50 pts.
Problem 1. In each part, find the general solution, or solve the initial value
problem.
A.
y 00 + y 0 − 6y = 0,
y(0) = 1,
y 0 (0) = −18.
B.
y 00 − 4y 0 + 4y = 0.
C.
y 00 − 10y 0 + 29y = 0.
D.
x2 y 00 + 2xy 0 − 2y = 0
40 pts.
Problem 2. Find the general solution.
A. D2 (D − 5)3 (D + 1)y = 0
B. (D + 4)2 (D2 − 4D + 13)3 y = 0
60 pts.
Problem 3. Use the method of Undetermined Coefficients to find the
general solution
A.
y 00 − 3y 0 + 2y = 2x2 − 1
B.
y 00 − 3y 0 + 2y = e−x
C.
y 00 − 3y 0 + 2y = xex .
1
40 pts.
Problem 4. Find the general solution by the method of variation of parameters.
A.
y 00 − 4y 0 + 4y =
e2x
x2
B.
x2 y 00 + 2xy 0 − 6y = x3
In part B, you may assume the solution of the homogenous equation is
y = C1 x2 + C2 /x3 .
30 pts.
Problem 5. In each part, give the form of the partial fractions decomposition,
with undetermined coefficients. Do not find the coefficients.
No computation is required.
A.
s2
(s − 1)(s − 2)(s + 2)
B.
s2 (s
C.
40 pts.
s4 + 1
− 1)3 (s − 3)
s2 + s + 1
(s − 1)(s2 + 1)3
Problem 6. Find the inverse Laplace transform of
F (s) =
5s2 − 3s + 1
.
s2 (s − 1)
Do the partial fractions decomposition by hand, showing your work.
2
EXAM
Exam 2
Math 3350, Summer II, 2010
July 30, 2010
• Write all of your answers on separate sheets of paper.
You can keep the exam questions when you leave.
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 6 problems. There are 260 points
total.
Good luck!