EXAM
Exam 2
Math 3350, Summer 2011
June 27, 2011
• Write all of your answers on the blank paper
provided. Do not write on the exam questions
handout. When finished, write your name and the
section number on the first page of your answers.
Staple the exam questions and your sheet of notes to
the back of your answers. You may leave when
finished.
• You must show enough work to justify your answers.
Unless otherwise instructed,
give exact answers, not
√
approximations (e.g., 2, not 1.414).
• This exam has 7 problems. There are 300 points
total.
Good luck!
40 pts.
Problem 1. Find the general solution of the following differential equation,
using the method of variation of parameters:
y 00 − 2y 0 + y =
40 pts.
Problem 2. In each part, find the form of the partial fraction decomposition
of the given function, showing the undetermined coefficients. Do
not determine the coefficients! If you’re doing long computations, you’re
doing it wrong.
A.
s4 + 6
(s − 1)(s − 2)(s − 3)3
B.
60 pts.
ex
.
x
1
s3 (s2 + 4)2
Problem 3. In each part, find the inverse Laplace transform.
A. In this part, find the partial factions decomposition by hand.
s2 + s + 8
s(s2 + 4)
B. In this part, use a calculator to find the partial fractions decomposition.
2s4 − 10s3 + 13s2 − 12s + 4
s3 (s − 2)2
C. Hint: complete the square.
s2
40 pts.
s
− 4s + 13
Problem 4. Solve the differential equation, using Laplace Transforms. No
credit for using a different method!
y 00 − 2y 0 + y = tet ,
y(0) = 2,
1
y 0 (0) = 1.
40 pts.
40 pts.
40 pts.
Problem 5. Find the Laplace transform of the function


0, 0 < t < 1
f (t) = t, 1 < t < 2

2
t , 2 < t < ∞.
Problem 6. Find the inverse Laplace transform of the following function.
1
1
1
−s
e
+ 3 + e−2s 2
2
s
s
s +9
Problem 7. Solve the following differential equation, using Laplace transforms.
y 00 + y = U (t − 2),
y(0) = 0,
2
y 0 (0) = 1.