辅导案例-UA 262-Assignment 1

MA-UA 262: Ordinary Differential Equations Summer 2020
Assignment 1
Due: Sunday July 19, by 11:59pm EST,
via Gradescope
• Office hours Friday, Saturday, Sunday from 11:30am-12:30am. Location
https : //nyu.zoom.us/j/9179442616
• You can also email questions and I will try my best to help troubleshoot. Do not post
questions on Campuswire.
• Keep work within the scope of the class. Everything is doable using the techniques pre-
sented in class.
• Show all work. Skip out on steps and hand wave answers at your own risk.
Remember nothing in this math business is free.
• Failure to upload to Gradescope in time is akin to no submitted. As a result, a zero will be
given. Failure to submit properly, i.e. assign pages to problems, will result in a heavy
deduction of points. Don’t push your submission to the last minute!!!
1. (5 points) Find the general solution of
y¨ + 2y˙ + y = 0
3. (5 points) Use the Euler approximations (linearized version i.e. section 1.13) to ap-
proximate the solution to the IVP
y˙ = t+ y
y(0) = 1
at t = 3/2 using step size h = .5.
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MA-UA 262: Ordinary Differential Equations Summer 2020
4. (10 points) A 40 liter tank is initially half full of water. A solution containing 10 grams
per liter of salt begins to flow in at 4 liters per minute. The well mixed solution in the
tank flows out of the tank at a rate of 2 liters per minute. How much salt is in the
tank just before it overflows.
5. (5 points) Transform the equation
t2y¨ + 3ty˙ + y =
2
t
from the (t, y)-variables to the (s, v)-variables where v = y and s = ln t. Note that
I am not asking you to solve the equation in the (t, y) or the (s, v) variables. Just
transform the given equation.
6. (5 points) Transform the equation
y˙ = tn−1f
( y
tn
)
from the (t, y)-variables to the (t, v) variables where v = y
tn
. Explain how you would
solve it in the (t, v)-variables.
7. (10 points) Let a, b be constants. Show that the equation is exact
3t2 + 8at+ 2by2 + 3y + (4bty + 3t+ 5)y˙ = 0
and find the solution.
8. (10 points) Let a be a positive constant. Solve
y¨ = a (1 + (y˙))1/2
Hint: Let w = y˙. Solve the first order equation in the w variable. Then solve for y.
9. Assume that y(t) is a function that satisfies
y(t) = 1 +
∫ t
0
w(s) ds (1)
where
w(s) = −
∫ s
0
y(α) dα
(a) (5 points) Show that y(t) defined in (1) satisfies the IVP
y¨ + y = 0
y(0) = 1
y˙(0) = 0
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MA-UA 262: Ordinary Differential Equations Summer 2020
(b) (10 points) Using the integral equation (1), we define the Picard sequence
yn(t) = 1 +
∫ t
0
wn−1(s) ds
where
wn−1(s) = −
∫ s
0
yn−1(α) dα
and n = 1, 2, 3, . . .. Compute y1, y2, y3, y4. Based on these computations, what is
a formula for yn.
10. (10 points) Let’s find the general solution to the following ODE
y˙ =
y − t+ 2
2y + t+ 1
by following the outline described below.
• Begin by converting the the ODE from the t, y variables to the s, v variables via the
change of variables v = y − c1 and s = t− c2.
• Now select c1, c2 so that the equation in the s, v variables is of the from
dv
ds
= f
(v
s
)
• Now solve for v. After which you can solve for y. Note that you may leave your
solution as an implicit equation in the t, y variables.
Remark: I am testing you on your ability to solve the equation using the method
outlined above. If you want the credit for this problem, then follow the outline.
11. (10 points) Let a be a positive constant. For which values of a does the IVP
y¨ + ay = 0
y(0) = 0
y(pi) = 1
have no solution.
12. (15 points) Assume that x(t) and w(t) are continuous non-negative functions on the
interval [0, a]. Let C be a non-negative constant, and assume that
w(t) ≤ C +
∫ t
0
x(s)w(s) ds, ∀t ∈ [0, a] (2)
Show that
w(t) ≤ C exp(F (t)) (3)
where
F (t) =
∫ t
0
x(s) ds
Note that the constants C in (2), (3) are the same.
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