Name
ID
MATH 308
Exam II
Fall 2000
Section 200
Solution
P. Yasskin
1
/60
2
/40
HAND COMPUTATIONS
1. (60 points) Solve the initial value problem
dy
d2y
2
5y cosŸFt with yŸ0 3 and y Ÿ0 1 .
5
dt
10
dt 2
Then identify the steady state solution and any resonances. Be sure to follow each of
the following steps:
U
a. (2 pts) Identify the homogeneous equation.
dy
d2y
2
5y 0
2
dt
dt
b. (6 pts) Find the characteristic equation and the characteristic roots.
5 2 25 5 0
®
5 "2 o 4 " 20 "1 o 2i
2
c. (3 pts) Is the homogeneous equation overdamped, underdamped or critically
damped? Why?
It is underdamped because the characteristic values are complex.
d. (6 pts) Find a fundamental set of solutions of the homogeneous equation and
verify they are linearly independent by computing the Wronskian.
y 1 e "t cosŸ2t W
y1 y2
U
U
y1 y2
y 2 e "t sinŸ2t e "t cosŸ2t e "t sinŸ2t "e "t cosŸ2t " 2e "t sinŸ2t "e "t sinŸ2t 2e "t cosŸ2t e "t cosŸ2t ¡"e "t sinŸ2t 2e "t cosŸ2t ¢ " e "t sinŸ2t ¡"e "t cosŸ2t " 2e "t sinŸ2t ¢
2e "2t cos 2 Ÿ2t 2e "2t sin 2 Ÿ2t 2e "2t p 0
Since W p 0 the solutions are linearly independent.
e. (2 pts) Write out the general solution to the homogeneous equation.
y h Ae "t cosŸ2t Be "t sinŸ2t f. (6 pts) To find the particular solution using the Method of Undetermined
Coefficients, write out the general form of the particular solution. (Take the
undetermined coefficients to be the coefficients of sin and cos not the amplitude
and phase.) Then compute the first and second derivatives.
y p C sinŸFt D cosŸFt y p FC cosŸFt " FD sinŸFt y p "F 2 C sinŸFt " F 2 D cosŸFt U
UU
g. (8 pts) Using the Method of Undetermined Coefficients, find a particular solution to
the inhomogeneous equation. (If you can’t do this for general F, at least do it for
F 1.)
Plug y p into the inhomogeneous equation:
y p 2y p 5y cosŸFt UU
¡"F
2
U
C sinŸFt " F 2 D cosŸFt ¢ 2¡FC cosŸFt " FD sinŸFt ¢ 5¡C sinŸFt D cosŸFt ¢ cosŸFt Equate coefficients of sinŸFt and cosŸFt :
2
sinŸFt :
®
" F 2 C " 2FD 5C 0
Ÿ5 " F C " 2FD 0
®
2FC Ÿ5 " F 2 D 1
cosŸFt :
" F 2 D 2FC 5D 1
To find C:
2 2
2
Ÿ5 " F C " 2FŸ5 " F D 0
Ÿ2F 2
C 2FŸ5 " F 2 D 2F
2
Ÿ5 " F 2 2 C 2F
add:
Ÿ2F C
Ÿ2F 2
2F
Ÿ5 " F 2 2
To find D:
" 2FŸ5 " F 2 C Ÿ2F 2 D 0
2FŸ5 " F 2 C Ÿ5 " F 2 2 D Ÿ5 " F 2 add:
Ÿ2F 2
Ÿ5 " F 2 2 D Ÿ5 " F 2 " F2 2
2 2
Ÿ2F Ÿ5 " F 2
Ÿ5 " F sinŸFt cosŸFt 2
2 2
Ÿ2F Ÿ5 " F Ÿ5
D
yp Ÿ2F 2
2F
Ÿ5 " F 2 2
h. (2 pts) Write out the general solution to the inhomogeneous equation.
y Ae "t cosŸ2t Be "t sinŸ2t Ÿ5 " F 2F
sinŸFt cosŸFt 2
2
2 2
Ÿ5 " F 2 Ÿ2F Ÿ5 " F 2
Ÿ2F 2
i. (6 pts) With F 1, use the initial conditions to find the solution to the initial value
problem with yŸ0 3
5
and y Ÿ0 1 .
10
U
Ÿ5 " 1 2
sinŸt cosŸt 2
2
2
Ÿ2 Ÿ5 " 1 Ÿ2 Ÿ5 " 1 y Ae "t cosŸ2t Be "t sinŸ2t 1 sinŸt 1 cosŸt 5
10
t
t
t
"
"
"
y "Ae cosŸ2t " 2Ae sinŸ2t " Be sinŸ2t 2Be "t cosŸ2t 1 cosŸt " 1 sinŸt 5
10
yŸ0 A 1 3
®
A 2
5
5
5
1
1
y Ÿ0 "A 2B ®
B 1A 1
5
10
10
2
y 2 e "t cosŸ2t 1 e "t sinŸ2t 1 sinŸt 1 cosŸt 5
5
5
10
y Ae "t cosŸ2t Be "t sinŸ2t 2
U
U
j. (4 pts) With F once again general, for large times, what is the steady state
solution? Why?
At large times the exponentials all approach zero and the steady state solution is
2
Ÿ5 " F 2F
y
sin
F
t
cosŸFt Ÿ
2
2
2 2
2 2
Ÿ 2F Ÿ 5 " F Ÿ2F Ÿ5 " F k. (6 pts) Find the amplitude of the steady state solution.
With C A
Ÿ2F D C
2
2
2F
and D Ÿ5 " F 2 2
2
" F2 2
2 2
Ÿ2F Ÿ5 " F Ÿ5
2
" F2 :
2
2 2
Ÿ2F Ÿ5 " F Ÿ5
Ÿ2F 2
2F
Ÿ5 " F 2 2
Ÿ2F 2
Ÿ2F 2
2
Ÿ5 " F 2 2
Ÿ5 " F 2 2
1
Ÿ2F 2
Ÿ5 " F 2 2
l. (6 pts) Identify the resonance (angular) frequency F R (if any) by maximizing the
steady state amplitude.
A "1
2
U
®
2Ÿ2F 2 2Ÿ5 " F 2 Ÿ"2F Ÿ5 " F " 3F F 3 0
Ÿ 2F 2
2 2
3/2
0
®
8F " 20F 4F 3 0
F 0, 3 , " 3
®
®
FR 3
m. (3 pts) What is the steady state amplitude at resonance?
AR 1
Ÿ 2F R 2
Ÿ5 " F
2 2
R 1
2 3
2
5" 3
2
2
1
1
4
12 4
MAPLE COMPUTATIONS
2. (40 points) Consider the initial value problem
dy
d2y
b
5y cosŸFt dt
dt 2
for general b and F.
a. (3 pts) For what values of b is the system overdamped, underdamped or critically
damped? Hereafter, assume the system is underdamped and b 0.
b. (2 pts) Find the general solution in terms of b and F.
c. (4 pts) Find the specific solution for b 2 and F 3 satisfying the initial conditions
yŸ0 2 and y Ÿ0 1. Plot this solution.
U
d. (8 pts) Find the steady state solution for general b and F and then find its
amplitude and phase shift.
e. (5 pts) Plot the steady state amplitude as a function of F for b 1, 2, 3, 4.
f. (5 pts) Identify the resonance (angular) frequency F R (if any) as a function of b by
maximizing the steady state amplitude.
g. (3 pts) For what values of b is there a resonance? For what values of b is there no
resonance? Does this agree with your plot?
h. (5 pts) What is the steady state amplitude at resonance as a function of b?
i. (5 pts) Plot the resonant steady state amplitude as a function of b.
To Turn in Your Maple Computations:
1. Save your Maple file as lastname_exam2.mws
2. Print your file as follows:
a. Click on FILE, PRINT and Printer Command.
lpr -J ” Yasskin Maple Exam 2”
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_exam2.mws (or the exact name of your Maple file)
Attachment
Last Name
Exam2
Subject
d. Call Dr. Yasskin over to check your email.
Y.
e. Mail the letter by typing
^X
and