Math 308
Spring 2015
Exam 2
04/23/15
Instructor
Name (Print):
Aziz Takhirov
You may not use your books, notes, or any calculator on this exam.
You are required to show your work on each problem on this exam. The following rules apply:
• Organize your work , in a reasonably neat and coherent way, in the space provided. Work scattered all over the page without a clear ordering will receive very little credit.
• Mysterious or unsupported answers will not receive full credit .
A correct answer, unsupported by calculations, explanation, or algebraic work will receive no credit; an incorrect answer supported by substantially correct calculations and explanations might still receive partial credit.
Do not write in the table to the right.
Problem Points Score
1
2
3
4
5
6
7
8
Total:
10
5
15
15
5
10
20
20
100

Math 308 Exam 2 - Page 2 of 11 04/23/15
1. (10 points) Determine a suitable form for a particular solution Y ( t ) if the method of undetermined coefficients is to be used ( don’t solve for Y ( t ), just write the general expression).
y
00
+ 3 y
0
= t (10 + cost ) + 0 .
0001 .

Math 308 Exam 2 - Page 3 of 11 04/23/15
2. (5 points) Using the given fundamental set of solutions y
1
( t ) , y
2
( t ) of the differential equation below, write a formula for its solution, if the variation of parameters is to be used ( don’t solve the equation and don’t verify that y
1
( t ) , y
2
( t ) is the fundamental set of solutions).
y
1
( t ) = e t
, y
2
( t ) = t ; (1 − t ) y
00
+ ty
0
− y = 2 ( t − 1)
2 e
− t
.

Math 308 Exam 2 - Page 4 of 11 04/23/15
3. (15 points) Solve the given IVP using Laplace transforms ( any other method will receive zero credit).
y
00
+ y
0
+ 1 .
25 y = g ( t ); y (0) = 0 , y
0
(0) = 0; g ( t ) = sint, if 0 ≤ t < π,
0 , if t ≥ π

Math 308 Exam 2 - Page 5 of 11 04/23/15
4. (15 points) Solve the given IVP using Laplace transforms ( any other method will receive zero credit). The final answer must be in terms of a convolution integral.
4 y
00
+ 4 y
0
+ 17 y = g ( t ); y (0) = 0 , y
0
(0) = 0 .

Math 308 Exam 2 - Page 6 of 11 04/23/15
5. (5 points) Transform the given IVP into an IVP for a system of first order equations ( you don’t need to solve the system).
y
00
+ p ( t ) y
0
+ q ( t ) y = g ( t ); y (0) = y
0
, y
0
(0) = y
0
0
.

Math 308 Exam 2 - Page 7 of 11
6. (10 points) Find the Inverse Laplace Transform of the given function.
F ( s ) = s
G ( s )
2 + 1
.
04/23/15

Math 308 Exam 2 - Page 8 of 11
7. (a) (10 points) Find the general solution.
→
0
( t ) =
1 1
4 − 2
−
( t ) .
04/23/15

Math 308 Exam 2 - Page 9 of 11 04/23/15
(b) (10 points) Plot the phase portait of the system and describe the behavior of the solution as t → ∞ .

Math 308 Exam 2 - Page 10 of 11
8. (a) (10 points) Find the general solution.
→
0
( t ) =
5 − 1
3 1
−
( t ) .
04/23/15

Math 308 Exam 2 - Page 11 of 11 04/23/15
(b) (10 points) Plot the phase portait of the system and describe the behavior of the solution as t → ∞ .