281-456C-417B-9275-BB658A3CC566 final-5af47 #7 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 BA1DF818-20B6-480A-8C6D-99460EBCCDEB final-5af47 #7 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 042D1390-1FE2-4547-853D-AC6D2C56BC4C final-5af47 #7 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 48FB4F39-02D0-4F79-9989-5FFC45ADFA5C final-5af47 #7 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 17BEC6BA-2FAA-4299-ABD1-4D6613DFBD7B final-5af47 #7 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 9FFAF44A-B829-4D02-BC7E-DF58DE02ABD6 final-5af47 #7 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 5334A2D7-347D-41C8-BA7B-06DF9A227CA3 final-5af47 #7 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 88292F79-7352-46E6-8837-F12664A14221 final-5af47 #7 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 342F82A9-09C0-4D63-8507-9D8AEA9E355E final-5af47 #7 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 BFCF06D1-1F19-44F3-8F7F-6587E7A05817 final-5af47 #7 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 ABEBAAB1-219B-4EC8-86CF-DB3DADD3D0C4 final-5af47 #7 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 129E7D7F-8339-438D-AFBF-CDB3CD6544F3 final-5af47 #7 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 2D3F9CBE-B9B2-4B02-B6C5-EDBA13E22A1F final-5af47 #8 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 6921FB1C-6637-4014-86A3-6CE6BC5BCE2F final-5af47 #8 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 30833615-6109-4DAD-97C1-77CFD30E8862 final-5af47 #8 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 444AF682-DA4A-4D17-8A15-F22C2DAC4042 final-5af47 #8 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 E71366A7-8C34-4584-98ED-CFF3B7CC00A2 final-5af47 #8 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 C1D1A629-5CEA-4CB6-B2F9-123240B9C607 final-5af47 #8 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 2FD570F7-33AA-4FCF-9AA9-0A318F691F58 final-5af47 #8 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 2CC3B1FD-D7BE-4F58-B625-6D3F8C81822E final-5af47 #8 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 BD40C63B-2146-42F6-AA1C-023FF73DED4A final-5af47 #8 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 3BB5D330-8D2C-4DBA-BAD9-2C57287EB96D final-5af47 #8 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 895F31FB-B57F-46D6-B813-406AF2E503C7 final-5af47 #8 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 852291F4-0434-439F-9A8D-CF99C764341E final-5af47 #8 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 45BE0B79-AFB3-478C-B1AE-EA6AE337CAC3 final-5af47 #8 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 FC396AF9-F444-4006-A837-2246E1F180B1 final-5af47 #8 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 870060E2-31EA-45F9-BF12-1B2710ABBA7B final-5af47 #8 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 60EEB9FA-1DD7-46AA-8A6B-958AD02638C5 final-5af47 #8 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 8CA524B7-55A0-4DA9-B81C-F344F1710BC1 final-5af47 #8 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 03812866-FF5D-45C2-A013-DD3DBAD14018 final-5af47 #8 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 C6AA4485-9167-45BC-9446-C157E17C15DB final-5af47 #9 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 10F5D1E5-3B7F-49C8-8896-9175C7E2A262 final-5af47 #9 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 13D09833-09C3-4BA0-8557-6CAA0354A247 final-5af47 #9 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 7D8C73BD-D11B-45C2-9A3C-05059D6AF668 final-5af47 #9 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 DD4C7BCF-63BE-4C6D-8FFB-E40FBB4524AB final-5af47 #9 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 59BD650D-C110-4C60-8F6F-9EC95600B3B6 final-5af47 #9 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 37D83F28-1C08-4503-ABF6-27ED8BC110F0 final-5af47 #9 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 5CDB8594-7378-4C75-BCDE-E453D3605322 final-5af47 #9 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 6DF5D7F8-209E-4FF1-AAC7-0B82E2086840 final-5af47 #9 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 4377DA05-BB49-4E0C-AD40-DDBC6027AB30 final-5af47 #9 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 698BCA73-B56F-42F0-8A66-DB5B6AAEF084 final-5af47 #9 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 F9F815C6-EE2A-4888-B723-5F7A7BFEB99F final-5af47 #9 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 B88426A4-C606-4135-AC24-6B155410F841 final-5af47 #9 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 FEDDD533-9E76-4F54-AC4E-0278A9A5B8FE final-5af47 #9 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 9D7C5C95-04D3-4459-B0F7-B632C85DF957 final-5af47 #9 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 CED05D5D-5DC8-449E-B6EC-332EB9322581 final-5af47 #9 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 2DD0AD91-476A-4A42-BFC6-3CFEE12B1815 final-5af47 #9 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 4E80B867-04DC-49DE-AC76-95F6A45983D7 final-5af47 #9 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 15C49476-EF6C-4040-8B07-08A6EFCAB550 final-5af47 #10 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 6C697CF4-CBF6-4ADF-8238-2B0E94DA553D final-5af47 #10 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 F6C5A289-328A-46FD-9627-8C51990706F9 final-5af47 #10 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 79B8AA37-6433-4D32-AD6C-E0CC67194F9B final-5af47 #10 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 2B57F443-0F3B-42E2-98D8-226BE4AC2577 final-5af47 #10 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 0E9D2A13-2929-475D-B604-E3D54FA73F09 final-5af47 #10 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 67679B03-3ECE-4218-A566-CD981B5067BE final-5af47 #10 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 5BD6179B-0DC8-4D6A-8770-673425B07B68 final-5af47 #10 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 1E665F49-E753-4C2E-A663-E952C7AEE361 final-5af47 #10 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 58C57D31-6310-4208-9C67-8F6E708A4E15 final-5af47 #10 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 10CD06C1-AFB3-4030-9593-DF6215AA6CFE final-5af47 #10 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 098AB1E2-2580-4A9F-9D61-D3AE114218A1 final-5af47 #10 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 162E76C0-CA43-4A3A-B6E1-01E9A64E9E4A final-5af47 #10 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 3A81775A-0BEC-4736-B23D-B4698F177BCF final-5af47 #10 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 21BCA262-00E6-4A12-AEBF-C6F31E757C22 final-5af47 #10 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 590A55F9-C73B-413C-B8FF-78CD34A082E5 final-5af47 #10 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 47C57B16-158A-4178-BE0E-555F795E0910 final-5af47 #10 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 46C1B4B4-6E35-4F69-986E-BA058A77EA2C final-5af47 #10 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 2D26A424-4324-4428-A23C-326C9FAD9AE1 final-5af47 #11 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 82919FFD-B4A7-467F-AEA3-E46DC5EAAD09 final-5af47 #11 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 CAF0B9BF-31EC-404C-91F3-6FA0B982A3D3 final-5af47 #11 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 3B8108B4-DF98-4A0A-B469-168353A9F852 final-5af47 #11 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 2E9FEE75-BC6A-4BED-8EA5-178D85C8692A final-5af47 #11 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 8E31E44E-E0EB-40ED-B543-32B2A10ED5D6 final-5af47 #11 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 AC6F7DAB-4582-445E-815F-47943032A772 final-5af47 #11 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 9159C010-A395-428A-B071-B657EAF2F012 final-5af47 #11 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 C503EB52-8243-4423-8821-EF9BDA7329C9 final-5af47 #11 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 6AFDD92E-B2BC-4B48-8A56-F85A71EA5FC8 final-5af47 #11 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 7ABFCEDA-65AF-4881-86DA-62DD02F27459 final-5af47 #11 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 9C4E6255-DB37-4919-AA39-0E696597DD7B final-5af47 #11 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 55094197-2FF1-4C7F-81CC-81D546296558 final-5af47 #11 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 7CE5C798-F783-4338-A9F1-ED15937F12D2 final-5af47 #11 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 E709BB2B-FA4E-4AA2-B45A-A09DCBBEBCF8 final-5af47 #11 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 A2ADB149-BE06-4417-BA38-91872BE7405A final-5af47 #11 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 DB2DA29F-F3B1-446C-A572-761B54582E3F final-5af47 #11 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 79D34BEB-7FDB-4F08-89BB-A448053A4076 final-5af47 #11 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 94DAE0C6-B73A-46A0-AA2A-633135B39A1A final-5af47 #12 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 6718AF1F-627C-4855-B2D1-3B8DA235E516 final-5af47 #12 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 2B57300C-04F0-4A25-A56F-A7167B3201FC final-5af47 #12 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 3883B893-3715-4A7B-9023-B56E0162C170 final-5af47 #12 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 071CA759-B8DA-4B6F-BFB6-0A603FA15F7B final-5af47 #12 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 A56E5DF6-6DC9-4EBA-BF8C-724B5CD88E1C final-5af47 #12 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 FC598EE6-B370-42E9-BAD7-69CB0FCD412E final-5af47 #12 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 966F1D42-07D7-4F0C-9075-D2D730C28D67 final-5af47 #12 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 A2CDC7A9-11A7-48B3-8F02-755B1EB7598F final-5af47 #12 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 D9FAB3DE-ECAB-4ECB-AA0C-191D03807FC6 final-5af47 #12 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 138B3E64-671C-4B21-A9DC-FB77605531AE final-5af47 #12 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 02E80FBC-2C2A-4D95-B73E-672724278E7B final-5af47 #12 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 983FF4FF-917C-4E74-990E-75AF8DCA4959 final-5af47 #12 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 D6E22F09-6F54-4496-91B6-E756650D2B4B final-5af47 #12 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 E7039609-D50A-4861-B8AD-34254E05741D final-5af47 #12 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 3659A6A5-F988-4FF3-9B8A-0082138C05B9 final-5af47 #12 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 E23D1136-439C-4E14-AF03-8835A301384A final-5af47 #12 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 3756A22C-6CCA-4FBC-82C8-04DC535987FA final-5af47 #12 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 DD5ED50C-0AED-4AF5-9054-F34DA8A20FF1 final-5af47 #13 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 1496EA7B-93B9-4B03-A74F-CD1C064B411C final-5af47 #13 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 BDE85E51-2FC2-444E-BF27-820DA5BE41EA final-5af47 #13 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 35E7BEBD-01F6-48EF-A0EE-D580B0F6A528 final-5af47 #13 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 D7D5A778-582B-40D6-AB0F-A4C33B4BAE0C final-5af47 #13 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 F60D6F72-D2FC-4A6A-8075-744D37D756A9 final-5af47 #13 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 3B3673DF-C659-4C9F-953E-CF17E48C86AA final-5af47 #13 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 E6EF0E56-CBD3-4722-A6DB-9C5119CD1F9C final-5af47 #13 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 6CD507A5-0E8C-483C-88CD-379441377B7B final-5af47 #13 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 5EE886D1-AFE3-4D8F-A65E-AF48E35BA77B final-5af47 #13 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 A60EFD9D-8881-4BA8-A7A9-B176B28669FE final-5af47 #13 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 7FFBAC3C-E62B-4D56-984E-51487A99D673 final-5af47 #13 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 152D41D4-FE97-48C8-98E5-589BB82BBD3D final-5af47 #13 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 AB969C05-E53F-4E34-BAC8-BDD5138BAE77 final-5af47 #13 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 AA778DA3-ECC3-4683-AB3E-255D31860881 final-5af47 #13 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 EE48CE81-BD16-4AE8-B16F-09FE3B8D1A16 final-5af47 #13 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 FF19146E-98B9-40A9-A120-43333FBF258B final-5af47 #13 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 9C7E77AA-F3B0-4FDA-A7FD-F69E9AA74413 final-5af47 #13 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 8A48C1C9-4D7D-43CD-8562-590BA9BEA8FA final-5af47 #14 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 4F09BB7C-0C95-439C-ACA5-040AE60AD0D7 final-5af47 #14 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 BDCB0573-4F6C-4A1F-B7FD-78CC49CAECD1 final-5af47 #14 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 150559D8-7349-4C70-8DF8-2AA3BD008B7E final-5af47 #14 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 FACEAE79-26C0-4274-AB31-CB434AA31FAB final-5af47 #14 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 2B43DDED-0F4E-4739-BD3D-CF04FE9B6AA2 final-5af47 #14 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 DB815751-4F2C-4549-9D94-595362AC151C final-5af47 #14 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 509D29A6-C84E-41DD-9BD1-BCBD049C3CDC final-5af47 #14 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 CB3ECD6D-8FEA-4A27-9C95-B15C44616CB5 final-5af47 #14 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 B81CF8C7-160C-43BA-8D19-75013962D0AC final-5af47 #14 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 8F421911-DD0E-4BC9-80F0-6C0AC86B97EE final-5af47 #14 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 6157E122-9249-4150-B69E-C6F583F55B20 final-5af47 #14 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 1D2452E7-B4D2-421D-9E6B-C9B99AEC0C20 final-5af47 #14 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 DBAD703D-7D73-4718-B816-7B0722DC41B0 final-5af47 #14 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 2F4843CB-4420-475A-92D3-A61180FC1B9A final-5af47 #14 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 0492C8C6-09B6-4C0A-8B36-B9531EA97538 final-5af47 #14 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 BD21F778-1E08-4E70-A260-FF7B4072E5DD final-5af47 #14 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 99036D12-4125-492C-9391-70656F472958 final-5af47 #14 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 B7B19A93-2414-4378-ABFE-FA8192CB208B final-5af47 #15 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 6618A9C2-0F2E-4542-AD4C-F1EBE34895D8 final-5af47 #15 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 830DE820-2E98-4928-B302-2DFEAE219488 final-5af47 #15 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 3A50CB0A-434E-48CB-95DD-99328F9314A1 final-5af47 #15 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 F658ED21-68C6-4FF9-9AB7-9FC7BB4F1548 final-5af47 #15 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 190FFD0A-8E7A-43B3-B349-C3DB3507C582 final-5af47 #15 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 D3144F68-9197-4414-975D-84478AE0BCB8 final-5af47 #15 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 79A30DE2-7410-4AF8-9D8C-E8BF5E627DD2 final-5af47 #15 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 340EDA29-E8B8-4FDF-ADB7-C440ED956E87 final-5af47 #15 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 A139621E-3F28-4F80-8435-5BB9A7DA58BF final-5af47 #15 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 5FFCDAA7-BC9C-4F8D-A31A-6C4E6395E87A final-5af47 #15 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 CCCA2AF7-B48E-4504-8FF9-F1C060A25785 final-5af47 #15 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 146D1E12-7066-4DCB-B93B-20BF7FACEE68 final-5af47 #15 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 13A96324-F924-4758-88BE-96247947079D final-5af47 #15 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 16F07651-9172-4FE5-AE73-BED7C2210DEC final-5af47 #15 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 13C2634E-28A8-4851-9640-0CDFC69B890D final-5af47 #15 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 77B7C1A4-509D-4F88-B42A-9A5F1931F79B final-5af47 #15 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 D755C203-8808-402F-AE86-1C34F63023C2 final-5af47 #15 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 B27D271F-F675-47DB-927B-B7B0A2BE2554 final-5af47 #16 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 D63FC527-6234-4AD3-9382-CF696188F8CE final-5af47 #16 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 FAEF9AE4-DE3B-48E9-9892-FA98BC307E0D final-5af47 #16 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 F7B2736E-697D-44E8-B3EB-0C92A5749BD0 final-5af47 #16 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 53D49E9C-9AD7-42FA-B5B0-9D13C62CF977 final-5af47 #16 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 54F4E4D6-F2B1-4FFB-9C36-ED48637CA1F0 final-5af47 #16 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 8B05641E-B583-4A06-9BA8-8CE184E827E2 final-5af47 #16 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 93D5FF8A-1177-46AE-8F3D-CEA3FA15BE6B final-5af47 #16 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 286999DF-2616-47C8-B246-E7E7AE900287 final-5af47 #16 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 D996DFD4-56F7-4099-A738-5F207A5A3F2E final-5af47 #16 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 431F5534-FBF4-411A-A244-05B36B1A38B1 final-5af47 #16 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 B19E498B-A8E6-47CF-9F5B-F84CC2347231 final-5af47 #16 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 30A51BC4-7454-4838-B846-C053835BC792 final-5af47 #16 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 4EC22B03-85C7-400B-A85C-F9013355E0AD final-5af47 #16 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 707F868F-B49C-452B-9983-187232D9AF52 final-5af47 #16 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 E7B3C638-71A3-4DCE-92AE-974F5966687B final-5af47 #16 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 47712C57-0E61-44DC-9E2F-B0BD358DFCB1 final-5af47 #16 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 281E9F50-1127-4A63-9677-FE9F3DDAA9E1 final-5af47 #16 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 521B0333-A426-4ACE-BBB8-B5181AD90BED final-5af47 #17 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 1A55F21D-ABB5-4128-966B-A0EB899430D5 final-5af47 #17 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 3E0CFA76-B84B-4126-8432-F140998AD047 final-5af47 #17 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 7FF61375-7A5E-4DCD-B76E-A270D7B101D4 final-5af47 #17 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 565A1905-E3D5-46CB-B80D-BDD2AAFFEC41 final-5af47 #17 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 E180D107-CB9D-4088-A15D-94CE8286248B final-5af47 #17 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 36C2745E-8BF1-490C-B2A5-F4949A34EEAC final-5af47 #17 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 4B862BB6-4C56-4C72-91CF-197BA78E2B6B final-5af47 #17 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 DF107E6E-1247-4FD2-992C-CBE9E87339A4 final-5af47 #17 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 59DB11D7-9C49-4A02-80CF-57621B285757 final-5af47 #17 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 50D0A54C-F7BF-4157-B4C3-EEE7C96EEAED final-5af47 #17 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 FF8702E6-FB08-4457-8654-E578E52BD453 final-5af47 #17 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 925966F4-178F-46B0-AA40-0541E75CC988 final-5af47 #17 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 AA228751-AE02-4A24-B030-159BF056FC16 final-5af47 #17 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 3543EEB1-A534-42E7-BCA3-24C0B3ED0D17 final-5af47 #17 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 C4954F6E-322B-4489-9611-935E7CE09CE8 final-5af47 #17 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 3C3A180C-3AA7-4C86-B8AB-DD048C394CFB final-5af47 #17 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 66E19F8A-6679-4086-8E22-57CFF8B7EEAA final-5af47 #17 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 E6F77B59-BEE1-4FD0-B7F9-8F4EAAD68371 final-5af47 #18 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 962E84E9-5CC9-4DAE-86DC-4E3DFDE8800A final-5af47 #18 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 8FEE88F9-0776-4F77-B059-2ED35399D2D2 final-5af47 #18 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 F0BACFA8-761D-47F7-9D34-64107E5E13AF final-5af47 #18 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 5A553EE2-FDD8-4CC8-B9E1-983668212D39 final-5af47 #18 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 7F978508-7221-440E-9D75-E0701617E912 final-5af47 #18 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 1902645A-5D12-4032-9890-C4BBF9AEAFA5 final-5af47 #18 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 B0EC175A-D141-4638-A1C3-86A9BACE0786 final-5af47 #18 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 9B2C0B76-2D72-4D22-B270-0D208C08415C final-5af47 #18 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 AC7994EF-C64B-419E-9F9D-38708808AD12 final-5af47 #18 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 430F22CD-DC27-4E61-9ADF-331D89168357 final-5af47 #18 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 798C7465-44F7-41A0-8F69-CB7C0F85686C final-5af47 #18 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 5E1124D9-7E76-471C-B848-D99E354DE8B4 final-5af47 #18 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 860C8763-FD67-46DA-83F3-0FD1D72A7225 final-5af47 #18 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 37D9C733-491B-4EA1-AAA6-724BC2ADED96 final-5af47 #18 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 A8F644D3-98C4-4982-B663-910221B791A4 final-5af47 #18 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 36CA015B-8CC9-4F8C-9A58-B80BEAA9F250 final-5af47 #18 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 2BDEBA2C-2587-48FF-A215-421F8943DABA final-5af47 #18 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 DF352D57-FFBD-4B23-9173-5F1B979663D3 final-5af47 #19 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 368815D6-DCDC-4BB2-8C8B-43F6908C1868 final-5af47 #19 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 22936A49-E88D-4767-9A0D-7F014C36A3B4 final-5af47 #19 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 581664AA-C3E2-4517-9F42-D4DA85F0F2C8 final-5af47 #19 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 709C774A-1471-4E15-95C2-84199DB6BDF0 final-5af47 #19 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 A7629B32-C636-4732-9EB5-424573E0FC5E final-5af47 #19 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 B5CCCACD-3B63-4C35-A3E0-CB3E1C2A8D48 final-5af47 #19 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 F1A67848-5545-4FF0-8FED-6EC9FBD9F88E final-5af47 #19 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 09457265-F697-452F-B3FF-35796A9C7C6B final-5af47 #19 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 5777AD00-13A7-406F-8498-2E7AFEE8298E final-5af47 #19 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 FE3E3594-D08B-439E-A218-2DCA7DEA2EC7 final-5af47 #19 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 9EDB9E6D-1E05-4662-9F4E-8241E9FF8062 final-5af47 #19 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 CCCBF2F5-05C1-4FA4-B1A0-F28A262EED6B final-5af47 #19 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 8C3B0891-3E1B-4334-BD46-70CFC54166AA final-5af47 #19 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 53B0E743-1971-424A-8A41-E3B693B0759C final-5af47 #19 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 E2E3D4DC-9B65-4A26-A9EA-FD8024348827 final-5af47 #19 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 AC0D8FE1-CB6C-4527-84A7-542CE9E22E54 final-5af47 #19 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 B9C3260F-EB5C-4F6E-B9CF-72BAB8DD6EE8 final-5af47 #19 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 EC09BC51-58C7-44C0-AEA1-F6C7BC7AE80E final-5af47 #20 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 5DDD9E7E-04C8-499B-B562-B27885BCC7EB final-5af47 #20 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 ACF06123-9CFE-4936-8B14-D3D6AF4BFA3E final-5af47 #20 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 CC73CEAC-749D-4CCF-8203-30764C80FB91 final-5af47 #20 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 DDF7FD6B-5B15-48F3-BB86-76CBE9261C9D final-5af47 #20 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 4A703F86-B1F9-44D3-A946-9F1A0B00099A final-5af47 #20 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 5F1A117F-8757-4928-B072-75426633F19C final-5af47 #20 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 86CCF24A-559B-460B-9563-CD93859708EC final-5af47 #20 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 68064928-05E5-4C81-8CE8-D47D1F39C80D final-5af47 #20 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 2BE72732-E72B-4F2B-87FC-43D61611F262 final-5af47 #20 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 BBB96B55-BFD8-4C2F-BA96-D52AC6D075A0 final-5af47 #20 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 8050F06A-6CBF-4F55-B6BA-9546D2680B2A final-5af47 #20 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 9066519B-5926-4086-8965-39C068EF2824 final-5af47 #20 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 D62DA24C-B72D-4EA0-A5C5-174D816E5580 final-5af47 #20 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 2428EC84-5B3C-4D6B-A097-A10D644BAEC3 final-5af47 #20 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 0C60CD24-FE9F-4F44-9483-E687099730E5 final-5af47 #20 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 D437A42E-5C2B-476A-9CE8-6951923533BC final-5af47 #20 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 43B78E4C-1A0B-496C-9C46-9B1908C9C8E6 final-5af47 #20 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 203FF132-CCFD-402E-9344-89A70E9050A1 final-5af47 #21 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 4F7C2AA5-9601-47F8-813C-8F53EB46D44C final-5af47 #21 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 89339C02-1874-4DD9-B09C-03456D29F4F8 final-5af47 #21 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 CC7497A0-15E8-4317-8B55-169FB1A0D9D5 final-5af47 #21 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 6ED62E17-2892-4673-A6E5-7BF7546EBAF0 final-5af47 #21 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 574E518B-F772-4EEF-B124-E03BC7AE7ABE final-5af47 #21 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 12B5D340-A051-498A-8906-771A14D0EAD9 final-5af47 #21 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 733BFC9C-88EB-48A8-A0A1-630D48351679 final-5af47 #21 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 099F3D21-1385-4BEE-83F6-AFB99B1F4B59 final-5af47 #21 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 A031FE7D-134D-4F01-AC82-602A1B035034 final-5af47 #21 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 D2533ED4-2F3C-4A21-8AF6-AD9C181D1CBE final-5af47 #21 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 9E380E96-135E-4326-A18C-6D68045B1D83 final-5af47 #21 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 F84D8AFA-30CA-4618-8921-F7978D2F71B8 final-5af47 #21 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 17F09198-37F5-4F53-A5D5-CC106C844049 final-5af47 #21 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 C20DE87D-D71F-41B3-98F1-06241E4D3413 final-5af47 #21 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 B02878D4-E620-4E46-8708-1D725CDDFD25 final-5af47 #21 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 BB2BBFC2-E7A1-4E36-A50D-9102A7AE9CE5 final-5af47 #21 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 D31A5EA0-0226-4F12-A563-57027EA58AAB final-5af47 #21 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 40A5B267-21B4-4CF1-A782-01B90C599CBB final-5af47 #22 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 23EC9F75-8194-4E9D-B6A7-85621CD8A614 final-5af47 #22 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 FF5B128A-88F5-4632-8F1B-E577B866EDA2 final-5af47 #22 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 F82C9474-B1D2-4BD7-829D-F251CE633F33 final-5af47 #22 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 122404B8-EF24-4B8F-838D-A22262E4E8E5 final-5af47 #22 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 4AF278CF-0A51-42AE-8017-3EC1331581A6 final-5af47 #22 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 30F4E879-BDD4-415C-BB18-D4AB87913E4B final-5af47 #22 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 624D216F-7363-4B73-8315-9BD9A945152D final-5af47 #22 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 3415FF32-1A24-47A6-B6A0-0CF1F5CEF3C2 final-5af47 #22 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 7E0FAC01-BA6C-4F4B-8C82-97E5B6FF0B35 final-5af47 #22 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 1873FA31-42B2-486C-8DD1-6BB1D0512F0E final-5af47 #22 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 E878D12B-B3E5-4BE6-B4C2-51397F3B21B4 final-5af47 #22 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 B2B73BFA-8A16-4089-A9F4-5D229A3751AE final-5af47 #22 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 58C9F166-FD88-43B5-8456-037B1A12060C final-5af47 #22 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 170C412A-A139-4340-A00F-FC62F56C4389 final-5af47 #22 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 DBB0EE30-C73D-4629-8822-C82BD6C5E829 final-5af47 #22 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 F0CE608B-6FA0-4013-B68B-68B452CB95BC final-5af47 #22 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 47471389-D438-408B-9E41-F2340A4334F2 final-5af47 #22 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 3648C94C-5B33-44AF-B963-1C3AA37437EA final-5af47 #23 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 8ACBE6F3-3090-4092-8C97-A53EC66A83E6 final-5af47 #23 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 A5250CFF-286C-458C-8650-A1ED4ACB108C final-5af47 #23 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 90A3F002-5969-4BA4-85DD-9EDF978B3447 final-5af47 #23 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 309F21E0-BD7F-49BF-A07C-9A5D1103C94C final-5af47 #23 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 9661D219-30F0-48E2-A4AC-1320778BBCB1 final-5af47 #23 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 6FF999C4-EA9C-43D2-A503-E8F8E40C3B72 final-5af47 #23 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 2B6029BB-BAD8-4387-AF6C-FBD921E034E4 final-5af47 #23 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 706DCF1F-8C2E-4D36-86C4-7A6CD170B390 final-5af47 #23 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 49E48F8E-3DDF-4917-8534-9ACD41268FD1 final-5af47 #23 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 B42E4396-0B1C-4902-9E99-6D652F5E1F69 final-5af47 #23 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 DF9B9A01-1DD3-42F7-9FB7-95351CF734E8 final-5af47 #23 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 BAEF9908-C489-4D0F-9A84-5CDF73D55229 final-5af47 #23 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 91EDF54A-DF08-4BDB-A0F1-FC2497DABEDE final-5af47 #23 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 510CEA5D-0FEC-44A2-963C-B1A705F13D7D final-5af47 #23 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 5355EC14-D814-48E8-BB69-6DE73C235A7B final-5af47 #23 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 DAC96138-2D77-428A-AA32-F310E6D0070E final-5af47 #23 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 B4243D24-274E-4A68-8E50-F3F62AF3DC33 final-5af47 #23 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 FA6B7FFD-6CBD-41B5-A767-5CD350F913D4 final-5af47 #24 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 76E38458-EDDA-4A96-8829-B6A69FB15100 final-5af47 #24 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 1FECF424-4CD4-4431-9273-08E2A9C412E7 final-5af47 #24 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 C85DE03C-8FCB-4900-BB71-F753A7319A16 final-5af47 #24 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 7A1180A6-61A1-40AF-9535-C14AB6B818AC final-5af47 #24 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 3D9044FA-1787-4A61-B9A0-360B87A45D85 final-5af47 #24 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 11386735-EB7E-4229-9157-5AE579D0F522 final-5af47 #24 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 DF1295CB-BBB6-4A4F-95B0-7DFF91081154 final-5af47 #24 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 6672D28E-2B4B-4769-BF89-8B230A2216A4 final-5af47 #24 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 99C20107-10E7-430A-8584-5CBED8DC31E2 final-5af47 #24 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 80C74D99-FFF9-41B4-A50E-CAA798E0133F final-5af47 #24 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 BA6DCC99-B32A-4861-BC9C-3F59DCBEDDBD final-5af47 #24 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 7E37E16B-8481-4F41-B50B-9E7D4FAFA259 final-5af47 #24 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 E06C20A8-5DFC-4A3E-AD1F-65E50A7DE758 final-5af47 #24 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 E41E31C3-D026-4282-8283-41A5296EDD6E final-5af47 #24 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 0470FE06-9727-4B36-943E-141493B7353A final-5af47 #24 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 D408245C-0172-420B-94FE-D28EF429170F final-5af47 #24 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 61BF3DDC-42B8-4BA4-A9B1-CA85C9F8AF99 final-5af47 #24 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 57E1D1E4-80A4-4F57-92C5-2BD7898F4DBC final-5af47 #25 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 7C8A8721-BF9B-4171-94A5-30687D8015EA final-5af47 #25 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 F3EDDF78-F9A3-4865-95C3-7945CF27AB26 final-5af47 #25 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 6A3688D6-1EA0-481D-88A2-3AFFCF49B9D8 final-5af47 #25 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 875FA7E5-82EC-4046-B48D-EA4918670E21 final-5af47 #25 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 A3AFD634-FE4B-4CB6-B1DF-2431EACB8299 final-5af47 #25 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 5C2EBE42-C7D6-4C6E-8691-BC9364024A23 final-5af47 #25 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 F1B0E89B-0ACF-43C5-BDBC-1C8609B66B1F final-5af47 #25 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 11CC54F1-C687-4483-B144-B892FFFC0AC2 final-5af47 #25 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 5FF9664B-6694-4AE4-BE8B-A8B6E9EFFD90 final-5af47 #25 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 2B7B41F2-0A0B-48F4-8307-D0537E3A5424 final-5af47 #25 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 96B3BEC1-BF08-47F4-99A5-0FC0C46EBF0F final-5af47 #25 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 5D4CB86D-C8CB-453F-92DE-043DDFF7C13C final-5af47 #25 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 CE4688D6-93FA-4491-ABC3-D3B0E29F2377 final-5af47 #25 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 4FBE233A-F66C-4CA4-BFBA-A01C4C23733F final-5af47 #25 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 F4EB576E-429D-43F2-9544-BB555F8FEE71 final-5af47 #25 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 9919F281-C4B1-4BD5-B37C-E15712F46A3B final-5af47 #25 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 347EAE11-9301-411C-981E-091CFEB4220D final-5af47 #25 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 E643A2A2-A8BA-4BE3-A9AC-236C9B8E9568 final-5af47 #26 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 298D646C-6D24-4F5C-B3CC-453D97039387 final-5af47 #26 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 EA0F07D6-DBA7-449D-912F-21F2CE0333AE final-5af47 #26 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 2DE116FC-4586-4CA4-A8DC-5F8F9644CFD5 final-5af47 #26 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 8E70BF74-3BBB-4452-93ED-A316442215F1 final-5af47 #26 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 2A3D9878-57F7-4332-B182-8BB38B11694D final-5af47 #26 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 E86F014D-6D2A-4CA4-9CE1-C221A762AFC6 final-5af47 #26 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 32F29E40-7E1E-4210-B38A-4735147489E1 final-5af47 #26 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 991C4FAC-D769-4E4F-BD3D-D5B8C2C6E1E7 final-5af47 #26 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 870E896B-A0CB-488B-AFE8-12FA4545C4FF final-5af47 #26 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 4504D03D-986C-4DAE-91B0-CD6BD5287802 final-5af47 #26 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 E7626F0A-00DF-4FB8-8CF9-5408164194FE final-5af47 #26 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 FB502C14-276B-4456-A178-7BD5376B1AE5 final-5af47 #26 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 004022BE-E70E-4D7E-95A7-F62E718F5DC1 final-5af47 #26 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 729AB758-B0A9-4405-BD32-31A36D4A904C final-5af47 #26 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 5A1EC7A1-A387-4500-B5E8-E9042816818A final-5af47 #26 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 D97017CC-86AD-4B3C-BFEA-A3DD67F48962 final-5af47 #26 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 5452F809-729F-4B1A-8A88-1ACF0DC64A77 final-5af47 #26 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 3791EB01-8E54-4CA0-868C-FB40F7817A23 final-5af47 #27 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 086F6A89-22D7-45A3-ABE8-A8DA6076A476 final-5af47 #27 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 F94F40FE-E3E8-4E8A-BB37-43B1BC299820 final-5af47 #27 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 BFA796FD-BA6F-4815-A626-B0418072749A final-5af47 #27 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 0DB87B4E-C8A7-42B0-9AFE-01BBD4F71399 final-5af47 #27 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 3775146D-C5B7-493B-A525-3AA979367E01 final-5af47 #27 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 E1D0F080-67AD-4DB9-9B7C-9CF6A048C0FB final-5af47 #27 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 AA6B6077-668B-4A2D-8153-00A058E343FA final-5af47 #27 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 6BD7BDAE-3CCD-463B-93E9-916C556A8540 final-5af47 #27 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 1B3D0831-06F8-4820-BB74-3DD5D0AA74A4 final-5af47 #27 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 EBC1A90A-6F0F-4581-8687-30C4EAF3B489 final-5af47 #27 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 66168224-7B1B-4B91-AFD7-670CEA6CC2C5 final-5af47 #27 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 A009ED58-525C-4C01-8AC2-84633B6BC60B final-5af47 #27 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 F7F54DF5-2500-401F-A97C-C9545A6E7030 final-5af47 #27 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 CF041186-8A23-48AA-9540-7BAEAA88613B final-5af47 #27 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 7CEC424C-C188-48DD-9892-2776B5088905 final-5af47 #27 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 0340FF45-38D9-4C27-9CED-A5C835B3B3D6 final-5af47 #27 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 035077D9-B40D-482E-9EC3-5EEC3BD648FD final-5af47 #27 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 6844DE43-CA61-42FA-9936-EF69B58E440B final-5af47 #28 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 1E64D572-729E-4903-B7FA-0FEB4A92E912 final-5af47 #28 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 37657E37-7CA4-41DC-95E3-4AA495BE7095 final-5af47 #28 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 995F64F0-6D33-4457-BE6A-DDF21CCF95CB final-5af47 #28 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 5FD12389-5668-43D9-882C-A6F979FBA396 final-5af47 #28 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 6C1C55CD-9E85-431B-93CB-566A0B36464B final-5af47 #28 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 6B4800E6-C846-4258-8B73-A714C9F4B699 final-5af47 #28 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 097FB1B4-1EE6-4D86-B974-18476D4AF262 final-5af47 #28 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 43C45214-5C93-4143-8F0D-CC2C3F76341C final-5af47 #28 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 672F8AD5-3F47-4984-8D62-BAB408725578 final-5af47 #28 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 FC33C952-F01D-440B-A18D-1A9052FD1423 final-5af47 #28 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 3B8AC9CD-6A93-417E-8BB9-38AA244A78DD final-5af47 #28 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 530A0FD9-92BD-42AD-8B28-41D118568D5A final-5af47 #28 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 53D5AD43-7F00-4DF2-B9B0-D51A5B750B92 final-5af47 #28 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 C722427F-0807-4D04-954C-D021FB100AFE final-5af47 #28 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 8B87590B-F2E7-442D-8D00-799C84FDCC86 final-5af47 #28 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 419BA3A9-49B9-454A-ABD5-374E01461B7C final-5af47 #28 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 DD5F26ED-A9DD-4D81-9A73-927F7162B671 final-5af47 #28 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 63253AE4-C5F2-49A8-B8E9-918CA7111D0D final-5af47 #29 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 BAFBAED2-A864-45AF-871F-D9D6A678B462 final-5af47 #29 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 4413F7B0-D585-41D3-80C8-A8431ADDC804 final-5af47 #29 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 B86CBD07-4351-45A8-9FD1-B9DE118C5FC0 final-5af47 #29 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 00505A55-9FB5-4D70-9B48-CF1A9B57D47B final-5af47 #29 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 F59459A0-9916-4D3D-AD7B-2BC1715FE3B7 final-5af47 #29 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 935D4763-ABDD-4650-A94B-7208D951C0C2 final-5af47 #29 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 84011821-C48F-4FF1-BF9F-C62A7DB6CECB final-5af47 #29 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 16B5EB03-F8E2-4074-8577-73232B8DBCCE final-5af47 #29 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 B72EA978-7BC3-4E56-A4B2-158AFDE8B647 final-5af47 #29 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 2F1AB144-6303-4409-ACC8-A56348404B54 final-5af47 #29 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 03CD62C2-7912-4E9E-9693-6D6894157CF1 final-5af47 #29 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 CAE13964-1693-49AD-9E6B-B5A381303E1B final-5af47 #29 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 80B81FB3-A45F-4A31-9D91-55E0E8DCEC61 final-5af47 #29 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 DED2C602-7370-4BE9-95F7-FB072B8C3A8B final-5af47 #29 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 62349BFB-8975-4B41-8CBF-D7FAE6DA8333 final-5af47 #29 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 F18A99E7-5E77-491A-B3DF-4FD12A18EFFD final-5af47 #29 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 E22D06C4-74DB-4FBE-AE77-0C439010A1DC final-5af47 #29 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 B64C63B3-328A-4098-A2E0-E45F752BA1A0 final-5af47 #30 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 6BA9FC7F-5F06-4E60-888A-5BD468EA3B30 final-5af47 #30 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 B820ECF2-731C-4A1A-9B71-871DBD73C637 final-5af47 #30 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 DAEB8429-4A85-4E2E-857E-04273F653C57 final-5af47 #30 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 731EBCF6-DEF6-4630-AC38-BFBBC9F7B0B9 final-5af47 #30 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 426B4F58-BB4C-4A72-9A0D-7993F894F4A4 final-5af47 #30 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 13519EC2-858E-4AB5-8485-7BAB8D260987 final-5af47 #30 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 D2FF43C7-9D90-4D61-BCD0-A1BFB906885E final-5af47 #30 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 4C3C5425-5ADD-4B8C-9AD7-49A88FA05927 final-5af47 #30 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 28C060F6-B115-40B3-8EA0-18D4D21AEDF2 final-5af47 #30 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 4C2947B5-36C1-4B81-8946-2F835B44C6DA final-5af47 #30 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 C9951194-2B63-4A97-9823-05FADB1AB424 final-5af47 #30 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 EE6284C3-ADCB-47F4-8730-F5CB74BDA5A6 final-5af47 #30 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 10F86200-2FBF-4628-8D24-6512807B2561 final-5af47 #30 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 06643209-CCAB-4E5D-BED2-4DCB6F62D254 final-5af47 #30 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 3DE5509F-0182-49BF-9F01-057450051887 final-5af47 #30 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 8137F701-A39C-4232-A1DC-29C32B01D4D0 final-5af47 #30 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 72C424A0-8D70-41A4-A61A-4DE171A0991C final-5af47 #30 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 3F649138-5F7F-4E66-8FE4-427FCE467699 final-5af47 #31 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 04F1CBE1-613F-4264-BAA1-168F018C8D9D final-5af47 #31 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 2ED3E8A2-EC46-4C6A-8C2D-B464DB2120F9 final-5af47 #31 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 A8DCE575-EEFC-4979-849D-4F8D1C1BC3F3 final-5af47 #31 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 0C7DD927-CAE2-4D5F-AD11-984AB50D1D4E final-5af47 #31 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 4A6DF79A-0FD0-4DA2-B491-50A02E703091 final-5af47 #31 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 6248F7D6-8610-4AE1-9456-562BBA3C6C53 final-5af47 #31 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 90F964BA-E763-4696-842B-6B1E80F5F5B5 final-5af47 #31 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 BF2D98E0-9711-41BE-9BD8-5936118A9025 final-5af47 #31 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 9B37FF3A-5AA4-44B5-8DA2-BCB5C8F456CA final-5af47 #31 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 2737CEF8-1E7A-439B-80C0-3EB532441B1E final-5af47 #31 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 C6EA217C-2C56-4943-96C9-D01FAF1B681B final-5af47 #31 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 F86DE25D-A94A-47D2-8FC1-C0E1491622E4 final-5af47 #31 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 0D864B5A-A1F1-433C-9A93-63F6BE0B9FFC final-5af47 #31 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 FCF6F41D-27A9-4794-B832-9129FFA61C15 final-5af47 #31 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 BBE71F30-0FEC-4D81-B898-4C761B8299A3 final-5af47 #31 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 7A9A8E04-77C5-4AD9-A797-97C49379E34A final-5af47 #31 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 9950C30B-A527-4BFB-BB1B-CA20FCB320C2 final-5af47 #31 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 24219320-BD5B-4DFB-BCCB-0198B8C5B70B final-5af47 #32 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 EB69E11D-CDC3-4301-B590-70810ABF2243 final-5af47 #32 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 4E11B681-C608-49FF-9353-539C10DAF2AD final-5af47 #32 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 8AF1CB6E-693C-4B41-A6CE-08A4710223FA final-5af47 #32 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 2486ACBC-36CD-4045-A2F2-0B93EC21BBD1 final-5af47 #32 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 768A7347-F424-41D2-BDE9-83BD26736F38 final-5af47 #32 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 EF334A28-5483-470B-8FCC-F99042841437 final-5af47 #32 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 148E5D40-0C5A-4883-9B77-9DD05F675012 final-5af47 #32 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 FA04EECA-FAFD-46EE-90FA-469FAA141FA1 final-5af47 #32 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 B2AB0DA5-1AA9-49D2-A12F-C8456F3D4C06 final-5af47 #32 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 9844559B-4FC4-47B1-98A3-72DA007BCB03 final-5af47 #32 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 478C1CF3-88AD-4A57-BBCB-32D69E7D3000 final-5af47 #32 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 482DBA23-E29B-485A-96D4-A16CB295A43C final-5af47 #32 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 6467E7F3-BFE7-4ACB-ACA7-AD7E9F2ADDEB final-5af47 #32 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 D163E837-99D9-4533-9A11-4B95EC3DE185 final-5af47 #32 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 1D2D7E51-94DD-457D-B299-05D6F159DA39 final-5af47 #32 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 87325697-DE89-47F8-9159-A2D291F5B42D final-5af47 #32 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 A3E0C5B7-024C-41EE-BB8F-8F8C598F4672 final-5af47 #32 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 85F7B788-9457-4D23-A471-C00B0753D6A1 final-5af47 #33 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 A7A784CE-A0F3-437F-BD41-242674CAF268 final-5af47 #33 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 7D9292D7-659C-4ED5-A770-BA12E0583EEA final-5af47 #33 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 0AF468ED-FAE8-493C-9AA4-3F834AC6FDFD final-5af47 #33 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 23CB226F-355D-4FDE-A91E-8BCB51633F66 final-5af47 #33 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 F7294F00-A8CD-4644-A005-216ED0FAFF24 final-5af47 #33 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 DDF50BA2-EF6B-47E6-A5E7-0058F4FEEA9B final-5af47 #33 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 6945084A-D431-4DFA-B3E2-EBB1BF924478 final-5af47 #33 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 80DEF8CD-1B70-40BA-98F9-62031630873D final-5af47 #33 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 DA905785-F5C1-46D9-AF54-27CF0723BAD3 final-5af47 #33 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 9587704D-9A27-4E31-9458-E63B4DC21899 final-5af47 #33 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 B9E5E49C-1529-4480-A25C-B78E7B53AD4D final-5af47 #33 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 722C07D3-4F83-4C09-81D0-275911D66E64 final-5af47 #33 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 0FA389E5-5E0A-46CC-8672-DA2CED3A970B final-5af47 #33 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 573AD231-9528-4C15-A0A1-25E245B12F1D final-5af47 #33 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 3874FCD2-F384-4E46-B0DE-25BE03FA67C1 final-5af47 #33 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 6D4FAA21-6A4C-470B-9044-31872078FDC7 final-5af47 #33 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 B56905C3-42BD-41B5-A583-8B88865762D6 final-5af47 #33 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 F86622AE-9401-4417-AF4B-7CD46A76A400 final-5af47 #34 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 1628516A-835C-4242-8F21-410A7C50D4D2 final-5af47 #34 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 F23BC556-8B14-4039-A4C8-20527AA814F9 final-5af47 #34 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 CD36CB2B-4646-4145-B95E-A26D83759592 final-5af47 #34 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 70854982-CF71-41F1-9873-C8CF7BD7AD47 final-5af47 #34 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 75F6539E-F193-4B56-A23A-8CF8F1FE48DF final-5af47 #34 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 46E988A0-C14D-4742-B1EE-D77C5ECCD08C final-5af47 #34 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 FD74CCB0-F7D9-4103-A0E5-E0D5554518DA final-5af47 #34 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 31D01C34-D4B8-4446-ACF1-F33F5968E02C final-5af47 #34 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 D32CA2CD-3EED-4884-8F0F-9F10B598EC17 final-5af47 #34 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 0AD0A4A0-B010-4604-ABAF-068328DA5926 final-5af47 #34 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 C2888D96-C980-4A63-9463-B78AADB67457 final-5af47 #34 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 DF3D8DFF-4B74-4C89-AFDE-C7A7F08802ED final-5af47 #34 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 F9CCC17F-E342-48CC-92EB-D926850F146D final-5af47 #34 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 382CEF16-554F-4C3E-A0E9-44716181A600 final-5af47 #34 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 1C0BA8AE-A2A9-40BE-9593-2DC994EE944E final-5af47 #34 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 17F33168-57BB-4C4F-9CC7-82A7186831AD final-5af47 #34 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 233E1A89-7665-4531-AF55-DC2DD08C37A8 final-5af47 #34 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 13858D7E-D661-47CB-97E6-2BAFF887BDEF final-5af47 #35 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 AAD47B73-E0A2-4720-8159-604576843D70 final-5af47 #35 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 0F63FDEA-BCA7-4A94-A7C7-92F115B1F214 final-5af47 #35 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 106A672C-4F3E-45D1-87F1-B70572C27379 final-5af47 #35 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 EA80AD42-C23D-44BB-93BA-B0914CA725CE final-5af47 #35 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 689552F6-1DE9-40EE-94FC-3F6FD7B71011 final-5af47 #35 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 9F61BDEA-F5E7-4B22-8846-92356AA1328F final-5af47 #35 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 CE803620-A6A2-48A3-839C-D1C42B202B5C final-5af47 #35 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 952AAC65-8D8C-4970-B2FD-221A71ED62DE final-5af47 #35 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 F4F87311-BAD5-437B-838E-1E48E7DECBB3 final-5af47 #35 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 C093CA09-FAE1-4B8A-9D8D-786A78E5C9EB final-5af47 #35 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 E17252D6-AFDB-491C-8AF7-ADB3E05FEBA7 final-5af47 #35 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 91A11426-F28A-4B98-B339-E8E3B4DCFDEF final-5af47 #35 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 42050134-213B-436D-82CC-E447D78F189D final-5af47 #35 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 042729E6-DAF0-4DDD-92DD-4FEB492196FD final-5af47 #35 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 0B5541E0-C1B2-4923-BEDD-68BAC86A539E final-5af47 #35 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 250D19C9-9C58-4F86-8808-14EEACC26A7A final-5af47 #35 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 57A6E930-FEA3-49F9-B8B6-A1ACF4827346 final-5af47 #35 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 003D83BD-FCD5-4669-87A6-BC847A0D16E5 final-5af47 #36 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 C5E02828-8AFF-48AE-832D-5651D65014F0 final-5af47 #36 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 E6A3E0D8-EB61-4FAC-98A0-B72CEF87CD58 final-5af47 #36 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 D4AE5379-CF3F-4717-8E45-A0141D6F0029 final-5af47 #36 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 AA934A3A-478F-4464-AB59-4CDEBA32B1D4 final-5af47 #36 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 C6B244B4-9B4D-49A2-8C0E-85A06DE66718 final-5af47 #36 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 799BE34A-632F-4661-8280-9685CCEA214E final-5af47 #36 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 15839DE5-E988-4C6A-A49C-0A7D087F52BE final-5af47 #36 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 480B7EC3-DA4E-4A8A-B803-ED93B7FD9400 final-5af47 #36 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 A9EA3607-2F9E-48A4-AA55-D538850FE207 final-5af47 #36 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 6E8CC7AC-32C8-43A6-9A84-3B8720CAE767 final-5af47 #36 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 022A0E0B-4C65-4A2B-984E-FF0E388D0D00 final-5af47 #36 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 AD696BC7-1549-4B6B-9C6E-23CF3BAE5485 final-5af47 #36 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 F7977166-EA75-45DF-91E2-CA7805DD13F7 final-5af47 #36 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 366CCFC2-6DE9-42F3-B50E-8665399A7BA6 final-5af47 #36 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 99DD3CD8-5991-49F3-BC2B-6CD1A18A3139 final-5af47 #36 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 9BE53A29-01D4-46A4-939E-13973CC5D24B final-5af47 #36 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 E7558DC7-D2C5-407E-8951-6DACDD98B27E final-5af47 #36 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 826FAAC0-F53B-4C51-A9F0-4BE699992D7F final-5af47 #37 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 4C05D5C7-3527-4410-A6F7-4F0EC52AD2AF final-5af47 #37 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 AF0A8478-3AC0-4A55-864D-29A33A6BD94F final-5af47 #37 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 C46F3F10-EBBE-4245-8588-B25F06027CE8 final-5af47 #37 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 329C17ED-E034-43DE-8A98-485A5E18BEFD final-5af47 #37 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 6694C3EB-22DB-4E3F-A9CB-8705FC492996 final-5af47 #37 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 31B8452B-213E-4B98-8094-48E9E9EC3184 final-5af47 #37 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 0C0E69AD-FA19-4EE0-83FC-3D941C61AA98 final-5af47 #37 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 159EEFAD-E23A-4B31-A21E-565B3CA5FBA4 final-5af47 #37 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 F833DACA-34EC-46AE-9E88-25443DD33D72 final-5af47 #37 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 5ADF82C2-9AD8-4053-A5BA-75690658AB4D final-5af47 #37 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 ADE1BBA8-4F3A-4C39-9FF4-FCBDD7131DBE final-5af47 #37 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 F6F8762C-864A-400F-BEC2-D57E47A15684 final-5af47 #37 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 63ED5612-87AE-4D34-8456-2E2035DC2B36 final-5af47 #37 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 E35CA4EE-F6E0-4592-84A8-3B21045E367F final-5af47 #37 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 C75DBE75-265D-456C-BB13-8BA154A8D903 final-5af47 #37 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 E919FE3A-1550-4ABB-AA82-DDD32C782107 final-5af47 #37 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 2271084F-BA85-4882-9FBD-6E35BDB62909 final-5af47 #37 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 DD65AE7C-7E6F-493E-9B07-91974BDED753 final-5af47 #38 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 0610CEBB-CE63-4B52-9300-FE914B398AF4 final-5af47 #38 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 E26C29B2-BFF8-4D67-B3A1-8B8B4BA8782B final-5af47 #38 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 76DEF430-108A-4C20-AF6D-5631BA1A68A9 final-5af47 #38 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 71CE30B8-7BF8-4BA2-8783-FC9E4075735A final-5af47 #38 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 60A09EB0-1C18-42DA-A26C-0C0E46D3566C final-5af47 #38 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 E217A6F7-A1B4-412F-8223-34B221C8DF15 final-5af47 #38 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 BD7256E6-C846-463A-8326-6D813182B5F0 final-5af47 #38 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 50E2D698-1572-4BD4-B110-6C1C3FCD2522 final-5af47 #38 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 FC4753EC-366B-4D3A-8C97-91ABB7129608 final-5af47 #38 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 24183C49-8881-481C-87EB-FCA6BB39DB45 final-5af47 #38 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 617255D3-F8DA-4DD4-BA93-25A9745BC658 final-5af47 #38 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 1F3CBDC7-ED11-493A-AE1C-987931DA102D final-5af47 #38 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 04048CFA-1CC0-4537-B473-4F9E46ACE59E final-5af47 #38 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 C94DABFB-911D-415A-A471-8C069EB84093 final-5af47 #38 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 AAAC0F1F-304C-4C09-81E6-2FE840BC2E32 final-5af47 #38 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 95022951-CB2F-490E-A5C6-D4AD91DDE172 final-5af47 #38 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 167902B1-5366-4E0E-91FA-A6F7A749ECDF final-5af47 #38 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 BB1DAA78-0364-4699-A60B-8E51AA4E4382 final-5af47 #39 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 AC465F9E-D5B9-4DBC-ADFD-DB3CA04D7EC2 final-5af47 #39 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 004F49C9-6F74-4C9A-8693-C24BAFFAAE3E final-5af47 #39 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 D05E4644-3544-46A6-A311-FD054A141129 final-5af47 #39 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 0A874262-535B-4F99-9908-20BBE4A1289D final-5af47 #39 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 968A999A-656F-49D8-AF25-E78D32BFBE29 final-5af47 #39 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 3DDB731A-944E-4C93-9E8A-B18F152FAF11 final-5af47 #39 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 7826EC4F-D134-4180-8AD3-7846B3C947FB final-5af47 #39 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 00BE7426-E937-4CEA-8A25-AF4186BC0DF5 final-5af47 #39 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 B6385535-9F2A-4E1A-9230-720D03F32942 final-5af47 #39 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 BEC5A469-BAB1-481E-A6A8-11B1997CBED0 final-5af47 #39 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 751DE789-782C-4466-AC95-57679816B86A final-5af47 #39 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 72715003-E993-4717-BE58-83352E204A7A final-5af47 #39 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 6C2D3CE2-D23A-4DA2-B6D7-196C6DCD212F final-5af47 #39 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 8B25B66A-20A4-4666-BB9B-93B6268515EB final-5af47 #39 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 F2750769-B56B-49E9-82FE-0AC868988566 final-5af47 #39 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 5F0143CB-0145-46AC-BEC2-118B2A273314 final-5af47 #39 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 933D14B8-A639-453D-BFBB-20860CA9F365 final-5af47 #39 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 07689B87-D17A-40D6-849D-3A38BE75A6BB final-5af47 #40 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 73201D17-1C31-4226-9DE1-B56F12D26E3E final-5af47 #40 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 1AA7FFEA-0198-45CB-A46C-D25E5EC17391 final-5af47 #40 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 91074708-7C8A-4971-855C-57B8B674C3E8 final-5af47 #40 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 6508AF7A-EF63-4FA6-B841-43017FB1765C final-5af47 #40 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 2EB8C4CE-87E3-44B3-8C12-1910AE80F5D5 final-5af47 #40 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 0F7B354F-4882-4E47-9FE4-12B69E1B74CC final-5af47 #40 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 BE4FF40B-FB50-452F-AA13-8DAE3BC083FA final-5af47 #40 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 0E98A325-D47C-4E2F-A8A1-06E987321C23 final-5af47 #40 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 FE08F05B-EAA1-4333-B41D-2EF3D0E20D3A final-5af47 #40 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 8BD9E862-261C-4574-8048-18FDD2C0D796 final-5af47 #40 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 F2842F42-F7B1-44E7-AE48-6A1E826E5186 final-5af47 #40 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 868D6041-3467-4CF2-B36C-9FC72F5F5082 final-5af47 #40 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 9A4C6913-8A4A-47DC-8825-FB64D9ADA746 final-5af47 #40 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 62D5DEEA-0219-4C95-9377-55380695B91F final-5af47 #40 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 80E250C8-8385-473C-8D9C-A2B0B177EB07 final-5af47 #40 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 DDC160A9-0531-4CD9-8938-475DC3BF5234 final-5af47 #40 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 F1FC58A6-AF3B-487A-BC3A-620495836C77 final-5af47 #40 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 897FB475-A084-4459-9547-B88A7043C01D final-5af47 #41 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 59933A97-C7ED-4E3A-9676-C65E0051D914 final-5af47 #41 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 EB9C0805-DAA2-4B7C-A74C-969355249981 final-5af47 #41 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 B1DE0BBE-0ABE-4E3B-A363-35FE485E7F62 final-5af47 #41 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 13CCA65E-699B-4935-A001-39D6EA3FF77A final-5af47 #41 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 ED198118-0E9B-44E6-A322-62E2B084E064 final-5af47 #41 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 119052C8-0972-4A72-8A47-3B81B69E2460 final-5af47 #41 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 F8203E0C-7C54-4258-892D-F16E6F0B7813 final-5af47 #41 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 2F45CFCA-483B-454A-8347-A75C79421890 final-5af47 #41 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 B46F271A-2A98-4443-B85F-2CBC5BDFED08 final-5af47 #41 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 E2DAD3A8-01F6-4BE6-8394-A8FE89F59E14 final-5af47 #41 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 01C134EA-B02E-464C-962E-770A9B7CFC3C final-5af47 #41 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 13F7056F-5D95-47CA-A6D3-DABA8BFC6482 final-5af47 #41 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 89AFDCB6-194D-4EE9-9BA4-E00C5BC512CE final-5af47 #41 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 0D573CC7-A6C1-41C0-AF5B-5C9C41AA3039 final-5af47 #41 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 592A1CAC-059B-4936-B3EC-CAF0B4539C36 final-5af47 #41 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 C4CE5835-9DBF-4C1E-9C4D-44CC2C350305 final-5af47 #41 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 B9B80345-8D8B-46C7-A3C0-B812A84C25B1 final-5af47 #41 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 C7E6D7C4-3F98-46AB-B0CC-DD563A10F7C0 final-5af47 #42 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 0253073D-892B-4BE1-97F3-DB8D654DC3D4 final-5af47 #42 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 F0A0302B-596A-46C6-8DAC-849B8FE7EA67 final-5af47 #42 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 AE52C0E3-0CE3-4883-B820-FD2C24B6AC71 final-5af47 #42 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 5AE910E4-D3B4-4FD2-986E-229416447401 final-5af47 #42 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 BA5F987A-028D-46BD-B813-184340C586C1 final-5af47 #42 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 03E353AF-7CD5-4261-ACE2-0B0EDE749B72 final-5af47 #42 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 DD4BC5ED-3FA7-4C9E-A435-BDFE7BBD1E88 final-5af47 #42 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 7140F2B3-415C-4C62-859D-3589767C89CD final-5af47 #42 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 A033510F-F0A9-4D14-B547-6607A927B1B0 final-5af47 #42 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 DD43B323-F35C-4250-9AC1-15DE6DFD9A32 final-5af47 #42 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 5990FD67-D3EB-471B-A586-A217C104CF1A final-5af47 #42 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 6D9DDEAE-B827-4FC5-BC60-C17D246545A5 final-5af47 #42 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 15E44781-57B7-424B-B21C-F5786FFE4E04 final-5af47 #42 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 6C81C8E6-C8AC-4282-B0F5-B114480E5335 final-5af47 #42 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 66C41147-BBD0-4A6B-A487-E10FA1B8AF4E final-5af47 #42 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 2E3C4685-9222-4399-8A5B-99DADA697AFB final-5af47 #42 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 6A3B991F-4573-4C4A-BF11-3C307FDD39E0 final-5af47 #42 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 68AC1093-759B-4135-A8E7-C9B30AAF8A1C final-5af47 #43 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 0D591064-B492-444D-A41D-061DD944C81A final-5af47 #43 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 7C464898-B565-46B0-84D4-35943B94424C final-5af47 #43 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 3A4AB937-AABB-4C68-966F-12AE656F485A final-5af47 #43 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 8477F1B6-A424-4E05-B298-F6BF61753D60 final-5af47 #43 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 FA0D93CF-ED88-40D8-8775-CDB12C6A8ABD final-5af47 #43 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 864183B4-1444-4A2A-9CE2-2216292A6A20 final-5af47 #43 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 2F2654FF-7E05-4E88-BC3C-C412014FF064 final-5af47 #43 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 C0712661-93D7-4D1C-8E3E-C2BA92268D48 final-5af47 #43 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 FABEC6DF-7DE1-46BF-AC5B-DE8DEE88552F final-5af47 #43 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 D4AB5379-D343-4C1A-9787-72C0508B57F1 final-5af47 #43 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 99319C3B-502E-46D3-BCBC-DB85EF6591E4 final-5af47 #43 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 0EC0DB6B-8E19-4F9C-8AD4-1B9DE09139A1 final-5af47 #43 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 9FEE2AD9-32B1-4DEA-87FC-6FFBE7C8EE1F final-5af47 #43 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 6E0075CE-33CD-411B-9CA1-86CB0980FF05 final-5af47 #43 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 66F181F1-AEE9-4430-8F84-EFEA90E7472D final-5af47 #43 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 07AFFD5D-8B14-4B48-9132-5C87A9720D0F final-5af47 #43 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 3D1C7AFE-9B6F-463E-9D72-2F7736D4294E final-5af47 #43 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 907A3516-E575-4CE7-A45D-FDC56A087499 final-5af47 #44 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 B234E525-BDE6-4592-A000-595A1EAF11A8 final-5af47 #44 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 A004FABC-C3D6-4BBF-A736-DF6324E9D594 final-5af47 #44 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 9F055FF5-02F2-4D5A-8541-FE4C34CBA4E3 final-5af47 #44 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 F5230E9B-F250-4861-982F-028C73ABD36B final-5af47 #44 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 6CB966C5-ED0E-4976-808D-25F984229766 final-5af47 #44 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 427FA061-6E79-4D27-9225-2BEF03066ADE final-5af47 #44 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 B3E01FE9-BFDD-434B-A6DE-423C24BB3B33 final-5af47 #44 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 37BFDC3A-FDEB-4850-AEA1-1AD0FF7B6075 final-5af47 #44 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 4B214463-0356-4DFF-8009-9AA7E8D3C9CD final-5af47 #44 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 5E54D2C8-51B5-4708-9055-A23F0702F033 final-5af47 #44 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 3B4B36FC-4294-4133-A767-C4BB5055423C final-5af47 #44 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 2D9D232C-EF45-4F27-9522-08E46AE1B0F8 final-5af47 #44 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 BAD12A62-4FA1-4BB0-95A7-6FC4635CC4F9 final-5af47 #44 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 ACE914C2-E89C-4092-B32A-8C8B12C76969 final-5af47 #44 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 1DA5DB05-0512-473A-B835-B1B1AB54FBB0 final-5af47 #44 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 70AE8342-756C-4A89-A105-D09CB3CFA36F final-5af47 #44 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 5505286D-6BFE-44D9-BD2D-6015DB4AC4B9 final-5af47 #44 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 55198969-13A2-4CB0-8B65-FD0DD76DB4AF final-5af47 #45 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 9A24AD90-5B3B-46C3-B09F-BFED95C12522 final-5af47 #45 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 C8B3E411-D894-4CF9-96DA-4F97A4952C54 final-5af47 #45 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 665B97D1-328A-46BC-9D9A-156EADA16B8C final-5af47 #45 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 D7304965-242A-4666-BFE1-5790A434C781 final-5af47 #45 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 38953683-2DF2-4949-BE63-9193D4019235 final-5af47 #45 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 DF819ECA-DE8A-439A-84CE-9A015EEEAE3A final-5af47 #45 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 77B7F493-5ACA-4A3F-B77C-5FE58AA77409 final-5af47 #45 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 C747745F-D43D-4635-A5F0-85C8C60A6FCA final-5af47 #45 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 938842EB-9515-46D9-8728-E7FC49C4F1B1 final-5af47 #45 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 F4136B4C-B608-4D3D-9224-BA11F47C94E4 final-5af47 #45 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 5F1DBBE4-DCCE-44B1-96A2-30CD3360B7B8 final-5af47 #45 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 490A7B0D-EE6E-4CF5-9CEC-3D7C03D2E812 final-5af47 #45 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 A632DD1F-0243-420B-848A-725D95557BDF final-5af47 #45 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 742FCACE-9DB2-4C9F-9FDD-1DAA8EE47823 final-5af47 #45 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 8F0311D1-E50A-41F1-9F42-E4EA28046F5A final-5af47 #45 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 25CAB603-DE1F-4B6E-A572-E0C07F41980E final-5af47 #45 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 20709E20-FF38-43D6-870B-FC6DB6A0BF2A final-5af47 #45 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 5098370E-460B-4C47-881B-CAA40664523D final-5af47 #46 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 41925D49-408F-4BE4-AB2F-5EE014CF49FD final-5af47 #46 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 A024C237-1E0E-42A1-A3C7-0D8F6C1D85D2 final-5af47 #46 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 65735107-3861-408D-BCD8-79C1C7E39D3C final-5af47 #46 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 055BFEBF-311F-4A07-B076-2A6B1CF10BB1 final-5af47 #46 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 5B06E0B2-9847-4C4E-8BB1-A65C363C0736 final-5af47 #46 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 882954C4-53E7-4D3D-90E5-FEFF00BF4412 final-5af47 #46 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 2EF576E9-88CA-4C67-AA4F-F6F41482ECBE final-5af47 #46 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 63F20547-2365-4818-B133-15F1512CE220 final-5af47 #46 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 150EE7FE-4D01-4487-9B2F-8477359F4CCD final-5af47 #46 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 4A693C15-73B0-4771-A251-AFE2D6D4A123 final-5af47 #46 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 4A2C2F08-BC59-4CAD-B04E-F252194C78EE final-5af47 #46 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 2B51CA63-54EC-4324-82A0-8580C01A8037 final-5af47 #46 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 1C60A10A-EE2A-4B68-BF60-829FC6693419 final-5af47 #46 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 049AAA2B-F7AB-4C78-8EA8-4A551F06DCFB final-5af47 #46 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 162BC08D-02CB-42E4-A80C-0F48291CFE72 final-5af47 #46 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 6973FEBD-B5F5-45DF-B0F4-68B8E428A642 final-5af47 #46 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 99DFE8FE-B86E-4B29-97CC-30215C266A57 final-5af47 #46 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 5EF00CB8-1064-49DF-8D15-6C19F0D7F65E final-5af47 #47 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 6715AFD4-B22C-40CC-8637-489D30C8F2B9 final-5af47 #47 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 DC7111F7-5EA7-4A08-8061-6D5E713C4EDA final-5af47 #47 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 B5662DFC-64B3-43FA-A68E-D1D7C38FA7AC final-5af47 #47 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 506E35D4-B1EB-4050-9A48-709097EF226E final-5af47 #47 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 621E13CA-7C86-42E9-B7FC-EA0355FE61AB final-5af47 #47 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 47BF8555-7A6D-499B-B914-72FB26D94ABB final-5af47 #47 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 012C1E39-6F0D-49C3-8FDE-2AD29AE4A146 final-5af47 #47 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 CC180CA5-2A5E-49F5-8F88-9AFFC5B02B38 final-5af47 #47 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 4F2D6E72-BD13-495E-9056-D0F9078F276E final-5af47 #47 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 F5CA97D4-F1CC-483F-8CBE-7D4A9B4C8B5A final-5af47 #47 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 5BDD356C-D887-49B4-A37D-71008C688DB0 final-5af47 #47 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 E4789EBC-0096-48B1-84F3-44DCF481D370 final-5af47 #47 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 13CEC50E-1F49-40EB-A8E2-653D398CA81C final-5af47 #47 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 A9EC3239-19B6-4AC8-9EB6-E7D26BFF3690 final-5af47 #47 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 EB48F74E-0CFE-4C33-96CB-ECC6CD634FD0 final-5af47 #47 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 1481F389-814C-486A-9B07-1917913EC452 final-5af47 #47 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 94C27E43-29A7-4A6B-BC2D-05A8D09C3140 final-5af47 #47 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 C471B7C8-5239-4993-81BB-6396A451E0E1 final-5af47 #48 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 4C5F0E99-6FE7-4519-B404-5D25EB8BCD32 final-5af47 #48 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 274A4963-D11B-4C71-9DB6-DC97848D80B5 final-5af47 #48 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 63564D28-FE7C-4A61-97F9-019E5F0870F2 final-5af47 #48 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 5349135F-AC8B-4861-B10F-1E6C00315640 final-5af47 #48 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 686512BA-DF31-4BC4-9316-62AF68C1642A final-5af47 #48 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 7239A5AF-779F-42EC-933E-B1D84D7BA6D1 final-5af47 #48 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 6DAE1EDB-8930-4027-8944-18F8E3AB20EC final-5af47 #48 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 7129AA02-6791-4724-9D2D-2B54DE4A4DD1 final-5af47 #48 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 D0A43C61-34F2-456B-8323-47EA81C31514 final-5af47 #48 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 8F86C83D-D135-4785-AE95-839FEAF7E99F final-5af47 #48 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 5C75D09D-D420-4BF2-A3EF-9FD67A81FBA1 final-5af47 #48 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 1A2099DB-5316-4E16-A769-7C05151CC7AD final-5af47 #48 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 C9838A89-BD9F-455A-ADBF-E5EED22857C6 final-5af47 #48 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 B5222B3E-CEFA-47C2-B00A-8A19AF4600A8 final-5af47 #48 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 3A04782A-A222-4F36-9C32-D1D1BD3CF1B8 final-5af47 #48 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 B27B309B-84A1-4C77-9585-1196DD364349 final-5af47 #48 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 9413E04F-7855-42D9-AEC0-3B6D608E2DD9 final-5af47 #48 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 03700113-81E5-469B-AC93-25FAC97B9598 final-5af47 #49 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 F9320957-588A-4600-BD06-60177CE1B48E final-5af47 #49 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 7BF730F8-97B5-4039-8D91-6D1B70FFE8D4 final-5af47 #49 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 C8FEA430-2807-483C-A588-F6072BEAAABC final-5af47 #49 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 58C3B25F-CA6A-4917-975B-006D42B3021D final-5af47 #49 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 60B6AEC1-96BB-44B3-AFDF-7602782B03A7 final-5af47 #49 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 9166D57A-583E-4C95-86B9-5DB0923CDD7E final-5af47 #49 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 28AED80F-6854-465E-A119-7692B9AC6667 final-5af47 #49 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 04299C21-9992-436C-AA04-46FE9743EE71 final-5af47 #49 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 4313DA26-6819-4C7B-96DB-BEF972B45CE1 final-5af47 #49 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 2801BBD4-4BF4-4992-A8BF-FC0A20EE155A final-5af47 #49 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 63AE7445-041A-4072-96BC-B36968938BCE final-5af47 #49 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 08E77AD9-3079-4442-B580-6AEC3A998A49 final-5af47 #49 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 6CB2A016-F52B-4D66-A57F-54F15C09DA5C final-5af47 #49 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 16A13242-5342-4D3B-8CE0-3E676BEDB490 final-5af47 #49 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 61ABB2C9-6077-411B-B248-B3057057D0CA final-5af47 #49 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 8D931003-6CB3-49F2-98D2-3D1DBE0AB908 final-5af47 #49 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 13BC87A8-D81A-400A-BE69-09239AA2C4B5 final-5af47 #49 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 D7134D75-B37E-4834-8BAD-1052C2271F13 final-5af47 #50 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 F7A06B6D-CD3B-48BB-A595-01CFCD06CCEC final-5af47 #50 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 24BEE734-F041-4795-A994-E915480DE1C4 final-5af47 #50 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 9C5989C6-3B85-4CC9-9534-C774CAFC786D final-5af47 #50 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 93278652-3269-44CF-9491-3898FED04660 final-5af47 #50 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 478CD9B7-7CA7-4871-9C4D-3CCE93863FEF final-5af47 #50 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 9B041A3D-3FF7-4DB5-8794-8E171B568576 final-5af47 #50 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 89BDE51B-39C2-47F2-85D5-E098F4BAF929 final-5af47 #50 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 5A713927-2E81-487A-8C02-4AABF6B9B03D final-5af47 #50 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 87665F0D-5FB0-45A8-A6AE-BD69F43E8819 final-5af47 #50 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 ECA74DC3-D16B-4749-871A-BC13E8BD7E24 final-5af47 #50 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 C4FB4C85-39EE-4DEB-928C-78D1024ADCCA final-5af47 #50 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 A8A61EB2-20DF-4768-9AB0-1D302FF302D6 final-5af47 #50 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 88948221-333C-4F07-AA19-F83DC6DF334B final-5af47 #50 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 257F8B42-5521-432D-A66B-2DE108EE007F final-5af47 #50 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 FC92AC0B-EFAD-4833-9B2C-6179660439A2 final-5af47 #50 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 1F190EDA-A4F1-4ED0-B55C-77679E159101 final-5af47 #50 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 9E5B460B-39DC-42FA-B56F-6FC8676EB331 final-5af47 #50 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 17339972-8625-4D65-BE02-96DDB54636C7 final-5af47 #51 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 0843F070-99FC-4CBA-9BFB-D44D9D8DB5F6 final-5af47 #51 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 8EA6C61C-627E-4434-A095-53BD469424C7 final-5af47 #51 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 3B086896-ADF6-4832-A0FD-2AC913F82A31 final-5af47 #51 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 EFD3E5C1-774F-4C0A-B6A4-E63E4CC0B156 final-5af47 #51 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 B6703F54-A6DA-4D22-A57B-4BF0A54D8647 final-5af47 #51 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 D93F173B-56D2-4B85-93A2-0AA8F65D385B final-5af47 #51 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 62072A2C-415C-45CC-BFB7-DEAB84D41380 final-5af47 #51 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 0C122CF8-4341-480B-8C61-D66946D2CCBC final-5af47 #51 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 3D974274-F94A-4EB7-A55C-EA1475B9DDDA final-5af47 #51 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 22A51375-9EE6-4B6B-B566-8B0E0B8C8BAD final-5af47 #51 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 11462A61-004D-4905-97FA-A59E78615DD9 final-5af47 #51 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 48E0E528-1301-4680-A2D1-CA94713A0B70 final-5af47 #51 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 9D2CCCBD-77D5-474F-B511-A6A375C8531F final-5af47 #51 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 0BB70951-FC90-456B-A2E1-F29FB3C0E5E3 final-5af47 #51 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 9CED404F-C6DD-4FCC-9B02-B02939E688AC final-5af47 #51 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 A691D42E-4E2B-4BBE-AADF-8BA0A0C39080 final-5af47 #51 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 EEA5BA2D-DC7E-4548-9FA1-E3C975260765 final-5af47 #51 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 F84358EE-C390-4028-91A7-D970158281F1 final-5af47 #52 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 D3DEB5F8-110A-43B1-BE16-1BC9C3ECD8A4 final-5af47 #52 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 D643AE46-3176-467B-A67C-121FBE469743 final-5af47 #52 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 60D06FDB-E824-4C5D-81EB-4BCAE678509E final-5af47 #52 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 A41CEF8B-4A2E-4EA8-8A64-CA01B508D380 final-5af47 #52 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 B7716653-D394-46EE-8940-50038574809C final-5af47 #52 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 32940151-98B8-4D5C-AC0F-1395693B5062 final-5af47 #52 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 D39E8509-2617-4F02-B726-640CFEAF5D78 final-5af47 #52 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 02F82D31-7F7A-4546-B70A-FB2253C21F56 final-5af47 #52 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 44938E47-8ACB-4434-A391-B724EBE1CDF5 final-5af47 #52 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 D958A8A8-BB01-4BFB-844C-B03BED52800A final-5af47 #52 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 099AE153-5B91-4EC6-9CA8-4618A36F4E57 final-5af47 #52 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 38FFEAC9-5EA9-416E-A256-F59E0FEE6755 final-5af47 #52 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 013131CF-8EDA-4F65-A385-9A726BD7A4A8 final-5af47 #52 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 A97FBA12-8777-4C7F-9061-D7C2EF383904 final-5af47 #52 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 57AD149B-8882-45A6-B889-C7C18C729C9F final-5af47 #52 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 E21F0FE4-F943-42E8-B3E9-06680744C31D final-5af47 #52 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 34E5E2A3-713D-42B3-89CF-7D5D0499B7FB final-5af47 #52 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 96E58F10-EC0F-4F73-A8F3-49A873BF254E final-5af47 #53 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 66A49FEC-B292-42FA-B941-E361D0E16F7E final-5af47 #53 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 F790CA3C-7685-4676-B025-EBBD19343ADF final-5af47 #53 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 A2385627-BB75-48C8-9A66-76CCE5D88152 final-5af47 #53 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 F7251500-C78B-4C40-AB53-468361DE1BF1 final-5af47 #53 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 B35983B5-C905-482C-9BE2-C38136B971F4 final-5af47 #53 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 F3228556-7A07-4450-BE4B-E6384E80A02F final-5af47 #53 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 F6D3A049-3BAD-460A-9CC5-076800F6AE2F final-5af47 #53 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 56665511-9D29-400C-A6A6-17A023798B0D final-5af47 #53 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 BD73890F-C5DF-4E32-A5A6-1AA88200E5DC final-5af47 #53 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 9C932D04-DF6A-42D3-9078-48794F576C9B final-5af47 #53 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 4F39651B-82FE-47AD-89FB-5B552BD4A966 final-5af47 #53 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 C46B559A-D17C-49EE-B9F8-9084CFDD5249 final-5af47 #53 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 30E17FA7-E896-42F7-B11B-15074E22D8EE final-5af47 #53 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 0CF27B1B-6728-424E-8BBB-74F29E7057F5 final-5af47 #53 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 161BB33F-7438-4CCB-A491-FF4F4B382164 final-5af47 #53 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 D194112A-9547-4B60-92E4-F7691779072E final-5af47 #53 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 3AB1DB4E-BD1F-4E7B-9613-15F6807EDEED final-5af47 #53 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 A3575F30-84E3-48E9-9FCC-BF42E2F4E930 final-5af47 #54 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 4F28016E-2205-4DFE-B98A-2FB1F9E74043 final-5af47 #54 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 8806EBE5-2637-4037-8BE1-A95D5CA17FB7 final-5af47 #54 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 2C9DA4E7-918D-44BE-9015-6FE0A77B260D final-5af47 #54 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 9AC0C346-E077-4342-A9ED-B6947A67ED9A final-5af47 #54 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 6A7DFFF6-B751-46E8-9478-06C3934D20B9 final-5af47 #54 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 92B3E908-6058-4AFF-AB02-31AB05EAEF50 final-5af47 #54 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 416740AB-8D57-4269-967D-3A07EA5324FE final-5af47 #54 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 9E62A740-674F-4F10-A99B-371A2DF1640E final-5af47 #54 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 603ADE54-6F48-41AA-B4E5-C825ADD945C2 final-5af47 #54 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 4D0410FF-E543-4CC6-9F4D-F469865BA143 final-5af47 #54 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 CC71E79E-1257-4AAE-BA35-0DBB6EFD071B final-5af47 #54 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 6ABCF79B-8870-4BF4-867F-9AB78A1F835F final-5af47 #54 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 121650E3-C291-4549-941B-681EA790EBC4 final-5af47 #54 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 A5DF9E86-954F-49ED-8C13-9641626964DE final-5af47 #54 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 4C4F1249-A05A-45A7-A0D2-9479A8CCF099 final-5af47 #54 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 81B8725B-3F33-461D-999A-63F67F71CB8A final-5af47 #54 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 4FF5002B-5BA2-4DAB-8DC1-DF103F16F3B7 final-5af47 #54 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 998583F2-29BD-41FF-96AC-4BBC42BB3894 final-5af47 #55 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 894FA407-0DEC-4DB3-9893-4695F90A0086 final-5af47 #55 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 2B3D837F-A4BB-4903-855D-7E61E826052B final-5af47 #55 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 5C7564F4-04C7-4E8B-89C0-3F172CD8076D final-5af47 #55 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 B0DB0247-69D4-4FD6-9DDF-1ED06996637B final-5af47 #55 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 678B81B6-DCBF-481A-A493-5F712E3F8137 final-5af47 #55 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 77553C62-173C-458A-9294-124FAC6F7C0B final-5af47 #55 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 BF1574DF-05DA-4E90-A42A-98AA58220E74 final-5af47 #55 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 57EDFD54-10B3-4155-A02A-2181F1336D12 final-5af47 #55 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 637C12B9-D676-4685-9EF6-5300822E011B final-5af47 #55 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 D48C7A88-3930-4F0B-8C79-5CFEB4A0CCEB final-5af47 #55 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 ECFBB6F3-86E3-4E13-B4DD-9141D69B3598 final-5af47 #55 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 C643874C-11C5-4944-A05A-EC8172676C7C final-5af47 #55 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 DF43D018-AE47-4FE1-8494-E882AF74E444 final-5af47 #55 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 CA063FC5-2714-4250-B513-1AB62EA3F2FA final-5af47 #55 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 80410FAE-AA34-4503-8BFC-22115B8A474A final-5af47 #55 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 E8A22725-9576-4171-BBCC-30B4AABB6AAD final-5af47 #55 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 DEC0784A-0D2F-4626-B5B3-06AC6B7A87F6 final-5af47 #55 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 2BC7C90F-360A-4360-A70A-29DCB13A03C6 final-5af47 #56 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 93BC2525-0B84-4A03-961D-24BEDF7E078E final-5af47 #56 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 C8B23ACD-3608-45CE-88E4-D104CDB4CFE9 final-5af47 #56 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 19C12349-16A9-4F93-90A5-3B53BCDE24BF final-5af47 #56 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 8FB40A44-AD01-482B-A7DB-45B72211C3E9 final-5af47 #56 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 E61FF2F0-BD2B-45F6-BC39-4D96DD608CBA final-5af47 #56 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 820205FD-AE67-4362-AAD0-49FC24F49E55 final-5af47 #56 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 BA4F1B2A-A474-4F08-BD12-5925C16DFDD5 final-5af47 #56 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 03C9B4D9-7FB2-414E-9BAC-C21AAA47EB65 final-5af47 #56 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 3E7CF1BE-9067-4081-8A5F-752FCB133A95 final-5af47 #56 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 F8E615B5-C985-4E7D-81C2-3BF1F23904E1 final-5af47 #56 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 B7D3318E-36F9-4181-B7E2-4B7855E1458E final-5af47 #56 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 981F91CB-BD1D-47BC-BE67-68A2C6CCA341 final-5af47 #56 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 898C7CDC-3E69-40A4-9173-24FED6B152B8 final-5af47 #56 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 A8B21C52-1963-4254-B050-3020DE71DD66 final-5af47 #56 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 AF58A445-5DAC-4D1D-A364-73A42577B7BC final-5af47 #56 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 E41752AB-DC0F-4484-B889-C94AC4DB11CA final-5af47 #56 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 B848F808-1471-4EF8-845D-C35AEA721381 final-5af47 #56 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 CEA0C8A6-F525-46FC-A64F-36A27DA859F8 final-5af47 #57 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 FB8B2931-F47E-4FAF-A148-65A5CEF478CA final-5af47 #57 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 6ACDF02B-1DD9-4A34-ABFA-50A9CEE53D7D final-5af47 #57 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 97DD8B08-1918-42FE-8B65-8F5862818387 final-5af47 #57 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 E1160B58-EA88-402D-907B-A95F86E01EEE final-5af47 #57 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 EC84EB5A-C70F-45CC-9F02-C3A63047C74F final-5af47 #57 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 648C7FB1-B960-4B45-96BD-11380880C14C final-5af47 #57 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 6ED4A8BC-B38C-4643-A022-4F7B40FA0B51 final-5af47 #57 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 257D318E-2282-4DDE-BD05-62A56246681E final-5af47 #57 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 B2D040B7-F3FF-43E2-A1EC-1D1D2467D236 final-5af47 #57 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 47B6BC5C-9262-4008-B7A2-19888716AB41 final-5af47 #57 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 6448ABA5-36E5-4D74-A2E8-DF8485C8EF7F final-5af47 #57 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 33A52539-35DC-4655-8302-B9A39394A7E5 final-5af47 #57 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 A86BCDA6-7B46-47B1-A946-35B71E722EEF final-5af47 #57 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 014E2805-2021-47FA-9136-76E17D690B05 final-5af47 #57 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 A099DF84-2993-4359-9F9B-158DE26C28F2 final-5af47 #57 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 0E60A286-9846-4AB4-BA92-30842C36183B final-5af47 #57 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 E5C4EBEE-9C9C-41F0-826D-9F3226518BDD final-5af47 #57 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 005FB64A-F6BC-4027-8A7D-1D92FA8F1479 final-5af47 #58 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 72E2B4D7-4ECB-4C94-82D7-6A0E3D3703A7 final-5af47 #58 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 81A65071-AFC7-4889-A9FC-278E311AD40A final-5af47 #58 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 39416DFA-4D66-40EA-8365-5E23DD76944F final-5af47 #58 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 8DCA08E6-F8BE-4A9F-B20D-EA22174EB535 final-5af47 #58 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 77356DB1-E0CC-4A42-B537-07514589DD8E final-5af47 #58 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 C27D47D2-968C-4B08-B9B0-052E96AEB25D final-5af47 #58 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 66F53FBF-3063-489D-B76E-4CBC3BFB2110 final-5af47 #58 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 E1FF0148-3A67-422F-B0B1-64FE59EA41B6 final-5af47 #58 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 76F70330-6727-4E98-96AB-BD0F76B43F88 final-5af47 #58 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 5BD8E1F2-18B0-4D01-A338-DE7BE4C54FB8 final-5af47 #58 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 F4CB9961-050C-4279-B122-DB601CEF51D7 final-5af47 #58 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 B0233A85-954B-49F2-9FC2-DCAC63557DE2 final-5af47 #58 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 6DC95BE3-133B-46CB-93AA-26C775DBC7E0 final-5af47 #58 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 94D6B9C8-4395-44AB-B23B-D61880AD0103 final-5af47 #58 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 DFB41D66-1084-478A-8723-D8C1AFF4D4C6 final-5af47 #58 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 8D681DA2-39D3-4294-9C36-5556008DA092 final-5af47 #58 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 6EFFC9CE-CC3E-4092-AC1D-36783338AFB1 final-5af47 #58 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 DCDC8C56-69D2-4FAF-9458-1C4465BD45C0 final-5af47 #59 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 46B79131-3F3F-4EF8-A7FB-8B6BBEB97FFC final-5af47 #59 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 9BE665AB-B538-4BB2-8DB0-A4605F68B3C8 final-5af47 #59 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 38C48233-FC8D-4905-850D-EAF8D787D127 final-5af47 #59 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 B7028000-67EC-4032-B024-14638128B2E1 final-5af47 #59 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 E669930F-4B1C-4B7D-B99F-D059BEB2D32B final-5af47 #59 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 2822BAAA-346E-4A6B-901F-095E51E1A65A final-5af47 #59 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 62D613D9-1DBA-4471-A18C-5509886187FE final-5af47 #59 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 73C1E8C2-CA99-4707-A27B-EBDFA48F466C final-5af47 #59 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 78D62CD2-49BC-41FE-8B37-D56F71CBFBE1 final-5af47 #59 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 BB558181-5BA4-461E-AABE-CC8CA14228C6 final-5af47 #59 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 AFEE291F-CA22-40AB-82CB-28F30EC47C1D final-5af47 #59 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 9843E809-23E6-452E-AC29-69ABEB66ADCE final-5af47 #59 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 D739A9D0-2D61-4F57-B2FE-B79DB5203EA6 final-5af47 #59 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 0C5340B9-49BB-4CA3-A760-6F24C7061992 final-5af47 #59 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 1117D93C-8758-4B7C-90BB-1653A2896EE5 final-5af47 #59 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 AD9D1AEF-2A92-40DB-9440-FDA0AFBC21A7 final-5af47 #59 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 C9A52D26-9181-4F31-922C-4BB17358D326 final-5af47 #59 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 19BC4B04-C86E-4D2F-A7FF-B38DA971A11A final-5af47 #60 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 B521ED5B-87D8-41AD-96E0-36700651780D final-5af47 #60 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 044474BF-B3E2-4806-8D44-E12D6176B50A final-5af47 #60 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 149AE10E-8829-48ED-9D6F-003F51BA9FD1 final-5af47 #60 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 F95BA8EB-7DAA-4395-8EC5-7C5DB9041E04 final-5af47 #60 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 6C837BE4-D06D-4B87-8819-620425A105AD final-5af47 #60 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 F12DA6F0-660A-4B70-8A1F-949AB58D6542 final-5af47 #60 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 275723E3-9151-461A-930E-0BCFD6FB3091 final-5af47 #60 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 9970E813-5389-457A-A84F-58A293A9B019 final-5af47 #60 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 23D997F9-3868-49D6-890D-7A21C11471AC final-5af47 #60 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 62FDD58C-C948-48F7-9A2C-7C6E7A58D015 final-5af47 #60 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 98016027-1693-401C-BB9F-87ED206ECBE9 final-5af47 #60 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 3A4F35B9-3063-4F6E-B7BE-30F91EA43360 final-5af47 #60 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 A9839278-BFC0-462A-BE3B-821303B3156E final-5af47 #60 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 E9311D80-3ED4-48F1-972C-363C3D60C08D final-5af47 #60 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 312198A7-8687-4FA2-8D87-0F081989BE1E final-5af47 #60 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 4BA7D82D-67C4-45FA-98A0-C5DCF35D6F24 final-5af47 #60 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 73B8C358-A81D-44C3-ABE1-5765FBA14CCC final-5af47 #60 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 28806E97-B57E-437C-923C-3449156657BE final-5af47 #61 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 DC23FAAD-543B-4F94-A64B-B9716BCE8EF8 final-5af47 #61 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 6A03DF58-6CC8-43BD-9041-95821F38435D final-5af47 #61 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 7BABA9E3-6DC0-4397-B599-03976BE2C1EE final-5af47 #61 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 4A5F719A-2067-4567-8A82-80125F09D562 final-5af47 #61 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 3DF0AE76-7052-4663-B538-21D8E03A41BD final-5af47 #61 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 E108B0F5-BB80-44CF-843D-D91F11FF5EE3 final-5af47 #61 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 5D305FAF-9903-4C37-AEB2-9F2C5A296C14 final-5af47 #61 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 6DBAC62C-D9BF-48DE-A221-F07E7516EDF0 final-5af47 #61 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 F9700375-3BD7-4948-B051-581D5E0B0C25 final-5af47 #61 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 98904609-61A3-44ED-963F-D8052071B752 final-5af47 #61 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 93D46A32-8974-4B53-96A2-AB2EE051BDEB final-5af47 #61 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 BDDD5E5F-B84D-4B0E-98C3-7998A00E9189 final-5af47 #61 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 7BE03F66-64FF-441D-960A-BE9B07F6E560 final-5af47 #61 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 EB7AAD69-EDA9-4F55-A4A3-48536624EC99 final-5af47 #61 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 04BF2801-CEE0-408A-90C6-D6D243C308C5 final-5af47 #61 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 3B51E900-210A-4B70-A0FB-930F206B29E9 final-5af47 #61 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 7CF5501E-AF86-41DA-8225-9C63729F8F99 final-5af47 #61 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 73CB7BDC-F0D4-40A3-8782-80AC945EE483 final-5af47 #62 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 54B05C7C-12DD-4B14-9258-021357D10E7E final-5af47 #62 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 0BD837EB-993D-4B61-839A-C2B894BF763C final-5af47 #62 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 FE21951F-9DCB-45A7-AA2F-0A9D6C2F1EF8 final-5af47 #62 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 AF148864-D3EA-48BF-AAAC-DF69094F3E04 final-5af47 #62 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 FCD2B7D7-FCAB-4229-9244-A84282731E96 final-5af47 #62 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 F14965E1-DD32-4737-9122-A21CBF55C6DB final-5af47 #62 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 1527F2E8-A1A7-4B35-A201-53296A06CF91 final-5af47 #62 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 0E8A0A06-E6E6-4E90-A4A3-208AF5C19AB6 final-5af47 #62 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 B9000350-6FB6-4110-9F2E-D279CA567D5C final-5af47 #62 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 A2E00142-9338-42A1-9806-CC6D459B8CBB final-5af47 #62 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 C265595A-FFD1-4F50-B4CA-34CB8B345844 final-5af47 #62 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 F30FFECB-F449-4413-A64E-7BDB280D6FDA final-5af47 #62 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 61FC90AE-0FEE-4CE4-A0F8-1EB047EC5F5E final-5af47 #62 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 DBC74C0C-3CFA-48A6-9801-256BEDA2E94D final-5af47 #62 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 2AC2BABE-C3BB-4167-AEAC-F4EED6F07E73 final-5af47 #62 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 7038B0A7-354C-4479-AB2B-F2ECF6D38291 final-5af47 #62 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 4444B588-888D-415D-A543-C17E2DEBB0D0 final-5af47 #62 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 E20B289D-C6F9-4065-A49F-C1A512232237 final-5af47 #63 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 153642D3-7A7D-4DFD-B361-D6A86A89BC83 final-5af47 #63 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 6935C3A4-9562-45F9-A775-1946CE01CC70 final-5af47 #63 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 709B004F-5CD1-4888-AA45-0C5AF0EDC98D final-5af47 #63 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 50C57C0F-38D3-4F72-8172-1DA3A45CC21B final-5af47 #63 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 431ED349-5EF8-4E6B-824A-FC5046940585 final-5af47 #63 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 002113A5-6944-4AC4-BB0B-A25E446E117A final-5af47 #63 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 D7D95418-ABDF-4078-BE61-AD35B66BFB8A final-5af47 #63 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 2395B06F-3CFF-4D69-9845-58342356693F final-5af47 #63 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 3C7E8A88-DDE7-4359-84F2-6989B28F18CD final-5af47 #63 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 D11BB6A0-59DA-4CB6-B4FA-C7C1D9BCB160 final-5af47 #63 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 C7253B5A-2A42-4D88-B792-B2934FDD04E5 final-5af47 #63 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 A2663323-4E6B-4368-9B6A-DF49D0A0EF99 final-5af47 #63 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 9699FA36-2841-45B2-8A6F-132AD5ABFBA1 final-5af47 #63 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 9953F06F-A3CC-49B9-A5AE-160349207CF6 final-5af47 #63 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 C938C36B-8E53-4AC6-AA7D-B254920F9900 final-5af47 #63 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 486C6E7D-CBD9-49F0-880B-A76B10A4FC44 final-5af47 #63 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 9966F487-07C8-4E0D-8B48-2B8B92358CD9 final-5af47 #63 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 556BB7BA-9F41-4CD4-8678-3D16E2658A58 final-5af47 #64 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 E3E103A6-104F-4DAF-94EA-482065421A26 final-5af47 #64 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 4B47B9D0-A73C-4D74-8BC3-ECA7644E61EC final-5af47 #64 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 4610A688-682D-46E3-A6D1-559855B7D17C final-5af47 #64 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 F10C2A6B-7B40-40B3-827B-17F26B707182 final-5af47 #64 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 F9FBC014-2BB8-4038-9F96-492105AF9D3D final-5af47 #64 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 A66CBD52-9244-4AA1-935A-270B01D66DFC final-5af47 #64 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 0B7A61BC-50B5-47F4-9A56-C17CB48634B8 final-5af47 #64 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 2EA24BCE-923F-498A-86F0-15E728A97CEA final-5af47 #64 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 C0DBF2CC-3570-44A5-B350-6057B4016946 final-5af47 #64 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 DA09A8FC-F689-4BFA-81EF-26CEA2C89B1C final-5af47 #64 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 3EB80139-5614-4EA2-8D74-5D33ABE19A02 final-5af47 #64 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 D1326B8C-B393-46AF-97A9-850DB9D3A089 final-5af47 #64 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 E159C79B-1E37-4A7A-9E50-BFC9C475B595 final-5af47 #64 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 9D451BAB-3B03-4657-BC9C-C02AAA212775 final-5af47 #64 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 AA0B3587-AFDC-4E18-8D76-FEC3DDEE768A final-5af47 #64 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 9C2668C7-D9D6-45F3-BD0C-6938E0057491 final-5af47 #64 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 94660A35-B049-43F2-A70A-C557DB46F385 final-5af47 #64 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 5B2694C6-8901-4E11-8661-3B0782A2665D final-5af47 #65 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 87851CBB-9F78-479F-9416-3BFFAB4E3882 final-5af47 #65 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 F55B7C4A-66A1-4055-BDAF-88318B09FE95 final-5af47 #65 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 529537DF-A9DA-4F6F-BEC1-591CF0E43D53 final-5af47 #65 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 0A510E16-D5CC-4C8A-A74A-EE42B79D916D final-5af47 #65 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 82227286-CD33-4301-832F-44CCB2B5B786 final-5af47 #65 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 ECF5D2A6-BB09-4A44-8F9E-ED8FFB5E71FB final-5af47 #65 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 729DEBF2-F702-4506-9F9E-8C26A25A10DB final-5af47 #65 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 96C9D8C9-D615-4326-9FBA-4F62C5035E15 final-5af47 #65 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 D72BDDD1-8CD1-458F-A59C-206C44DD6ED9 final-5af47 #65 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 9D46EBEF-2731-4675-B1A8-143C0F449663 final-5af47 #65 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 829553F0-520B-4E19-8DF1-9F63159D1F59 final-5af47 #65 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 7FC40566-4976-4BBB-8E24-8F5133E6D296 final-5af47 #65 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 4C6BA351-5CE1-4929-A900-2F264A40E5AF final-5af47 #65 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 5C25856E-8CAE-4660-B95A-6AB9698E5789 final-5af47 #65 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 FEF1331D-B250-4AC5-9257-783BF2EE3C7C final-5af47 #65 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 F2C2560F-AE94-4788-BBFE-B4D411A87C8F final-5af47 #65 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 A7A0FE87-6DCA-41CE-8312-CEB871A6911A final-5af47 #65 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 7048ADD4-0DF1-4DF8-9760-16EBB3E2CBAD final-5af47 #66 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 15BA2902-4B7F-4FB9-ACA2-1C007EFF61C5 final-5af47 #66 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 3C839A50-D021-45C9-AC03-7DEEA8556088 final-5af47 #66 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 00FFBEB3-41DA-4734-9D67-2079F7929D8A final-5af47 #66 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 1891E7E7-CEC5-40DC-ACB9-4A45369EAE39 final-5af47 #66 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 DD144D47-7C85-40EB-A1AE-A0232C5B12F8 final-5af47 #66 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 FFC91186-9698-49E5-8861-D4A82F57828A final-5af47 #66 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 33AE3007-FD43-4B0C-8700-9D914B8B51D5 final-5af47 #66 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 6BB0C5C5-6B67-41FD-9DD6-4D3C602F7D91 final-5af47 #66 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 021BCFB3-8DB6-4EBA-9557-8DF03EE63E5B final-5af47 #66 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 D395B34D-FD28-4567-B89E-79BFE6ABF6E1 final-5af47 #66 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 6724C683-9A54-4450-8989-DA7D99FED5BE final-5af47 #66 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 BB919953-D61D-47C4-95F9-178A31709C65 final-5af47 #66 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 376AF757-638D-4D19-8DB6-D4A53C95FCAA final-5af47 #66 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 B241E127-9A59-41E1-B60F-88E6ECFF46DC final-5af47 #66 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 5AE1BCDB-F707-484B-AF50-33F59679D297 final-5af47 #66 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 E1B8B3D5-87C3-4123-B115-A0DFC3CA88DC final-5af47 #66 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 94906087-14A1-49C5-857E-97F655A12FE3 final-5af47 #66 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 37325516-B7EA-4877-A730-71E994BE4FDA final-5af47 #67 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 D971693A-4FEF-464A-826D-39D7B0E63A8E final-5af47 #67 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 98309DB3-3AB1-4EF8-8FCE-2A416BDA9CB1 final-5af47 #67 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 325F4D01-FD3D-44F4-A015-3159FA477EB7 final-5af47 #67 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 D1DC1F9E-07C3-4D07-8E60-24C061075384 final-5af47 #67 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 FCF318D0-C811-40B0-A303-956A3BE21B35 final-5af47 #67 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 11866CBB-DDE0-4889-9E24-810C7B5489AB final-5af47 #67 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 04295236-D7DF-45E3-8BE9-406570DF71CE final-5af47 #67 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 627374CB-57BE-4D89-9DBD-C02BAED9BF1B final-5af47 #67 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 D75D10F7-1EF2-4604-82F6-49CA96DBDBA5 final-5af47 #67 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 5AC8CF5F-CC45-41EF-9D28-7F56780A3A37 final-5af47 #67 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 731C16E5-065C-4F77-AF63-56C4F18D2CD0 final-5af47 #67 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 1BA2ED66-B7B0-4A68-BCD3-1CA671A2417F final-5af47 #67 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 7ADA136E-9062-4686-A2A7-5D071466E576 final-5af47 #67 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 DD91E89A-8590-480A-A8D8-47FB9502E0FA final-5af47 #67 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 50FB93D4-DD50-4014-A8D9-C464B35C58DA final-5af47 #67 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 48842EC5-6BC3-4177-9005-F75907B209E4 final-5af47 #67 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 F0939064-36B3-4E1C-92FC-4C7FCAA41B54 final-5af47 #67 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 0B1227D3-B9D1-4310-9E6D-244160D11853 final-5af47 #68 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 21E4DBBC-FF29-4D55-9595-4E5F68418A57 final-5af47 #68 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 D91B3E51-734C-4496-A754-AD0B56494762 final-5af47 #68 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 42494D16-B650-4E97-8A93-CC25CDDE6462 final-5af47 #68 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 C459FFA1-4CC1-4BE2-9543-AC0EB67D5382 final-5af47 #68 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 D43D9BFE-1A7D-4A70-A895-57093FD2575E final-5af47 #68 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 2BBA7454-D2A3-413D-A992-3CAF1C78CBA9 final-5af47 #68 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 E069748A-43A2-4619-ABE9-27DE8FC0A1AF final-5af47 #68 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 365B6066-E70B-48C0-BC37-9695477AE8E4 final-5af47 #68 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 1FE33715-3B41-4247-B119-B51A34468E82 final-5af47 #68 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 DA24969B-B8BF-43BD-9211-0501EBDEA9DC final-5af47 #68 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 71D9D4C5-8469-4593-965B-8A92F02CE557 final-5af47 #68 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 A2F97C0A-5F03-4C36-B32B-49CE740AB3F8 final-5af47 #68 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 7D9BD6FE-C2B9-426C-B798-55E5AEA11ADB final-5af47 #68 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 43D9C75C-5C94-4059-A3A4-865FF743EC72 final-5af47 #68 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 15F2D61F-9907-456F-9EAE-348F5F06320C final-5af47 #68 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 EC3A11EB-44AA-4FC0-AD0F-A44D58BF4C88 final-5af47 #68 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 8D6C1825-BC19-4344-A885-EA3A5236DBD4 final-5af47 #68 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 3C704CDC-DB62-47C6-B382-62377B8E1F7F final-5af47 #69 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 2E3E88C7-BD7F-4B62-9CA2-C49D66AF5AF1 final-5af47 #69 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 C3C9B038-AD55-4611-9055-5E65545EB7DA final-5af47 #69 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 70376EF2-4085-42CB-8FAD-9194B3691657 final-5af47 #69 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 563D8032-70F2-49AB-B267-C56C0135806B final-5af47 #69 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 0AA42CC2-0E22-4D5A-BE67-1E1A8FC93E7C final-5af47 #69 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 AECE61EB-F660-4ADF-9148-365301C1634F final-5af47 #69 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 B6F32CCA-143F-4F79-9123-7E6FC89F8FA6 final-5af47 #69 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 A32D00A3-77A9-4EA2-A688-6E568F7BF641 final-5af47 #69 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 42D5D78B-D75B-45CF-A1C7-2F5015FBD51B final-5af47 #69 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 47DA73D4-FC47-4768-8631-3EEB76CDFC96 final-5af47 #69 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 40F90C2D-C939-41EC-B822-AE3E5E069809 final-5af47 #69 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 4468C76E-0473-4FE2-8341-8A0627E5B6D7 final-5af47 #69 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 3DCCDF60-7093-42FA-A1BB-87B72FE96BC3 final-5af47 #69 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 62FA2783-144A-403B-A988-C6A4466272DB final-5af47 #69 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 1CD03A05-62C0-4273-B254-DC4B4CD7BAE0 final-5af47 #69 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 948F59DB-855A-4A32-8392-8B721B83708E final-5af47 #69 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 C0922D12-3997-4627-80FF-965373BAA87C final-5af47 #69 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 7331975A-7153-4417-B974-05EB2A959349 final-5af47 #70 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 0D0BCE6D-DF7E-40A7-90F9-32680E011B4D final-5af47 #70 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 80D43642-A035-4E38-97A1-7326DF8D9E57 final-5af47 #70 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 2267DBD6-4DFA-4A30-AACA-223FF401561E final-5af47 #70 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 F2B26913-64A3-4498-813D-427689244C3D final-5af47 #70 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 917F66C9-0286-4FE2-B186-B7996533E459 final-5af47 #70 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 2083A84C-63DD-498C-9086-FD32456B2997 final-5af47 #70 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 B9E669D1-3FA3-4649-8E57-E0AAAA2C7769 final-5af47 #70 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 486A2EC0-0796-46AE-9672-4BBA3F598EFB final-5af47 #70 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 8A0B1DC3-75F5-4EC5-8B61-641216A4D80E final-5af47 #70 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 29697F0E-0BA4-4C02-983B-AF8F5DFE3490 final-5af47 #70 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 96E386FA-D95C-47CC-8186-B4DE018D1626 final-5af47 #70 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 AF35C78A-5546-47DF-AB58-3CDA5D25FA3C final-5af47 #70 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 EC2B5A60-CF65-4DCA-A323-E38FEC8A8771 final-5af47 #70 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 255403A4-1CDF-4165-8A8F-6B413C5B71B6 final-5af47 #70 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 45A49AD5-A3B1-448C-B95A-7D97E776162D final-5af47 #70 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 661FA27E-5A90-4A9D-9EA6-99DBB4760E92 final-5af47 #70 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 4C73F836-3BCC-4DC5-A612-44473D3FC58B final-5af47 #70 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 A6278A31-566D-4E91-B35A-7CBED2213C94 final-5af47 #71 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 CED4C63E-8AAE-47FE-97C8-AFFFEB2E8363 final-5af47 #71 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 B16F72D6-D509-4ED0-8191-7E5DC9F1D938 final-5af47 #71 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 67AFEDA9-ECF0-49AA-A459-3EA67D45FC07 final-5af47 #71 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 EDDC993B-E09A-4997-BB04-4B49D2ABFA5E final-5af47 #71 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 4E76413F-CEB5-4668-B974-DCD52F269FBC final-5af47 #71 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 6283D993-7752-4B13-9842-157755447D9D final-5af47 #71 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 2CE0F5F7-A766-4F4B-8B91-5314711288E0 final-5af47 #71 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 50860366-6135-46D6-9DBB-28EACAA57940 final-5af47 #71 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 7D651081-272D-41FB-8ED0-33669F372076 final-5af47 #71 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 F08F5739-515F-4D9A-A57A-F671EA1D5C15 final-5af47 #71 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 F6C37A50-688D-4E51-A93A-5B621CD5FAD1 final-5af47 #71 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 2011D0C1-C4CA-49D8-9317-94E6E5F20DDE final-5af47 #71 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 7FFFE736-E808-4F59-BCA5-A72390741ACF final-5af47 #71 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 F2CDD419-09DD-4562-B1B1-7BBA6B04812E final-5af47 #71 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 47D05F47-B4BC-419E-B49A-D3DE5A8163F1 final-5af47 #71 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 FD4AD1BB-BAB8-4609-A7FB-B5BC669732AF final-5af47 #71 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 809788EF-9870-4D78-84A9-636C8BC28B7D final-5af47 #71 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 D8A27CC8-4BC9-447A-8911-EEE295A55540 final-5af47 #72 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 E68D326C-FB33-40FE-82A4-01E284C3E515 final-5af47 #72 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 243754C5-8E09-4201-88FC-FD948D568DA3 final-5af47 #72 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 1713B842-4D38-4952-8E69-B680D71CB3CE final-5af47 #72 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 72ED04FA-AD33-46A5-BEC9-1F2144502658 final-5af47 #72 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 4EB2E294-4C88-4404-A1A3-A9EE67E74302 final-5af47 #72 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 99F0AB53-08FD-4A07-BBB0-BF9C49DC6C05 final-5af47 #72 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 A5AB3B26-DAB0-4249-8EB3-DC63FDAFE6B9 final-5af47 #72 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 26F2B6B0-342D-41CA-8617-2CEB2D28BB3E final-5af47 #72 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 D7184974-9902-4847-8FF0-525E99038C81 final-5af47 #72 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 7CD565E8-ADCE-424C-A9A0-415E03653C6E final-5af47 #72 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 9BE71E41-E45D-4D1E-819B-E734A081FCDD final-5af47 #72 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 7C658E4F-450D-44B9-A9A0-9F249D7B014E final-5af47 #72 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 0A5822DF-73CA-4012-A678-156847635FAB final-5af47 #72 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 2F0C462D-3FA8-48CD-BDD8-64BC4E2F56C9 final-5af47 #72 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 9F47E40E-00AE-4C93-9AAB-E35F3BCBCE5F final-5af47 #72 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 D524B4D8-D35E-44E5-B8FE-01C2E7C5BD7E final-5af47 #72 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 E5D27E4F-1B46-461A-AC34-57ABEB17BB0A final-5af47 #72 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 E10FF6C9-DEE9-4500-9C30-05E8CA68A8ED final-5af47 #73 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 72A1E00C-C7AE-49F2-B6FA-4CE3FD69A587 final-5af47 #73 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 8B8F25A7-BBF6-4BE6-B2B5-2C1F253E9FE3 final-5af47 #73 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 EB27BDDB-D690-4439-ABD0-6A72937EC492 final-5af47 #73 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 571FFF8A-7FEE-4388-9A89-A8FF6A122452 final-5af47 #73 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 997DFCA5-6479-437B-B344-AC7BA654372C final-5af47 #73 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 7B243398-07AB-4A57-96C3-113806DC519E final-5af47 #73 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 ACAA57BF-A124-464C-AB2B-E721AD03F6BA final-5af47 #73 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 1672C5BA-FA4A-409A-9FEF-B1939234EF3D final-5af47 #73 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 59EC9E4E-047E-4F28-BAFB-15630CC6616D final-5af47 #73 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 74E6F6D9-FAC2-4A18-B64B-993A54E95D7D final-5af47 #73 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 CC9A7E27-5CA5-4946-8AB5-960BE616CECC final-5af47 #73 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 FB89F477-6F03-4A31-A639-AA07774A63CD final-5af47 #73 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 09164FCD-0CE0-478D-A29A-66BA6560D821 final-5af47 #73 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 5FBD2AA7-0958-4655-953C-6236562DA197 final-5af47 #73 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 268310F3-1AB7-4300-9C24-A496516460DC final-5af47 #73 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 7E1DE422-206C-40BF-AB53-947D2A4D8FD4 final-5af47 #73 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 81AB9024-BC01-40CF-A945-EABA2B12967B final-5af47 #73 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 FEF6343D-E527-49E8-B403-2D10F83E09A8 final-5af47 #74 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 A2A7B408-DEAE-4FBD-8C7C-B7A17EB647A3 final-5af47 #74 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 4CEA47D7-BEAE-4978-8C04-79BFE3823ACA final-5af47 #74 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 3570ADE1-1076-43A8-B732-0EB709C396AA final-5af47 #74 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 F3418850-7471-464C-A8D4-F61E3A2F00B4 final-5af47 #74 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 3366B9B4-B347-4698-A239-AF32CB07DA53 final-5af47 #74 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 5E18813D-BF76-43D4-BFBB-DC7216F313FD final-5af47 #74 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 3F28A55E-64D9-4408-B6CE-698E0F49E414 final-5af47 #74 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 AB8C90F8-04A0-43B6-BDBE-2CBB463F4AD1 final-5af47 #74 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 BC9D99D6-F1D6-46A8-8602-4408D1CB21A6 final-5af47 #74 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 22C33D94-EBFC-48A4-A6D6-3094313BFBDF final-5af47 #74 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 58596394-48AD-4C3A-BF67-32B64AC36276 final-5af47 #74 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 C3192149-B85B-47A4-8613-B437677EE2BE final-5af47 #74 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 440F5F5A-4DA1-47DF-BA15-CC7619959559 final-5af47 #74 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 F1E0813C-5A3D-49AD-A1C2-3F37CACE5F46 final-5af47 #74 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 C0E53728-E51D-4DF5-93CF-52D3CC106373 final-5af47 #74 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 8F9408EB-6D58-4092-BB91-318E6B6C6E47 final-5af47 #74 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 C435D2BB-53DE-4B44-A0A9-978CD45AAD54 final-5af47 #74 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 4A9F3DC3-C135-4275-B55A-15E371FC55BA final-5af47 #75 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 6C91589A-ED1F-411F-8D03-4F72D34C0276 final-5af47 #75 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 6F6A2E13-D27B-438B-AF8A-E52E7A0F3877 final-5af47 #75 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 DD16AA5F-8857-4465-8C13-73C4777EBDD3 final-5af47 #75 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 18C88130-EE46-4BB8-805F-BF11F3A1A6DB final-5af47 #75 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 D61F7FFE-9AC3-49B8-B2C9-4F70506137C0 final-5af47 #75 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 8838B8E6-CB02-4044-B79F-946A88688C45 final-5af47 #75 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 4BD8537B-4541-4F8F-8A9D-C31EA15DAE92 final-5af47 #75 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 02862A1B-1EE8-4E59-B5CA-E3C681052D61 final-5af47 #75 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 670A50A5-05C0-4E3F-A6B0-7169A011AC5B final-5af47 #75 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 7721F430-C337-4307-9E6E-FDCBBA0879DC final-5af47 #75 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 6B0B9F59-F806-483E-8037-A0E1ECDDEB54 final-5af47 #75 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 D65567C1-653E-495F-BC0E-FD7A72C27633 final-5af47 #75 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 5B7465AE-E00C-451A-9860-ED236C5505A5 final-5af47 #75 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 B389BF38-9EE6-4F8E-B049-17F4D4C8425D final-5af47 #75 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 0A0DC104-0472-46F0-8F0D-23DD8996AA04 final-5af47 #75 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 D9F74A3E-A21C-4E51-98AC-0570AF3008B4 final-5af47 #75 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 34F43112-753D-4ED2-8C9D-4F3C9DF5BDCF final-5af47 #75 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 7E24AB0D-FA85-47D7-8DF3-EF447F7BD4D2 final-5af47 #76 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 9FCD0C23-D4CE-4BD5-8F80-0C4342D38602 final-5af47 #76 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 B7FE7F8A-EF77-4544-A5D0-D8EB770D7677 final-5af47 #76 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 DA085D18-ACE7-4E20-A930-ABA6A2DF85B6 final-5af47 #76 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 80D4C285-71BA-47FC-BF5B-5A6626624E09 final-5af47 #76 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 7CE2D035-A171-44D3-BD7B-14FAB948F37B final-5af47 #76 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 E1B5D19D-76B1-450C-B60A-DFC9B90B3307 final-5af47 #76 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 8EADBFF6-24FC-4310-BBC7-17AB490C6605 final-5af47 #76 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 66179495-2A11-49C2-9564-900494A6F2DE final-5af47 #76 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 F92E0D11-AAD1-453F-B43E-4665ED463652 final-5af47 #76 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 56220A3D-A94A-4BBD-9ECB-E0BD34020A89 final-5af47 #76 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 37FDE65A-2C9F-4344-A8AE-394C46E488F1 final-5af47 #76 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 08E6C05F-9255-4212-8595-79D0EFD745FF final-5af47 #76 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 3198D6F3-1439-4474-B227-D9300DEE305F final-5af47 #76 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 14F0D031-5FFE-4C68-8B25-C09B3BE9CE07 final-5af47 #76 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 49903E68-688A-4E9D-B0CE-B3A77E1660F2 final-5af47 #76 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 1C2461EB-C07D-49B7-B787-C962CEB0E3BB final-5af47 #76 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 10FFD68F-AB73-43F1-B237-9894B303050A final-5af47 #76 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 B47769F7-66BE-4A6B-8776-D1FDB6BD6BCE final-5af47 #77 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 95F29DDF-0E75-4FA0-8D4F-64579EF45FED final-5af47 #77 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 F0725A69-57FE-4317-902B-D63034C7BD97 final-5af47 #77 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 FE9279BA-1BEC-41B1-8784-AE32C825120D final-5af47 #77 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 818D31CA-BA12-4578-A28A-A0E5E8EBF1AD final-5af47 #77 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 B99CA6F3-0B7B-4E09-865C-ED9CC5D022E5 final-5af47 #77 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 4A974F5A-7627-4F1F-9957-0ABABC9D2684 final-5af47 #77 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 87672DE7-191A-491D-9163-49D452E1CA90 final-5af47 #77 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 9D034C2F-46B5-44D5-9A4E-B3529205F838 final-5af47 #77 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 DB2FCC61-1AE4-4D0F-83EE-3EF9935219D1 final-5af47 #77 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 B2CF3E77-A5B5-4757-9684-BE9FA641A643 final-5af47 #77 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 8708315A-41D7-4BCB-85B9-BFF3BD2A7928 final-5af47 #77 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 C8D7C1AC-D2FC-4FFB-84B9-56E6E288ED42 final-5af47 #77 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 CE67B287-80A1-4BE5-8C9E-F59DF70E6067 final-5af47 #77 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 EC7F7238-1FEF-493B-8F09-19C1587EF06E final-5af47 #77 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 2CB95DAB-57CB-467E-B37F-914C00416CE4 final-5af47 #77 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 2A1D29DA-D702-49E0-97E5-92F364BB4473 final-5af47 #77 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 CBA87228-4B1F-48A8-912F-8CF8FB63C1E0 final-5af47 #77 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 D3E3B494-2B56-4C35-A99F-2D481F790356 final-5af47 #78 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 001A61F9-8732-4512-A1A9-F90A55F33A87 final-5af47 #78 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 0861430C-45A9-405F-B0F1-DE19F6E39603 final-5af47 #78 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 D3F3DE58-290D-4613-92FA-AD5773D87C5A final-5af47 #78 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 F712E122-7297-4359-8CAB-BC626E576CA1 final-5af47 #78 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 69B0E908-DD4C-472F-B2AE-F587F363B464 final-5af47 #78 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 25DFE77B-8178-4BCE-A4BB-04005DAB90C2 final-5af47 #78 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 212988EF-271D-43BF-91E3-25D906E4B814 final-5af47 #78 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 89D4FC7F-E84B-4386-9838-DFCCE0C7B148 final-5af47 #78 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 CD677505-64BF-45F1-BD69-59B77C1C8542 final-5af47 #78 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 F95986E7-8538-42D9-BF40-9115F0A4FABF final-5af47 #78 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 3ADE74D9-49E6-4CE9-B90B-5325B3ADC904 final-5af47 #78 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 8BB11099-4CB7-41BF-8128-A6F5CA28A362 final-5af47 #78 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 B261A601-077F-4CC1-9565-A80326A4F053 final-5af47 #78 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 BCA85D49-C369-4EA6-8769-2449B9FB6B6A final-5af47 #78 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 7B75B4BF-3AA9-4792-BE6A-D4DB70DE6538 final-5af47 #78 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 212AE275-3ADE-4F84-8D8F-6CED6062F466 final-5af47 #78 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 CDB7EB1A-8589-479C-B697-A2CC05314A33 final-5af47 #78 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 F87D6F64-BF77-4FC1-86E9-2C35D9FD8C76 final-5af47 #79 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 5861D5B2-376D-491F-8DE9-09C146164422 final-5af47 #79 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 D15AC74D-8048-4012-BF32-16576A6AC62A final-5af47 #79 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 482EF9C0-EC97-4715-AAE2-CCE3A28D81CB final-5af47 #79 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 D628A741-F491-4F06-831B-F3E8415A29D9 final-5af47 #79 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 19FAD66C-B182-45DA-A629-27FD72A97F5A final-5af47 #79 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 2B1BCED0-FB9C-42B1-B164-5A7FF642EA78 final-5af47 #79 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 0CD9A9BD-D549-4DEF-AA72-4DCEDCD33208 final-5af47 #79 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 461A978B-1ADB-4E2E-A997-5364C9E80A6C final-5af47 #79 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 313DAF6B-CA46-4FA2-8855-F5040F88F229 final-5af47 #79 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 DC3F875A-F580-4F4C-8ADE-1EDD260B95A4 final-5af47 #79 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 9FF6630B-D1AD-4436-8FF2-51C4CE5791FD final-5af47 #79 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 C022B545-1AE7-464E-B30A-7F25587748E9 final-5af47 #79 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 BF0C7DC7-B530-41D1-929C-127CE8EFDC7B final-5af47 #79 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 55A0EFEC-56C2-4AD4-AA52-E835828876C2 final-5af47 #79 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 84C7AA6B-AB4E-4EB5-8EAA-0329498637AE final-5af47 #79 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 8C0104D4-6567-4C07-919F-445B9708BB03 final-5af47 #79 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 F8B89F01-BFA0-4583-9A10-C78E7F85E25F final-5af47 #79 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 ACA0503A-FCC9-4DBA-B13D-89478E58A13F final-5af47 #80 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 69B56DD6-36BB-4E24-9A5B-D674055A3E42 final-5af47 #80 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 482EE58F-8FAF-473C-A68E-E09E89F70640 final-5af47 #80 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 42927840-96A3-45CA-A624-EA61198CE435 final-5af47 #80 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 0315DAFA-0F43-440C-956E-C44DCE866FA0 final-5af47 #80 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 FB642F0C-2B9D-4EDE-A54A-7B6E1C151E4C final-5af47 #80 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 F5E15C24-CB7C-4785-A8E7-A15EF1C875FE final-5af47 #80 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 CB7CE727-C599-458B-B2C2-1B191164B12F final-5af47 #80 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 7E569CAD-A2D5-4346-88F8-24A9A8EFCBC2 final-5af47 #80 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 526602A8-5F03-4C47-8723-37208512F968 final-5af47 #80 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 36F25550-38DD-458B-8C45-E175B25C300F final-5af47 #80 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 494D51E8-A084-4876-96A3-50241ECB3022 final-5af47 #80 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 E5ED08A8-8D1B-41FD-AA6B-F827524BA6E5 final-5af47 #80 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 B7D0B563-6858-492C-9A35-4A537C42C702 final-5af47 #80 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 5C0253AF-D0F2-4545-99F0-3688219BBB2C final-5af47 #80 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 043D34CF-3ED5-4DA1-9F33-88E271C924B4 final-5af47 #80 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 FF30610E-BF5F-4042-A029-4E567159C71D final-5af47 #80 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 6A724597-6C08-4456-8C43-FF79DA296B3D final-5af47 #80 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 FDF31F0E-BFF1-43EA-96EE-F146E75222DA final-5af47 #81 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 EA11835A-976F-48DC-A6C2-30AF79FF04D9 final-5af47 #81 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 8BA7733C-7CEA-48AB-ABE8-366073EB484E final-5af47 #81 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 4FC96104-E74C-4DF5-ACB5-6C701D871EB2 final-5af47 #81 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 6EE6CCB3-0E8F-4DCB-A902-DC6E5A35EBE9 final-5af47 #81 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 6DA8FD65-2765-4CB8-90B4-4012C1E08AD4 final-5af47 #81 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 D7A96D4B-A9FD-4CCF-93BC-1F7FD13BBDB3 final-5af47 #81 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 CB5B3ED0-E8E6-4B9C-976E-85A4FD313F1C final-5af47 #81 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 36133EDC-AEB7-4B95-B778-21C9FE677CA2 final-5af47 #81 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 52F30D14-4BC5-4D4C-9761-5A4CF16D686B final-5af47 #81 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 37A414C1-EE57-4C8D-B254-500983A46C23 final-5af47 #81 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 8D72AEA4-0724-4580-A7CF-CD116A00441B final-5af47 #81 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 E22D2057-0300-4677-9D90-830C1A9CDF27 final-5af47 #81 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 4C95E231-299A-4F76-ABA0-434F6E061A28 final-5af47 #81 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 52FC8881-F4E6-4DB9-AB65-3C18FCD42AA7 final-5af47 #81 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 CC6580C9-C90C-42A0-A88C-FAEA06A78442 final-5af47 #81 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 DC7CF219-03C4-4481-AEAA-6A7B4C0ECB97 final-5af47 #81 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 B02784CD-6C4E-4E89-8D2E-FFA9121F36E8 final-5af47 #81 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 00937CD4-44F3-436C-9AF5-A56E0CA8CD8A final-5af47 #82 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 C3129798-F49B-4E06-994C-15A78286FBE2 final-5af47 #82 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 467B66A9-AAEB-4D68-9916-2ED1159783D0 final-5af47 #82 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 139D2CC5-FFCC-4ECA-8DDD-3E4987ED4218 final-5af47 #82 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 B0F1AD0C-E124-4358-8278-8C9EDE176CBA final-5af47 #82 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 BB1449E3-D85B-450A-AF51-8AB825444864 final-5af47 #82 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 95F73B26-7E63-4D27-A16D-F6331F6CCFE0 final-5af47 #82 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 218F1B7F-48B3-4171-B971-F0B112B0FF75 final-5af47 #82 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 759C9680-16FF-4F4B-8F47-C8074935F3BC final-5af47 #82 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 75320831-0B54-4BA1-B206-E72DCD312BD6 final-5af47 #82 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 776DB2CD-B22A-46CF-B5CE-06C5947F154B final-5af47 #82 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 00C65853-27EE-4369-9E48-95A1B71ADFA9 final-5af47 #82 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 061E2BBF-8504-497F-9321-E21C52C6671B final-5af47 #82 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 6CBD9230-62D4-45A7-8110-E9D7B8CED79B final-5af47 #82 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 ABFA9145-5596-43CF-80AA-0A3459BDCF22 final-5af47 #82 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 744FBF0C-7646-43BC-9216-BD1B9E787C06 final-5af47 #82 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 B82385BD-75CA-4603-98D9-FE0C50047C5D final-5af47 #82 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 0CD8E93D-AFC9-4222-BCBE-27603CAE2840 final-5af47 #82 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 429E3801-8B0E-49C7-9868-04E5F820354A final-5af47 #83 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 BAE9A0EE-4D4E-4848-9EF2-6CC483A8DAC2 final-5af47 #83 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 770E074F-0F3A-4358-89E7-B756501C2EB5 final-5af47 #83 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 688115E6-EDD3-4EE0-99C3-B7096E35B3BF final-5af47 #83 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 0A46DF4B-0D03-4FA3-9A08-75D421AF9AA5 final-5af47 #83 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 04DE7417-573D-4513-ACAC-D1A52478E9E5 final-5af47 #83 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 FF1302E5-9D7D-492F-BB60-7E1482E41414 final-5af47 #83 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 49A8EF8D-1F9C-4CE6-ACCD-96CAFE4B0FA3 final-5af47 #83 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 5B8B5602-CEB1-44BD-B731-FC637D80BF20 final-5af47 #83 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 8D1AB8A0-CB80-4323-8640-B77641660B28 final-5af47 #83 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 45BAEB27-DEE7-46FD-9161-5A8DE26388E6 final-5af47 #83 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 48ADF579-2E21-4916-8A3A-8064D5078C6C final-5af47 #83 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 186535F6-9111-4620-8AA9-BBE7EC4AEEF5 final-5af47 #83 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 F6DDF7A6-D9D0-42C1-BFC3-54559434E91F final-5af47 #83 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 4FDA8742-6C79-4193-9653-A3981F4A32C7 final-5af47 #83 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 E498F88D-0A8C-48FD-8C20-63E11C5947CA final-5af47 #83 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 7CC7C59D-30AF-4358-9636-11DA7067FC7C final-5af47 #83 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 BC7105FF-B625-44AF-AA32-A1F184CBAA2C final-5af47 #83 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 5616D48A-EB59-4184-AA85-7E04ED5BAC21 final-5af47 #84 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 63D57682-F4BF-4705-81F4-EC380F6485DB final-5af47 #84 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 91CF5EEA-2CA8-465B-A496-E53F91E42D7A final-5af47 #84 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 C74EBF28-138A-4B26-8434-A0F4E1877A5A final-5af47 #84 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 E8A69D87-35C9-479B-BF86-4CF4DD1E3B71 final-5af47 #84 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 D48B0E1C-10B2-4FB4-AFF0-E1259F3D9FCB final-5af47 #84 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 6A8D56E5-31D7-4AE3-88D8-DD0B212907BD final-5af47 #84 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 1AB87ECC-6374-4AFB-B9BB-200A4C850C00 final-5af47 #84 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 8EACD2CF-0263-47D6-B92E-1D8CCE20A546 final-5af47 #84 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 0291806A-643E-4B5B-8441-5ACD9E2BC918 final-5af47 #84 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 ED438513-600E-4F2A-B266-38024BE1C83B final-5af47 #84 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 E0100E8E-B78A-4957-A565-562913B747ED final-5af47 #84 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 74D1D79C-2BA7-4F25-9DFB-029F10CDADBD final-5af47 #84 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 21BBFDA5-C274-4CC0-A45A-A0B626516E85 final-5af47 #84 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 86E46A26-4159-4F60-8BBB-1121B82C28F2 final-5af47 #84 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 C8E39F6F-9365-4CF5-B0A7-B841B971E148 final-5af47 #84 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 B34853DE-BCE0-46D8-9C4C-1389A83B7107 final-5af47 #84 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 F12AFEF3-ABA2-46FC-8186-F29BDD32338E final-5af47 #84 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 5F86926F-3ABF-4FDC-8159-B73E7EDC5518 final-5af47 #85 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 4BDBC417-A376-4C07-83B4-BF42BD63E6DB final-5af47 #85 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 38EF2300-AE7F-47AD-92C7-B80FFD4333AC final-5af47 #85 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 438CA37C-D95A-4780-854F-E52B9D63F164 final-5af47 #85 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 AE1FD0F6-BE15-43EF-A48B-40EA45EAA667 final-5af47 #85 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 AB3ECBEB-6D88-48EC-B197-35D2C3C945AD final-5af47 #85 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 B7FB85F2-5914-45A7-9B8B-C5631BA0483A final-5af47 #85 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 F99437A8-1A97-4F79-8578-E68320FA3212 final-5af47 #85 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 8F8CE79A-AAB2-441A-8BC3-0D17D713264E final-5af47 #85 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 675B2519-0725-48E1-9490-F68B9D2DC1D7 final-5af47 #85 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 35B932C4-C9EB-48AC-BEF5-0D58F1FB2ACA final-5af47 #85 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 1C38B5E3-8144-407B-BB19-AF921D04FD4E final-5af47 #85 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 2E62B3C3-4017-4BD5-9256-6DD6AF5B33A9 final-5af47 #85 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 5BD66921-EC74-4E44-B7AD-17F35D69A221 final-5af47 #85 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 435677D6-EA84-4114-9F25-2EC21B06E652 final-5af47 #85 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 0FF8CC2E-6EF3-4156-8863-AE82E4E6AEE7 final-5af47 #85 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 BC6C2867-8B08-437E-A2D7-3D3C0ADB188D final-5af47 #85 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 ECAA795B-D121-4CDE-90B2-ED484CB1FDA2 final-5af47 #85 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 49367CB8-A32F-453B-B70C-74D56BD303BD final-5af47 #86 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 5DCA2363-B592-45F3-92A5-4D8494274595 final-5af47 #86 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 E45C7E90-12CB-430B-B79A-8CE4267A9F44 final-5af47 #86 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 31770386-D133-40EE-9D53-86A61F9E81EB final-5af47 #86 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 C8F4C0E8-A38C-4A27-8E57-B16BB1E2F0C8 final-5af47 #86 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 47083B44-95D6-4E7C-B5FA-E1399BF86EFF final-5af47 #86 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 D136C8A0-E7D4-45E7-9394-E1EC64B02CD6 final-5af47 #86 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 E0830FDA-3C1F-409D-BBC2-FE46F6A3C1EF final-5af47 #86 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 CD3DC95B-B6DD-437A-B9D1-8D0E17DF5F10 final-5af47 #86 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 BBC34B49-E179-4350-8008-BD1694D81361 final-5af47 #86 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 D792247B-26B6-4E80-A851-04A8B0E48C63 final-5af47 #86 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 0F84102B-816F-4C63-8505-B2235EF7D4B6 final-5af47 #86 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 C46D3A78-D748-4427-9037-FE3F396B9484 final-5af47 #86 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 D4C83459-4B53-474E-84B7-2A883361FABC final-5af47 #86 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 5A592CB2-B1FC-4DBB-9D84-E92F368B764B final-5af47 #86 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 85909DA8-6E35-47DB-BA16-BFA49FC1B6B4 final-5af47 #86 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 AD15FCB7-2CEF-4291-9DD2-ACB0FCBC1FA2 final-5af47 #86 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 C5FD8360-5087-49ED-AB28-B2142FCD7068 final-5af47 #86 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 D6D9B860-AEAD-479A-8860-C772FDFA7A28 final-5af47 #87 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 A327BECE-766C-4665-878B-C1C9171A7ACB final-5af47 #87 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 33F3D7EB-E637-4426-B1A7-1481D17A08D1 final-5af47 #87 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 1CCC3C1E-F064-4D2E-8B05-72A9DFDEE73D final-5af47 #87 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 B480A0D3-CC8D-422C-9F20-5A86CC87F3E9 final-5af47 #87 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 521C5E1D-ACC8-47B4-AB1D-45D8B240CC5A final-5af47 #87 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 5B703CCC-17F1-4E85-8A8D-2706FB2A72B9 final-5af47 #87 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 352B65D1-D991-48A6-A53A-FA11B8B165F2 final-5af47 #87 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 6DD6A2C3-E13F-4B48-A225-6B66EBE04896 final-5af47 #87 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 EBC09791-15C9-4848-9BA2-0C4BA38B10F9 final-5af47 #87 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 F9D164A2-C24D-4BA2-9EF3-8E8C031992FC final-5af47 #87 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 A1CA7458-ED9C-4173-AFF0-0592C30CCF2F final-5af47 #87 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 9E0F74E0-15B5-4374-AD28-CA986179E05E final-5af47 #87 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 954DFFAC-8F2D-4584-8F82-5F2148AC433E final-5af47 #87 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 845660DD-0B1C-48A4-91F8-0913710016A1 final-5af47 #87 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 7B79862B-22DB-4365-BE4C-4874A9C6E79C final-5af47 #87 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 FF34A32E-50BE-4DEC-9011-C691F6947511 final-5af47 #87 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 14F0F279-AB45-42F9-A2AE-48C829AB00DD final-5af47 #87 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 E25A904D-38E6-4A97-B271-4E8900D52C63 final-5af47 #88 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 65EF7CD5-29B8-4889-91FC-CE75C13ACAB3 final-5af47 #88 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 C1ED1121-2B1C-4B9F-805E-E26FD83A11AD final-5af47 #88 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 3CE4AAA8-AC23-4B97-8455-A34F4426D3E2 final-5af47 #88 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 E472F0F3-C626-4792-8B6E-3BDB7F7D42CE final-5af47 #88 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 1F37B9D0-27BC-48D9-840A-272B88BA3284 final-5af47 #88 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 0FB7976F-4D94-43FC-BCF5-0C451824E65F final-5af47 #88 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 1CDF32E8-4565-4B13-9E44-C48C6A38A0F5 final-5af47 #88 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 0BB2FC4C-933E-4F7F-84D5-B9AC262AB987 final-5af47 #88 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 219DE84A-93D2-4B71-BDC6-995A97172B80 final-5af47 #88 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 187E057C-4E6E-4A6D-B19E-14944EA17A2C final-5af47 #88 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 E1F86E89-3BA4-48D5-A807-769B87D6A65F final-5af47 #88 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 0BF8CBF7-F7F4-4A30-9E3B-38846D0B3200 final-5af47 #88 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 AEFE7FDC-7C77-44CF-ACA3-BB74C6221FE2 final-5af47 #88 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 E9374650-00F8-4886-9455-04F18BA1E43A final-5af47 #88 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 398E80A0-A3F4-44FF-BCFF-67AD40B28618 final-5af47 #88 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 F222C32C-1F94-4387-88AD-EC6BD3803883 final-5af47 #88 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 6061C3C8-174E-465C-88F3-A9633475A992 final-5af47 #88 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 BB9CAEF1-8CEC-46F0-A0E6-91638E268D7E final-5af47 #89 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 2EF2F73F-E824-4F4B-8EFF-4C5A375CB3FC final-5af47 #89 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 9CC957A9-4E3F-4FEC-A2CE-80605947B70D final-5af47 #89 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 1B8940D2-A4CB-43EA-AB23-286E4C255CAA final-5af47 #89 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 39BA0F35-57A8-437E-926C-244229886E6B final-5af47 #89 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 AD47368C-33AC-465B-B127-6BC522C1C616 final-5af47 #89 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 D33F2857-9745-4257-9DD4-7E0BEB7583D0 final-5af47 #89 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 A01F9596-CE93-42C8-AFD9-A5C5EBBD4479 final-5af47 #89 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 7241F151-ED33-41F7-9D44-11C6B5AF4FD4 final-5af47 #89 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 13B9053E-908C-44A6-AC50-6B2E464D10C0 final-5af47 #89 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 963916AA-D978-446F-B155-AAC24A1D8000 final-5af47 #89 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 21056CAE-7D7C-49B0-B122-5CA0879A21D0 final-5af47 #89 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 7AE37595-6E04-4DF7-B6A4-92C84393A5E2 final-5af47 #89 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 6FCCAEF1-2E16-4F97-9383-918D15344BAE final-5af47 #89 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 63A02C71-04B7-47D3-843D-0B85D88B14C5 final-5af47 #89 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 A9D572D7-4BDE-4CB6-83C4-D2F19886D75C final-5af47 #89 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 BEDAC0B1-1101-4168-A3B9-0286659C3C3D final-5af47 #89 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 2D006E40-628E-4A0E-A302-E639687FBBCF final-5af47 #89 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 12D50E71-6B00-4F78-A15F-2B39A387C7C7 final-5af47 #90 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 7FA9DA82-771C-4E70-86CB-D0A30DB2AD56 final-5af47 #90 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 58DAC46E-7BEA-4376-BBBE-5927AFEC24D9 final-5af47 #90 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 F21C6F33-AE53-46AC-B50E-A48571C4BBF8 final-5af47 #90 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 AB3CF050-0966-4AB6-B43F-5F0CA1517113 final-5af47 #90 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 4AD762C9-AE87-4B6F-877B-887D5F75EE5C final-5af47 #90 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 654C7551-16DB-4EEC-A32E-C75AB00968C0 final-5af47 #90 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 9AE16E0A-A440-442E-9915-90F669F46993 final-5af47 #90 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 87BAB4BD-63FD-4FAB-90A7-E1D24937542C final-5af47 #90 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 4E98C0B7-94C5-462B-8769-FF2298ACAEE8 final-5af47 #90 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 0F147701-7053-4788-93BE-C5C877643E38 final-5af47 #90 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 138D19A4-E8F2-4AD4-B954-141B752492DA final-5af47 #90 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 5A9F046B-226C-4F76-96C7-A454EB8BB927 final-5af47 #90 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 37DD3650-A4C8-4F9B-B9AD-61A5AA8A8CF3 final-5af47 #90 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 B5B68A1D-0A0D-4FBF-8B23-5DF569DBBC2D final-5af47 #90 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 D168DBA6-9268-406E-8E0F-A5CAD30FF843 final-5af47 #90 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 9A49043E-9E58-4EA7-8B6A-1D28F9531FA7 final-5af47 #90 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 12DD23D4-5319-44E7-937E-EFA1AEB49E42 final-5af47 #90 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 E08C94B0-E438-4FEC-8AD5-E599C472E2AD final-5af47 #91 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 80642D85-D499-4062-929C-A9F367F62F35 final-5af47 #91 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 5B6B829B-49E3-4922-8488-DDB2FA617657 final-5af47 #91 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 98837BA5-D277-419A-BA42-D547192A2AB5 final-5af47 #91 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 F0397AA1-1F00-4B5C-8432-31DFBAC7090D final-5af47 #91 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 38F34ED7-8959-4F91-94D4-415429CCE25E final-5af47 #91 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 C44B518C-8A9F-47BA-A6A2-9EAA09E164F6 final-5af47 #91 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 CC6A912D-D859-4520-9AB8-3E86583099AF final-5af47 #91 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 CC3EE55C-897E-415D-8582-E7A85E863A39 final-5af47 #91 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 00B04081-D844-4358-804E-C8BB9707AD94 final-5af47 #91 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 95B51456-E8FF-4AE6-80F8-75A3EA348163 final-5af47 #91 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 2E0F77BC-5A0B-41E3-8C03-9E973DEC72E7 final-5af47 #91 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 7DE167D9-66B0-42F4-AFA2-C5C076F5FCD2 final-5af47 #91 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 3D321E79-0548-4DDE-BD67-D453A4F24856 final-5af47 #91 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 7453C896-CD98-4696-907A-794DA48C1CA3 final-5af47 #91 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 FCCB6587-87C3-4213-8373-F2CBE197F6C4 final-5af47 #91 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 0E3D8EFD-067F-48B1-8992-5A16C4C53198 final-5af47 #91 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 00A9FDC3-6A3A-459F-9DE1-48D88918D5D0 final-5af47 #91 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 2AC0A0B6-D523-446F-AA94-4DF1981D0F4D final-5af47 #92 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 179905B9-0977-463E-A64C-4CB2FBAE321D final-5af47 #92 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 386C83C2-A40C-4227-A5CB-69E7331F5C9A final-5af47 #92 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 18D332FA-8137-4C77-B006-70438903221C final-5af47 #92 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 61548D52-0A0D-49FC-9668-23E9B7C8D54B final-5af47 #92 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 E585B478-BF37-41BF-A1E0-ED651A149A4F final-5af47 #92 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 434DCF3D-0DA7-434A-936C-B45EB2A700CF final-5af47 #92 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 BFC56C8F-7956-4B9C-978F-8CC2AC8AD8C4 final-5af47 #92 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 95BCE56F-00BB-4013-8681-F8B20F4C88FA final-5af47 #92 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 955E79BA-57B3-430E-BA97-8ED00B7C473B final-5af47 #92 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 BAD3D15C-3BD9-4296-988C-DA5B374CAEAD final-5af47 #92 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 6933511B-5C90-4E90-9FBF-598D21EDF2CE final-5af47 #92 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 F23C90C1-5BBB-47FA-8E5C-3170CABB1510 final-5af47 #92 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 2D6259AA-07AD-48AC-862F-7C7FB18B078B final-5af47 #92 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 23291730-2388-4690-8529-CAE572C193B2 final-5af47 #92 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 04823BB8-5E0A-4979-A610-A2C65F9DC182 final-5af47 #92 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 8CDA678D-423C-4902-91F8-6C7369FE1E55 final-5af47 #92 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 6F835E58-B233-4AB4-8503-6801962EE5F2 final-5af47 #92 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 5D0190DD-E8E5-41BC-9AE2-E76B5CFE4033 final-5af47 #93 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 503E7EE2-810D-4BA2-ACF1-3EB1A4A75C60 final-5af47 #93 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 D7A1F79A-2BB7-47ED-99BF-804FA47D1F1F final-5af47 #93 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 D306B5F5-8B23-4649-8433-B219C522173F final-5af47 #93 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 451E5511-89D5-40B1-8227-62E08C8B4272 final-5af47 #93 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 915D25A6-36D6-40F5-9C93-0B8443306F94 final-5af47 #93 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 00C08924-E36B-468E-B7BB-5AB9C932C42C final-5af47 #93 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 FCA3A82A-7A53-4703-BA84-1F278651B33F final-5af47 #93 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 F81D08A2-B378-422E-92EC-B1F3567DBFAC final-5af47 #93 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 577ACB69-35BE-4EC6-883B-A88BA835C718 final-5af47 #93 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 ECCA2557-18D3-470C-80B7-68AB0B01540F final-5af47 #93 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 3DB98837-17F2-4442-A25A-CAC840A63A8E final-5af47 #93 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 31E47569-46B0-451B-A3A0-85AE05FAA0CA final-5af47 #93 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 53434ECD-A274-478C-BD3D-DFBD099EF250 final-5af47 #93 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 2FBBEB27-C3CB-42BF-90FB-0DFA579DAB5C final-5af47 #93 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 E896C828-65CB-40A9-9D26-2683686DD929 final-5af47 #93 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 E0E59A61-A90D-4994-8B5C-6277941BB4D8 final-5af47 #93 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 525E3500-9FD5-40E9-967F-76749FCADDBB final-5af47 #93 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 1307B366-611B-4642-BB59-65FC51EB0CB8 final-5af47 #94 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 E192E509-5258-4BE2-BC75-303351A6A1F9 final-5af47 #94 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 DBEA51FD-ADC2-4295-85D6-C0047C1589D1 final-5af47 #94 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 E2F78402-EF6D-4A95-9E26-4C4F7B8A3750 final-5af47 #94 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 0CFF70F9-17D8-48B7-A3A5-70D1ECF07574 final-5af47 #94 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 CFAC0EC2-6D35-4C63-BFE8-B9FB36E3122E final-5af47 #94 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 8D4DC3C9-F952-480E-A73E-3FC9B68D317B final-5af47 #94 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 9FB194C8-9C5E-4807-A134-B8BE0917FF94 final-5af47 #94 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 3C9E54CE-2AD2-4651-83EE-4B1CFAC265B1 final-5af47 #94 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 77F1B1A6-DC6D-4C8A-834B-23145BAC5D01 final-5af47 #94 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 736871AA-331A-4D10-9BE0-2C1655E38B7D final-5af47 #94 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 B20D0CB0-6DB2-420F-A429-7BA7A6067C68 final-5af47 #94 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 D09D6B5F-D6FC-48C2-8105-DA5C8F65DC13 final-5af47 #94 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 573BF94D-4022-40C7-B9FF-B1C5B6E20787 final-5af47 #94 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 08E71DF8-14DB-4F43-B6D7-419299ED2B47 final-5af47 #94 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 4D6AAB1D-124E-48BE-8EF3-468985273D05 final-5af47 #94 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 466E0160-C333-4D13-A027-06C98A32B1C9 final-5af47 #94 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 C019F7DD-761B-47C6-8FC1-F7EC11B5AD26 final-5af47 #94 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 D4D67432-7103-488D-8AC8-562C7F5B82E1 final-5af47 #95 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 E8DA6BD3-0018-4B1E-8D37-FDF7EFE258AB final-5af47 #95 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 A5A7EC69-DCF4-4D52-8AC0-F7E5A931B181 final-5af47 #95 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 037BA247-5099-42B7-8C09-8A8A8E3DF5F5 final-5af47 #95 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 614AEB16-86E1-437F-9160-67BB031DCBA6 final-5af47 #95 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 2743F6EC-A1F6-4893-A77D-91047BB2B297 final-5af47 #95 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 CBD1F22A-F0A0-4007-A04E-AA59CD11FBFD final-5af47 #95 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 02FF52BA-7B77-48F5-A50A-3E008E44763F final-5af47 #95 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 7512EE1E-7BCE-4284-9FE9-85EF24EC2EE6 final-5af47 #95 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 C2A692ED-9CD8-410C-9816-9AF89EDF8D0B final-5af47 #95 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 166F6A20-D53F-4B70-8BA2-CAC6152FBAA5 final-5af47 #95 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 BCEA30BD-7BA6-4AEA-BDDB-68D12355FD25 final-5af47 #95 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 AFC74E24-786B-44BC-8AE7-962106064459 final-5af47 #95 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 5CAD17AE-5CEB-40DE-B5BE-F3ED7AC2CA1B final-5af47 #95 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 E1EA1650-BC97-4B9B-BA29-AFE801CA86A2 final-5af47 #95 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 20D43155-0B09-4E19-B113-E8AFC6648200 final-5af47 #95 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 9BB0F34E-25B4-4D74-B04F-93D02B7BAE00 final-5af47 #95 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 8D74439A-570C-4E3B-A553-A24D43365AD0 final-5af47 #95 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 A71C8F52-C608-4A22-8995-07BC77377814 final-5af47 #96 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 FC28B405-9706-40F4-A719-EA73F9F7ABA1 final-5af47 #96 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 29AB4654-4596-4A55-8ADF-F3007E46EF31 final-5af47 #96 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 78C13ADE-DD1F-4933-B30E-F36448429AF5 final-5af47 #96 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 11C6AC16-EA8D-4E5A-BD6A-77EECE540BA8 final-5af47 #96 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 20C654F3-6187-4228-B581-15381091F0A7 final-5af47 #96 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 769D3F81-3278-4810-BAC3-A2E6F7DF77D7 final-5af47 #96 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 A33064BF-6B05-4EE0-8E82-0D22E66EDEC9 final-5af47 #96 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 9E5F64BD-3F89-4501-9D1C-CE8DF04C145F final-5af47 #96 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 41FB519D-24C1-44B2-952F-4B9E46CC23AC final-5af47 #96 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 833E86DE-6A72-42D8-984D-C560396FFC4E final-5af47 #96 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 6CEFA1D8-E788-4AF5-9FD8-23DDC5E6F856 final-5af47 #96 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 4B67678C-BE93-4E23-A737-C598EA140A59 final-5af47 #96 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 3655A525-2D7A-43C4-B635-640951ACCE2D final-5af47 #96 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 EEC7F4DF-AB13-414C-9B7C-508939A2C680 final-5af47 #96 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 5DB31921-E3C6-4ADC-B365-EB4E7BE3D5F0 final-5af47 #96 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 7DEEFFEB-FE2F-4558-9112-B143732FD433 final-5af47 #96 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 5D080757-FDF4-4E79-B808-FB4AC916036B final-5af47 #96 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 449E279D-3384-459B-A5E3-3CD186BE4C3D final-5af47 #97 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 94891EE3-958E-478A-82C9-5F1015A8957C final-5af47 #97 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 4AE87C68-E129-40FB-A5E6-0001C98CB193 final-5af47 #97 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 E7EB66B3-3A37-467B-AA20-3D7AB4B173A2 final-5af47 #97 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 A4B1A8B3-2A87-4752-BF0A-A4BA30200231 final-5af47 #97 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 8FA1FF7C-FD58-4185-9A8D-698B344FBBC3 final-5af47 #97 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 E0E7FEC1-0D70-4371-AB4E-87AC5BE88C33 final-5af47 #97 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 D80DF47D-90D9-4FC2-B1E6-988E6D58B88A final-5af47 #97 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 39266363-E8ED-4979-BA61-EFC9705736E4 final-5af47 #97 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 A3657072-3ED3-4894-A1CA-E1B8894DE0F8 final-5af47 #97 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 FED59A8D-9386-4FB6-B8A6-C2C6D8555A4C final-5af47 #97 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 4A604EB3-D098-4D83-BF8A-35F2F184E978 final-5af47 #97 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 09AA0805-4965-492A-AF15-563D1731D8D9 final-5af47 #97 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 FF329812-76E4-40E1-8D29-CEDF52D0D1EA final-5af47 #97 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 540971BC-26D0-4F24-94B4-5B035BBC0C08 final-5af47 #97 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 E8BC9D35-4473-438C-82CA-BB064F998A6B final-5af47 #97 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 6657092C-FC15-459E-A4E3-EE4D934A8702 final-5af47 #97 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 F5585C67-B3F4-4FE3-8B69-1745D450DBAE final-5af47 #97 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 4F01BBF7-6F4F-4022-863E-E811A65097EB final-5af47 #98 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 84E1E917-A6F3-4288-B003-3C44959D21F7 final-5af47 #98 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 64A0289E-6E44-4CEE-8521-1434400ED434 final-5af47 #98 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 05C2F1D5-1EAC-49A2-B840-6812D627011F final-5af47 #98 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 7867F58B-FE75-4B57-B119-C5002A312DF0 final-5af47 #98 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 89ED9D73-6CA7-46C6-B4ED-59B8CE2E6F20 final-5af47 #98 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 22B40C49-0754-4879-AADC-A69EADCB1AA5 final-5af47 #98 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 D5DA071E-C804-4986-B80F-10873BFB5133 final-5af47 #98 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 BB056DCE-B5C8-4968-AAF6-DBDD66179A87 final-5af47 #98 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 C95AA124-83AE-4BC2-AB8B-097ED89653BB final-5af47 #98 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 B6A903A9-CA64-4B0F-A25B-F3E28B58EE6A final-5af47 #98 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 D69E68EA-9A05-418D-962A-36BDE523115D final-5af47 #98 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 962D2322-C1FB-4204-B7EC-C42864D57C2F final-5af47 #98 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 6167E1E1-E5AD-4C30-9DF4-643B1BE9737D final-5af47 #98 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 CA4AF806-B5C2-4070-BC88-F492840B8095 final-5af47 #98 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 AF35B6C2-94ED-4365-86EF-BFEB18FD0D4A final-5af47 #98 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 0C412292-DEB8-46B6-BAE1-908E7846B582 final-5af47 #98 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 A4579733-6D3E-4527-87A4-F4AFAA2F59A0 final-5af47 #98 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 901C4AD3-697C-42BC-A7FD-4002C0982764 final-5af47 #99 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 6457E534-41C8-4234-A638-7DCCB0F797D3 final-5af47 #99 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 D9F093BC-A4D8-41C7-926A-719709017897 final-5af47 #99 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 3A6E97D9-EA8C-4153-846F-DB79B3F0A716 final-5af47 #99 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 C7B16CAB-3C2D-4CB0-9AB2-EC0811F012E8 final-5af47 #99 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 976319E6-2382-47DF-BF88-B2B8EB086FC8 final-5af47 #99 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 BFCF03B3-1308-4888-B975-8B79ACEE9974 final-5af47 #99 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 A21C6D5D-4083-4A37-ADFE-31836187522D final-5af47 #99 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 AEF49A05-116B-42EE-A465-10D483A5EA06 final-5af47 #99 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 C551FAAA-DE6E-4C4A-AEE6-0E882EB08222 final-5af47 #99 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 312BDC0E-9720-4209-B4A6-93BA381C2199 final-5af47 #99 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 A8FCA027-01B8-4CA3-AD4D-753448D50D78 final-5af47 #99 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 8B9A391C-F4BC-44CF-9D1A-9D4C3E629515 final-5af47 #99 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 EAF236A1-5AEA-46D9-BEFF-3BCAE8EF75B3 final-5af47 #99 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 FC058C4B-CBB6-4A6D-8BE0-6B7C47D40C64 final-5af47 #99 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 8D884B68-9FE0-4689-AE9B-B11BCBA0267C final-5af47 #99 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 157CAD37-EBAF-496C-A0D6-B49114C89974 final-5af47 #99 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 1011C107-B60C-4A2B-9654-4039C6605286 final-5af47 #99 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 942B5872-1907-4D2B-B6CA-6AE7B691FC38 final-5af47 #100 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 C5035972-67AB-480F-B937-B67988945BCE final-5af47 #100 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 23AEB772-8BB8-45DA-A7CD-F78627ED8E1A final-5af47 #100 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 57F0E475-D45A-414B-B2E8-4A1637072EE2 final-5af47 #100 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 69C11595-017B-49E4-B582-3452C6098163 final-5af47 #100 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 4E30839B-D0D8-4D6D-BA57-CF79CD0EE319 final-5af47 #100 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 5EF29A84-D809-427F-A664-4CCFF64E8460 final-5af47 #100 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 D3E157D9-95D6-4EE4-B29F-63335BB09479 final-5af47 #100 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 ED578269-4F97-49DB-9E12-9C5246E0A94A final-5af47 #100 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 A3A77447-D4C4-4635-B275-1552F1B6CC0A final-5af47 #100 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 FA332B5D-4B46-4458-ADEE-022D902574B9 final-5af47 #100 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 F9181221-C69D-4AF0-BEDB-5C8F396FA1E8 final-5af47 #100 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 E3427671-281F-479D-88FD-6ECE20E3FCE6 final-5af47 #100 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 9CD99B02-16BD-441B-90B2-9D8B076628F5 final-5af47 #100 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 4AD50323-CF23-44DC-90B9-8F68F05AE202 final-5af47 #100 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 FCC6D523-F9F1-465E-A581-926AD4D143D0 final-5af47 #100 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 EEF5B76B-A005-43BF-8E41-86EB75C440E1 final-5af47 #100 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 553C3410-70E8-447D-8790-973A72A263D7 final-5af47 #100 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 8CA75EBF-2F96-4415-A52B-9353B2D3BBA7 final-5af47 #101 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 5D8E8761-ED88-4A8B-96B6-AC26F539FB61 final-5af47 #101 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 BA23CA9B-6593-423F-93EE-1CA7AF643174 final-5af47 #101 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 6EE449A5-48A7-4BB7-9899-127D5E2C9CA8 final-5af47 #101 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 7241E6AC-8FA5-448A-9F98-7B700CDF5996 final-5af47 #101 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 8398952C-F5AD-401F-85DB-AE72FCDB5ECE final-5af47 #101 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 FCA5A816-4F85-4901-BD04-64B21F942505 final-5af47 #101 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 7C7C335A-A9AF-4040-852F-B02482B66111 final-5af47 #101 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 77E0934C-F74F-493F-85D7-FA42092305F1 final-5af47 #101 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 38DD83EF-983F-4E77-93B4-278D05F374E4 final-5af47 #101 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 6BD98F92-9114-4464-AB8F-C20CEDB0F89F final-5af47 #101 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 39E5BC2D-5218-43F5-9FB8-74AC40879722 final-5af47 #101 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 D575A151-B095-48B7-A3CD-53CC28602D6C final-5af47 #101 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 63B7F2DC-7EB5-4AA6-8CD0-E756BFBF0197 final-5af47 #101 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 82E3F0C4-4C58-46EE-AE87-37FA53C89E55 final-5af47 #101 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 AFB5F012-A1BB-4C41-B0AE-55CD4272262B final-5af47 #101 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 A364F780-1727-4692-BDE0-77B50AB8EECC final-5af47 #101 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 2F6D08B5-68D1-42B2-AAFE-ACD9E4510C2E final-5af47 #101 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 95469E61-9206-42D7-B14C-CD88273D34F1 final-5af47 #102 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 6846C9AF-F2ED-4272-831E-EF5695B50D0F final-5af47 #102 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 609D6C56-6F69-4DC8-B264-148CBCB4CED5 final-5af47 #102 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 0A34F628-CB65-4EA2-ADC3-B25A41D3205D final-5af47 #102 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 546A5B94-E9A5-491C-9809-C999D4202F66 final-5af47 #102 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 10285173-33B9-4B4F-B65A-E1BF2755D1A4 final-5af47 #102 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 3D22D7F7-E9A5-4CCF-AA4E-14A3882CF635 final-5af47 #102 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 54D1D559-6B61-4230-8447-BBE043DAD5AD final-5af47 #102 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 3483690D-F4FB-4EBE-9EDE-337C5FA9D8B0 final-5af47 #102 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 15B17337-E01B-4588-B2B9-161991C1C2F8 final-5af47 #102 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 16B35301-FED1-470D-8B5E-3E9FE0FBF3C8 final-5af47 #102 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 284D38AB-3BD0-4D45-B2A1-63183F1225E8 final-5af47 #102 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 9F023A3B-E94E-4267-9737-AE8FD688042D final-5af47 #102 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 56995509-3F54-4EF2-8050-405DCD43CD9F final-5af47 #102 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 380A5257-4BC9-4A12-B1BD-30646364EA38 final-5af47 #102 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 B04CD326-A0D1-4EB4-B267-4023ED61A1A0 final-5af47 #102 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 D9700738-D3CD-460B-B573-ED4B2758B48F final-5af47 #102 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 FF304333-85F6-42E5-9593-B8B0001B45A5 final-5af47 #102 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 0CA05837-7F17-4637-8EB6-3799B9A2F0A8 final-5af47 #103 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 B3E6AB3A-CF3A-43D0-ABF5-92D74E3D995D final-5af47 #103 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 4BD45637-37FC-4EEC-961F-292D16A13AAD final-5af47 #103 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 1C99A053-52CA-4E68-A633-527E28817F82 final-5af47 #103 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 37645BD3-11C5-418F-B905-954CAB18540F final-5af47 #103 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 8735A512-2613-4B03-9940-A8C4AC0D532B final-5af47 #103 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 C4EAC705-03B3-4656-8371-C7479B001CA4 final-5af47 #103 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 644F7CD2-1E1C-4DEC-9BB0-766A8926F5A1 final-5af47 #103 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 FB9BF78D-D49A-4AF3-AAD9-30E855A77800 final-5af47 #103 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 896494D1-6753-445C-BCA5-430CB00BEA06 final-5af47 #103 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 363AC114-58B3-44EF-BA0D-0AB0BBD5C811 final-5af47 #103 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 FA98DF1D-756C-46D0-B011-F9034B8B9B56 final-5af47 #103 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 BFB997B8-E8C6-449D-8B12-E42441D8371D final-5af47 #103 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 2A08A232-3C45-4B8D-9E46-1BC405149687 final-5af47 #103 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 D77313F9-1558-49BF-9947-4B8636CB5815 final-5af47 #103 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 39218522-75A8-4D4F-8873-91B279873B01 final-5af47 #103 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 EB80FF18-7DAE-49DD-AD3E-E9BE7DABBBFD final-5af47 #103 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 33E3FE2B-C927-4CF5-A21D-D92B818971DD final-5af47 #103 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 F1AE285C-3709-4167-BCBF-D15ACCE32AB7 final-5af47 #104 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 36E3B877-757E-43E8-A55F-624F5B138916 final-5af47 #104 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 D9CB7CBD-E437-4331-A2D3-4D12EBBC64A4 final-5af47 #104 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 BDA87505-0DB0-4005-9811-954C9EC7C4B0 final-5af47 #104 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 D5850143-17E2-480D-95A7-5D9614EDFF25 final-5af47 #104 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 B13F15FB-6F96-4DF5-9DC2-2B8EC14855FF final-5af47 #104 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 80030085-E013-423C-9366-8B97EFE4EDAF final-5af47 #104 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 21A7132F-1495-4BED-A202-9F1B3D6F8986 final-5af47 #104 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 CCC400A7-0E39-4C6E-8DD7-BC2106E79D44 final-5af47 #104 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 6608B54C-9947-46E9-A1A8-3195196B9F2E final-5af47 #104 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 52D1D1F5-2675-41C6-BD30-5E09DFACCF1B final-5af47 #104 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 E219E180-ECCC-43FF-99AC-EF6C3DFADF13 final-5af47 #104 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 5C12F570-B33D-411E-A935-5B7E1A465B6C final-5af47 #104 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 85F647AB-87F0-4759-AFFB-E6E026785B25 final-5af47 #104 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 58D8354E-A4C4-43B6-B6F5-5BDE48CCD604 final-5af47 #104 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 3CCF1271-6C44-4434-97B6-716AAD270202 final-5af47 #104 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 23A9747C-45EA-4209-9A41-C02462428E13 final-5af47 #104 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 590624EF-12AE-439F-A064-B7CE57E03292 final-5af47 #104 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 301053C5-D20C-4CA6-BD38-9526AD61835A final-5af47 #105 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 F6D3776B-88FB-478E-988F-DD128702DE16 final-5af47 #105 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 B49ACC12-896B-4240-B31B-B7D8421A98F8 final-5af47 #105 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 6AFA8ADA-50D3-4D83-9F7E-793BD4C529CB final-5af47 #105 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 0F9875E5-7E77-4DEE-850F-2CD71F20F739 final-5af47 #105 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 1DDBDE38-00BF-4AD4-8B2A-57E86C95DEC9 final-5af47 #105 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 629140E9-5BA1-4B00-93E4-9593F1105C50 final-5af47 #105 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 B674C7C7-6E52-47E8-B627-3D9A20D9BCE7 final-5af47 #105 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 DA013427-F3ED-43E6-8C67-4FF2C8FF3B75 final-5af47 #105 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 426FEA1E-34DB-4F12-9199-E9222F97A284 final-5af47 #105 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 61491120-6E8B-4F0E-A625-012B7FEEEADD final-5af47 #105 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 AA621DC2-D72C-4333-B155-B9517D64387F final-5af47 #105 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 D4CAB0C3-0B71-4754-85E8-4EC3B18F8831 final-5af47 #105 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 8FD94557-3D6C-4070-A6FB-642EE4764A22 final-5af47 #105 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 232210F9-E387-4F17-B0CF-6BAAC2A42C37 final-5af47 #105 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 409086DF-9E01-4A87-B6E7-9BD56618DF8C final-5af47 #105 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 E4B4FCF2-F847-49AF-BBF8-9D313A9EE005 final-5af47 #105 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 A26E5B7B-A822-40F6-ADAE-1CE487486760 final-5af47 #105 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 7FD718F4-54F9-46FF-AD1A-1E37FD6113B1 final-5af47 #106 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 7BCDE43F-5953-49D1-A999-0020B48EE437 final-5af47 #106 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 EAD2BD7F-489E-470F-A756-63AAEDCA76CC final-5af47 #106 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 F596B13F-4E21-48D1-970D-7A8F47D81273 final-5af47 #106 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 D01AFA7E-D109-4985-9313-D311DD4E8E2E final-5af47 #106 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 6DCD4F99-4BA1-4ACD-884E-BA0243156C1C final-5af47 #106 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 CA67CD16-D82C-456D-AF83-2DAE6A60AE88 final-5af47 #106 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 3DF1865D-7808-4E57-9E5C-1FF62D97D57E final-5af47 #106 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 87993DE7-2B84-46E2-AB9F-EB1C13627A43 final-5af47 #106 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 392DDC0E-CE5A-46FB-BC7D-FC4C257A4F70 final-5af47 #106 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 257DEDA3-AD66-432E-BB55-7CA476803088 final-5af47 #106 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 E343964D-D8D5-407C-AE05-2DD9416384F9 final-5af47 #106 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 0132E46D-FD34-49DB-96DC-B812CE6BD754 final-5af47 #106 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 99A3A695-BDE9-4DB3-BA5A-512EBFAA1BE5 final-5af47 #106 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 BF0C472C-D907-4BE4-83D8-7DDC5C8BE724 final-5af47 #106 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 58F4E0EC-B8CB-42B0-80E6-73E2596B75FD final-5af47 #106 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 4AC94DDF-5570-4795-8908-79259F1001FA final-5af47 #106 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 221E448E-7C38-4710-BADE-3FE2DDC15016 final-5af47 #106 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 26B40F70-34F3-45C6-A3A0-C3AFC64BF941 final-5af47 #107 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 651505FC-5594-45BE-81C7-5AADF392EFBB final-5af47 #107 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 1D0EA926-4B95-4847-830B-57A3188ED5E6 final-5af47 #107 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 AFBDD8E1-10C9-4EE3-B35E-CECA0972A07A final-5af47 #107 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 32239344-0213-4011-B19B-6372EAF6F67C final-5af47 #107 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 792BCAAC-8E78-48A4-853D-0C5E4AEF4563 final-5af47 #107 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 241AAC08-7B04-43ED-A835-6DC415660511 final-5af47 #107 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 74B7291C-A785-41A5-93A4-11D9E12639A4 final-5af47 #107 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 65F17D85-4DA1-48A0-BDEE-7654152E17FB final-5af47 #107 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 6DF4D48A-8F79-4DC8-B72D-FE2AE9CF2DA5 final-5af47 #107 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 D6F8A2BA-4BC0-467D-BDE4-002E0FC6DE6D final-5af47 #107 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 E5F986A8-E6F1-48BC-8846-0CC6DD65332E final-5af47 #107 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 9CE6D87F-77D5-4A93-92F9-CC173DF9BAD8 final-5af47 #107 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 E434862B-252E-4FE9-8BD6-6E6314EECB33 final-5af47 #107 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 1C582566-DBEC-4864-B018-40D727AE92C8 final-5af47 #107 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 A9D2B64A-6774-4015-96E4-B29CED90D405 final-5af47 #107 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 721F25C0-C6F5-4883-8F8B-AAB3C732E62A final-5af47 #107 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 D2E8FC80-5E52-4F4A-8B80-7260EB4DF3D1 final-5af47 #107 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 C53EADF5-11B7-4557-A6E0-F0AA5CF6ACC0 final-5af47 #108 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 5A38D686-C9CB-45EF-9E41-5D162B5113DE final-5af47 #108 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 AED75203-F496-44F0-BA9B-31D57DF1B619 final-5af47 #108 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 24A70524-6624-4C67-A82C-4F266739E94F final-5af47 #108 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 2361187D-0ADF-4B5F-8E97-65181D94069A final-5af47 #108 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 70BC2625-7964-4835-83EA-6803B2A0684A final-5af47 #108 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 5F09CEA3-5D91-40B3-9B3F-9BA3953C09A9 final-5af47 #108 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 3AE51364-96E0-44EC-BE79-6C4F220DCC9B final-5af47 #108 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 F1DA4375-F2E3-4C67-A1FA-36B059DDEEDA final-5af47 #108 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 A68EEEE6-FD2A-4A5A-AD84-CF191C3CA597 final-5af47 #108 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 09C24E5E-6198-49B2-A010-79BDF3E240A6 final-5af47 #108 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 81900934-2A7A-4D2C-B64E-1B6EC2763753 final-5af47 #108 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 5FCA5C22-B291-461E-8E42-35EA70D5DE66 final-5af47 #108 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 F60312FE-F309-44B4-93AD-60D341E714F8 final-5af47 #108 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 8643C2C9-8CFF-43EC-BCE0-A6DDE6EC84F7 final-5af47 #108 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 8E4C921E-7F1E-44AD-AB9E-AF5E6C89A53B final-5af47 #108 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 5A7A95C1-1650-4059-A192-355E73D7CFE1 final-5af47 #108 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 F39B2FC5-13FD-4CF9-AA7C-268CF7207500 final-5af47 #108 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 E90C1B72-A583-40E7-9986-10388F723E7D final-5af47 #109 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 814CF8E6-4EB7-4EA1-BD96-DA51CF683641 final-5af47 #109 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 02BB010F-61E7-48D4-99E5-371668ACDC9F final-5af47 #109 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 88DB3678-00D5-4382-81FE-F341B681CD45 final-5af47 #109 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 45743E06-4CA2-4DA6-BC2F-711460AF8D6B final-5af47 #109 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 86AB7DFD-FE44-4B0D-A0EF-6EF0641F5E4C final-5af47 #109 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 9B0371E8-7B4B-43CE-AFE3-C4D3694D2531 final-5af47 #109 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 9ABE7B44-2DFC-4616-9B01-84B46765D450 final-5af47 #109 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 A7B77E42-46D8-45E3-841F-3CB8CE31BCD0 final-5af47 #109 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 56369374-A7FB-435E-85E7-03A2576FC87E final-5af47 #109 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 75F2A515-8FBE-4B6F-A0D5-4C11C9179B12 final-5af47 #109 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 21854F6E-321C-45A2-9128-3D23BD4FC40A final-5af47 #109 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 1810C1F3-93A9-44C8-B351-968B65766794 final-5af47 #109 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 4E9255C4-0892-45E2-9B7A-EBF8D1C4D78E final-5af47 #109 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 4217C1F9-B210-4B5E-A6FF-610D13F52DB0 final-5af47 #109 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 0476AB51-6283-4C34-B4B3-EF6EF3B256B7 final-5af47 #109 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 EC34CF8B-9541-4EE5-B323-34E9761A16A1 final-5af47 #109 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 18E912F7-2717-4DAF-B1B9-970DEDB4BA4F final-5af47 #109 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 B0F65C9A-D054-4261-9CA4-E37EEB76215E final-5af47 #110 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 F27CB6D9-8818-44F7-9A80-AA9767B91390 final-5af47 #110 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 C4606ACA-7711-44C5-A654-2324B7973FAB final-5af47 #110 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 EBB24047-1B8B-4A83-9957-10000DDF7D69 final-5af47 #110 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 07D3B644-C8A2-4C90-8CF7-E167EEB7DF73 final-5af47 #110 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 D9416BFF-A3CB-4ACE-B2CF-B7CF1E142AA7 final-5af47 #110 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 DD72C9A0-EFFD-4B7B-92F7-A0986141DD6E final-5af47 #110 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 5674AA58-3056-4E74-A7DD-01C14500C265 final-5af47 #110 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 95A21EEE-1268-407E-869E-AC0C2283DAED final-5af47 #110 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 319077F0-78AE-478E-9543-EB6D8EB17E53 final-5af47 #110 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 04EF7BA1-1D4A-4AD8-8C5F-F38AC573BFE1 final-5af47 #110 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 31EE17D7-FAD1-44EF-80B7-1488C03BC798 final-5af47 #110 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 3CC44C86-ACF6-4059-8961-C4CC53EF9237 final-5af47 #110 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 66D94FF5-4431-488B-B1E2-1E33A51E9110 final-5af47 #110 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 6B7F3CC6-ADEE-494A-9E3B-2367ABC50A07 final-5af47 #110 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 10FC76C8-2A90-465D-9869-B9C87ABC8211 final-5af47 #110 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 15BD5E57-EE92-4476-A5BE-90EDBC5ADAEA final-5af47 #110 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 35668C49-110A-4E0D-9D9D-44DF8FB4DEBA final-5af47 #110 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 9F957203-3ABC-4063-9D41-0CFD8E1B341A final-5af47 #111 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 4F300679-BBB0-41BB-8212-6771E29F4CF4 final-5af47 #111 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 CA83DBE0-8DB7-4D14-A59D-7D5EB8B15577 final-5af47 #111 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 934C287B-D060-4AEE-9847-2830D880E01C final-5af47 #111 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 407DACA8-0A5F-4891-BB66-206414BFD0D0 final-5af47 #111 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 18D3F24B-DFC4-4E4D-9200-E78A2A617606 final-5af47 #111 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 3EE9209F-BAB1-4D52-9CAE-81294B5FF673 final-5af47 #111 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 FEFD94A4-6B5A-4461-BCAE-AF2261E7ED61 final-5af47 #111 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 226530A1-7B40-4B73-9192-F2DAF972E715 final-5af47 #111 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 D5DC88CE-FA3D-48BB-84D8-06778789B6B9 final-5af47 #111 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 4340C870-95FD-4D48-94A0-0A9816B1D8D7 final-5af47 #111 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 CDED47ED-FF4E-46CC-A487-FA282560DA69 final-5af47 #111 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 1F7C3E42-6A46-42E2-B611-4BE326016FB1 final-5af47 #111 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 4BA9E82D-C0CA-462C-B932-980F544BE371 final-5af47 #111 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 C42FC73B-A0F9-456B-B874-CB69DDC9BA1A final-5af47 #111 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 844CF79F-F285-49E0-94F4-4AA7A5D66660 final-5af47 #111 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 ACEB86AF-8220-46A0-943A-AC6195484D7A final-5af47 #111 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 366159EA-4F94-429A-AACA-F836414EB3D3 final-5af47 #111 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 22420DE9-7D86-4392-BF57-B5AEC43A1F7D final-5af47 #112 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 4F3D0E26-66E9-4E5A-B338-FFBD9CEC1970 final-5af47 #112 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 C8669C44-16F2-4266-ADBD-DB187BE03B2B final-5af47 #112 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 A1610780-2BEE-40ED-9EB4-46AF71CE3952 final-5af47 #112 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 96F23569-D9BF-4F5D-82B1-3F5140960A29 final-5af47 #112 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 B05F9C0A-A551-4FE4-86FF-D1B144D3F381 final-5af47 #112 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 B6A702B9-6881-4E95-A19C-884B49B9CCC9 final-5af47 #112 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 4A204D5A-6F5A-424D-B14D-50677954343D final-5af47 #112 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 39613C33-77BA-4EAE-939F-D808981E1426 final-5af47 #112 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 70A1A728-BB74-4532-8F6A-928A5C2E3E90 final-5af47 #112 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 B8165D73-6BC5-4E70-A8F3-0EA038BE0BF0 final-5af47 #112 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 A6601C39-CD2A-4EDA-A081-0842C9E08C37 final-5af47 #112 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 3F4CA2BD-9EC5-42DF-8086-61C6CF046ECC final-5af47 #112 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 4E819BEA-FCDB-4B06-9485-50D64A392F2C final-5af47 #112 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 E1111244-D85C-41B6-811A-1D794B3B65C0 final-5af47 #112 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 AE21839A-0686-47AF-9717-A905617EE383 final-5af47 #112 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 D9867F8C-4CAB-4352-87F2-3E696B248ACB final-5af47 #112 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 A9E37FBA-A648-4E6B-91E8-0DD365A3F6A1 final-5af47 #112 18 of 18 NAME (PRINT): Last/Surname First/Given name STUDENT No: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA DECEMBER 2019 FINAL EXAMINATION MAT244H5F Differential Equations I Goncharuk Nataliya Duration - 3 hours Aids: 1 page(s) of double-sided Letter (8-1/2 x 11) sheet The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You are reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam. Clear, sealable, plastic bags have been provided for all electronic devices with storage, including but not limited to: cell phones, smart watches, SMART devices, tablets, laptops, and calculators. Please turn off all devices, seal them in the bag provided, and place the bag under your desk for the duration of the examination. You will not be able to touch the bag or its contents until the exam is over. If, during an exam, any of these items are found on your person or in the area of your desk other than in the clear, sealable, plastic bag, you may be charged with an academic offence. A typical penalty for an academic offence may cause you to fail the course. Please note, once this exam has begun, you CANNOT re-write it. The exam has 7 tasks. Points possible: 20+10+15+15+15+15+10=100. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 1 48898493-98A9-4A7B-AEBC-7923F56B37C9 final-5af47 #113 1 of 18 Please do not write on this page MAT244H5 December 2019 Final Examination, Goncharuk N. Page 2 9E74D264-F032-4682-B07C-060C6A860C0C final-5af47 #113 2 of 18 Task 1 (5+5+10 pts). Consider the following equation: y′ = √ y sinx. (a) Does Existence and Uniqueness theorem apply to this equation? Ex- plain why or why not. (b) Find all constant solutions y(x) = c of this equation. (c) Use separation of variables to find a non-constant solution with y(0) = 0. Provide two different solutions with y(0) = 0 using (b). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 3 1B8680BB-30F6-411E-AA8D-CAFF9B4D0DD3 final-5af47 #113 3 of 18 Additional space for Task 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 4 6DC1A534-62B5-45DD-AA73-21C5D524D3E3 final-5af47 #113 4 of 18 Task 2 (10 pts). Solve the following equation: y′ = y/x + 1. You may assume x > 0 if this simplifies your computations. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 5 C35644C5-7BD3-462B-A8B8-11D8241B2C62 final-5af47 #113 5 of 18 Task 3 (15 pts). In the following equation, choose k so that the equation is exact, and solve it for this k. Implicit solutions are accepted. (3x+ 4y)y′ + (ky + cosx) = 0 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 6 196CFD17-CF9E-463F-A06B-1EA92C068B4A final-5af47 #113 6 of 18 Task 4 (10+5 pts). (a) Solve the system of linear differential equations: x˙ = y, y˙ = 3x+ 2y with initial conditions x(0) = 1, y(0) = −1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 7 6D7B1137-8B77-405F-BF02-D11DAAFEFE01 final-5af47 #113 7 of 18 (b) Solve the system x˙ = y − 1, y˙ = 3x+ 2y − 2 with initial conditions x(0) = 1, y(0) = 0. Hint: compare with (a). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 8 88234B1A-D0C1-4FAA-9C31-6261DADD2F54 final-5af47 #113 8 of 18 Task 5 (8+5+2 pts). Consider the following nonlinear equation. x˙ = (x+ 2)y y˙ = x(y + 1) (a) Find all its critical points and linearizations at critical points. De- termine their types and stability. Sketch phase portraits near critical points (your sketches can be rough but should show stability, types of critical points, and directions of saddle separatrices). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 9 4F82A98E-0077-4EB3-9CFB-6436BADB1C99 final-5af47 #113 9 of 18 (b) Sketch the phase portrait for this equation using 5(a). x˙ = (x+ 2)y y˙ = x(y + 1) (c) Use your sketch to predict the behavior of the solution with initial condition (−4,−4) as t→∞. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 10 6131E91C-4EDA-44BC-849C-C0A225A41831 final-5af47 #113 10 of 18 Additional space for Task 5. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 11 87030F04-0131-4C43-AE12-B6C8E7466F3B final-5af47 #113 11 of 18 Task 6 (10 + 5 pts). (a) Find the solution with initial condition x(0) = (1,0,0) for the linear equation x˙ = Ax in dimension 3, where A = 2 0 10 1 −1 0 1 3 MAT244H5 December 2019 Final Examination, Goncharuk N. Page 12 C5D2AE0F-F3D5-4D51-84A8-A60CF63576B9 final-5af47 #113 12 of 18 (b) Find at least one solution of the equation x˙ = 2 0 10 1 −1 0 1 3 x+ 00 et (the matrix is the same as in (a)). MAT244H5 December 2019 Final Examination, Goncharuk N. Page 13 4D7FF465-7BD2-40EA-BB78-14F39066B58D final-5af47 #113 13 of 18 Additional space for Task 6. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 14 9D4E6121-2A5F-4718-A7C5-3312EC11ABB0 final-5af47 #113 14 of 18 Task 7 (5+2+3 pts). Suppose that a solution of the equation y′′ − xy = 0 is expressed as a power series: y(x) = ∞∑ n=0 anx n. (a) Prove that an = 1 n(n− 1)an−3 for n ≥ 3. (b) Show that a2 = 0. (c) Use (a), (b) to find the first 5 Taylor coefficients a0, a1, a2, a3, a4 of the solution y(x) with initial conditions y(0) = 1, y′(0) = 1. MAT244H5 December 2019 Final Examination, Goncharuk N. Page 15 B4220845-D69F-4677-9DE5-4161B4156590 final-5af47 #113 15 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 16 BA0A3842-CB4A-4C1F-8867-95E561D247DA final-5af47 #113 16 of 18 Scrap paper MAT244H5 December 2019 Final Examination, Goncharuk N. Page 17 A79900F0-2FD0-45D4-A394-7E36E9BBBFDA final-5af47 #113 17 of 18 Scrap paper End of the booklet MAT244H5 December 2019 Final Examination, Goncharuk N. Page 18 FFE29FE3-3A65-4D3A-9970-06EB56E061F2 final-5af47 #113 18 of 18
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