ENGR452/BIOE452/CHE452/ME452 Instructor: Parisa Khodabakhshi
Class 22 Review Topics for Test 2
Inhomogeneous linear ordinary differential equations. Method of undetermined coefficients
for polynomial, trigonometric, and exponential forcing terms; cases when forcing term satis-
fies homogeneous equation. Constant coefficient and Cauchy-Euler problems.
Variation of parameters. Reduction of order based on a known solution. Variation of pa-
rameters applied to linear systems of ordinary differential equations.
Series solutions of linear ordinary differential equations; regular points. Regular singular
points; distinct indicial roots; repeated indicial roots; distinct roots differing by an integer.
Gamma functions. Bessel Functions.
Laplace transforms; definition. Properties; calculation; inverses; convolution theorem. Ap-
plication to constant coefficient linear ordinary differential equations; linear integral equa-
tions of convolution type. Step function, delta function. Application of Laplace transforms
to partial differential equations; diffusion equation. Application of separation of variables to
the solution of the diffusion equation.
Sturm-Liouville theory; eigenfunction expansions; singular and periodic problems. Review
of Fourier series and their properties.
References: 2011 Lecture Notes, Professor P. A. Blythe
2021 Lecture Notes, Professor J. W. Jaworski
Michael D. Greenberg 1998 Advanced Engineering Mathematics (second edition), Prentice Hall
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ENGR452/BIOE452/CHE452/ME452 Instructor: Parisa Khodabakhshi
Sample 1 of Test 2
1. Find the explicit solution of
y
d2y
dx2
=
1
2
[
1 +
(
dy
dx
)2]
that satisfies
y = 1,
dy
dx
= 0 at x = 0.
Answer: y = 1 +
x2
4
2. Given that y = x is a solution of the homogeneous equation
x2y′′ − x(x+ 2)y′ + (x+ 2)y = 0,
determine the general solution of the inhomogeneous equation
x2y′′ − x(x+ 2)y′ + (x+ 2)y = x3ex cosx.
Identify a second linearly independent solution of the homogeneous equation.
Hint : You are given that ∫
ex sinxdx =
1
2
ex (sinx− cosx) .
Answer: second linearly independent solution yh2 = xe
x
general solution of the inhomogeneous problem y = c1x+ c2xe
x + 1
2
xex(sinx− cosx)
3. The equation
x2y′′ − (2 + x)xy′ + (2− x)y = 0
possesses series solution of the form
y = xr
∑
n=0
anx
n
(a) Deduce all values of the index r.
Answer: r = 1, 2
(b) For the larger of the r values, deduce the general recurrence relation satisfied by the
coefficients an, n ≥ 1.
Answer: an =
n+ 2
n(n+ 1)
an−1
(c) Obtain the solution of the recurrence relation corresponding to the larger r value,
and give the corresponding solution of the differential equation.
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ENGR452/BIOE452/CHE452/ME452 Instructor: Parisa Khodabakhshi
Note: You are required to find only the single solution corresponding to the larger
r value.
Answer: y1 = x
2
∑∞
n=0
n+ 2
2n!
xn
4. x(t) satisfies the integro-differential equation
dx
dt
+ x = 9H(t− 2) + 2
∫ t
0
e−2τx(t− τ)dτ
with x(0) = 3.
(a) Let the Laplace transform of x(t) be X(s). Determine X(s).
Answer: X(s) = e−2s
(
1
s
+
6
s2
− 1
s+ 3
)
+
2
s
+
1
s+ 3
(b) Hence find x(t).
Answer: x(t) = 2 + e−3t +H(t− 2) (1 + 6(t− 2)− e−3(t−2))
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ENGR452/BIOE452/CHE452/ME452 Instructor: Parisa Khodabakhshi
Sample 2 of Test 2
1. (a) Find both critical points of the system
dx
dt
= −x+ y + y2,
dy
dt
= −x− y.
Answer: (x0, y0) = (0, 0), (2,−2)
(b) determine the type and stability of each critical point.
Answer: (x0, y0) = (0, 0): stable focus, (x0, y0) = (2,−2): unstable saddle
2. (a) You are given that
4x2(1 + x)y′′ − 2xy′ + (2− 3x)y = 0
has solutions of the form
y = xr
∑
n=0
anx
n.
Obtain both possible values of the index r.
Answer: r =
1
2
, 1
(b) Using the lower index value, find the general form of the recurrence equation that
relates an to an−1.
Answer: an = −n− 2
n− 1
2
an−1
(c) Determine the solution for an from the recurrence equation obtained in part (b).
Answer: a1 = 2a0, and an = 0 for n ≥ 2
(d) Hence deduce the solution for y(x) that corresponds to the lower of the r values.
Note: DO NOT calculate the solution that corresponds to the larger of the r values.
Answer: y1(x) = x
1/2(1 + 2x)
3. (a) Determine the general real solution of
d2y
dx2
+ 2
dy
dx
+ λy = 0, (A)
where,
λ = 1 + µ2.
(Your answer should be given in terms of x and µ together with any arbitrary con-
stants.)
Answer: y(x) =
{
e−x(A+Bx), µ = 0
e−x(A cosµx+B sinµx), µ ̸= 0
(b) If
y(0) = 0, y(1) = 0,
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ENGR452/BIOE452/CHE452/ME452 Instructor: Parisa Khodabakhshi
obtain µ and hence the eigenvalues λ.
Answer: µ ̸= 0: λ = 1 + n2π2, ϕn(x) = e−x sinnπx for n ≥ 1
(c) Write equation (A) in Sturm-Liouville form and state the weight function w(x).
Answer: w(x) = e2x
(d) Give the set of eigenfunctions that are orthonormal with respect to the weight func-
tion.
Hint:
∫
sin2mxdx = 1
2
x− 1
4m
cos 2mx.
Answer: ϕˆn(x) =
√
2e−x sinnπx
4. (a) Use Fourier tranforms to solve
d2y
dx2
+ 4
dy
dx
+ 4y = 9H(−x)ex
given that y → 0 as x→ ±∞.
Hint: H(x) represents the step function.
Answer: y(x) = (3x+ 1)H(x)e−2x +H(−x)ex
(b) Invert the Fourier transform
e−3iω.e−ω
2/4.
(The inverse required for part (a) and for part (b) can all be obtained from Appendix
D in Greenberg.)
Answer:
1√
π
e−(x−3)
2
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ENGR452/BIOE452/CHE452/ME452 Instructor: Parisa Khodabakhshi
Sample 3 of Test 2
1. (25 points)
(a) Find the solution of
y
d2y
dx2
=
dy
dx
−
(
dy
dx
)2
that satisfies
y = 1,
dy
dx
= 2, at x = 0.
Answer: y + ln
2
1 + y
= x+ 1
(b) What solution satisfies
y = 1,
dy
dx
= 0, at x = 0.
Answer: y(x) = 1
2. (30 points)
(a) Find the general solution of the homogeneous equation
x2
d2y
dx2
− 2xdy
dx
+ 2y = 0
by first converting it to a constant coefficient differential equation.
Answer: yh(x) = c1x+ c2x
2
(b) Use the method of undetermined coefficients to obtain the general solution of
x2
d2y
dx2
− 2xdy
dx
+ 2y = x.
Answer: y(x) = c1x+ c2x
2 − x lnx
3. (25 points)
(a) Given that
2x2y′′ + (3x− 6x2)y′ − (1 + 3x)y = 0
has solutions of the form
y = xr
∑
n=0
anx
n
determine both possible values of the index r.
Answer: r = −1, 1
2
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ENGR452/BIOE452/CHE452/ME452 Instructor: Parisa Khodabakhshi
(b) For the lower index value, find a recurrence equation relating an to an−1.
Answer: an =
3
n
an−1
(c) Obtain the solution of the recurrence equation for an.
Answer: an =
3n
n!
a0
(d) Write down the solutions for y(x) that corresponds to the lower of the r values. If
possible express your final answer in terms of a simple elementary function of x.
Note: DO NOT calculate the solution that corresponds to the larger r value.
Answer: y(x) = x−1
∑∞
n=0
3n
n!
xn = x−1e3x
4. (20 points) Use the results in Appendix C of Greenberg to invert the Laplace transform
ln s
s(s+ 1)
Part of your answer should be expressed in terms of the functions
ϕ(t) =
∫ t
0
eτ (− ln τ)dτ .
Answer: e−tϕ(t)− γ(1− e−t)
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