MATH 308
Exam III
Fall 2000
Section 200
Solutions
P. Yasskin
HAND COMPUTATIONS
1. (40 points) Use the Laplace Tranform Technique to find the Laplace transform of the
solution to the initial value problem
dy
d2y
3
2y 9 6t
2
dt
dt
with yŸ0 1 and y Ÿ0 1.
U
Laplace transform of the differential equation:
s 2 YŸs " syŸ0 " y Ÿ0 3¡sYŸs " yŸ0 ¢ 2YŸs 9s 62
s
Substitute in the initial conditions:
s 2 YŸs " s " 1 3sYŸs " 3 2YŸs 9s 62
s
Solve for YŸs :
YŸs ¡s 2 3s 2 ¢ s 4 9s 62
s
9
s 4 s 62
s
YŸs s 2 3s 2
U
2. (10 points)
a. Find the Laplace transform of the function:
f 1 Ÿt sin 3t
f 2 Ÿt e 2t sin 3t e 2t f 1 Ÿt fŸt te 2t sin 3t tf 2 Ÿt fŸt te 2t sin 3t
3
s2 9
F 2 Ÿs F 1 Ÿs " 2 F 1 Ÿs b. Find the inverse Laplace transform of the function:
G 1 Ÿs 23
s
2
G 1 Ÿs 1 G 2 Ÿs 3
Ÿs 1 "2s
GŸs 2e 3 e "2s G 2 Ÿs Ÿs 1 3
" 2 2 9
"3¡2Ÿs " 2 ¢
FŸs " d F 2 Ÿs "
2
ds
Ÿs " 2 9
6Ÿs " 2 FŸs 2
2
Ÿs " 2 9
Ÿs
G Ÿs g 1 Ÿt t 2
g 2 Ÿt e " t g 1 Ÿt e " t t 2
gŸt g 2 Ÿt " 2 !Ÿt " 2 gŸt e "Ÿt"2 Ÿt " 2 2 !Ÿt " 2 2
2e "2s
3
Ÿs 1 3. (20 points) Find the solution of the system
dx "8x 8y
dt
dy
"3x 2y
dt
using the Eigenvector Technique.
xŸ0 6
yŸ0 4
Coefficient matrix:
"8 8
"3 2
A
Characteristic polynomial:
detŸA " 51 "8 " 5 8
"3 2 " 5
Ÿ"8 " 5 Ÿ2 " 5 24 5 2 65 8
Characteristic equation and eigenvalues:
5 2 65 8 0
Ÿ5 2 Ÿ5 4 0
5 "2, "4
Find eigenvectors:
5 "2:
"6 8
"3 4
u1
u2
0
0
0
®
" 3u 1 4u 2 0
®
u2 3 u1
4
®
u "2 4
®
" 4u 1 8u 2 0
®
u2 1 u1
2
®
u "4 2
3
5 "4:
"4 8
"3 6
u1
u2
0
Vector solution:
u "2 Ce "4t u "4
X Be "2t x
y
Be "2t
4
3
Ce "4t
Scalar solution:
xŸt 4Be "2t 2Ce "4t
yŸt 3Be "2t Ce "4t
Use the initial conditions:
xŸ0 4B 2C 6
yŸ0 3B C 4
®
Solution:
xŸt 4e "2t 2e "4t
yŸt 3e "2t e "4t
B1
C1
2
1
1
MAPLE COMPUTATIONS
4. (30 points) Find the solution of the system
dx "8x 8y
xŸ0 6
dt
dy
"3x 2y
yŸ0 4
dt
using the Laplace Tranform Technique. (You may not use the dsolve command
except to check your answer.)
To Turn in Your Maple Computations:
1. Save your Maple file as lastname_exam3.mws
2. Print your file as follows:
a. Click on FILE, PRINT and Printer Command.
lpr -J ” Yasskin Maple Exam 3”
b. Make the command read:
c. Call Dr. Yasskin over to check your printing.
d. Click on PRINT.
3. Mail your file as follows:
pine
a. Start the mail program:
C.
b. Compose a letter by typing
c. In the header region, enter:
yasskin
To
lastname_exam3.mws (or the exact name of your Maple file)
Attachment
Last Name
Exam3
Subject
d. Call Dr. Yasskin over to check your email.
Y.
e. Mail the letter by typing
^X
and