EXAM
Exam 2
Math 3350–015, Spring 2012
April 5, 2012
• 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 5 problems. There are 290 points
total.
Good luck!
100 pts.
Problem 1. In each part, find the general solution of the given differential
equation. If initial conditions are given, solve the initial value
problem.
A.
y 00 − 3y 0 + 2y = 0,
y(0) = 1,
y 0 (0) = −2.
B.
y 00 + 6y 0 + 9y = 0.
C.
y 00 + 4y 0 + 13y = 0
D. This is an Euler-Cauchy Equation.
x2 y 00 − 3xy 0 + 4y = 0
40 pts.
Problem 2. Find the general solution.
A. D2 (D − 2)3 (D − 1)y = 0
3
B. (D + 4) (D − (3 + 5i))(D − (3 − 5i)) y = 0
60 pts.
Problem 3. Use the method of Undetermined Coefficients (either version) to find the general solution
A.
y 00 − 3y 0 + 2y = 2x2 + 1.
B.
y 00 − y 0 − 2y = 4xex
C.
y 00 − y 0 − 2y = e−x
40 pts.
Problem 4. Find the general solution of the following differential equation,
using the method of variation of parameters:
y 00 − 2y 0 + y =
1
ex
.
x2
50 pts.
Problem 5. Consider the second order linear, homogeneous differential equation
2
6
y 00 + y 0 − 2 y = 0,
x > 0,
x
x
(actually, it’s an Euler-Cauchy equation). One solution of this equation is y1 =
x2 . Use the method of reduction of order to find a second solution y2
so that y1 and y2 are linearly independent. No credit for using a different
method!
Find the general solution of the differential equation.
2