Name
ID
MATH 308
Final
Fall 2000
Section 200
Solutions
P. Yasskin
1
/15
2
/40
3
/15
4
/30
HAND COMPUTATIONS
1. (15 points) Find the inverse Laplace transform of the function
FŸs s
s 2 25
s"2
F2 2
Ÿs " 2 25
"3s
Ÿs " 2 e
F3 2
Ÿs " 2 25
F1 F F 3 4F 4 3F 6
" 2 e "3s
3
4 .
2
2
s"1
Ÿs " 1 Ÿs " 2 25
Ÿs
«
f 1 cos 5t
«
f 2 e 2t cos 5t
«
f 3 e 2Ÿt"3 cos 5Ÿt " 3 !Ÿt " 3 «
1
s"1
F 5 12
s
1
F6 2
Ÿs " 1 F4 «
f4 et
«
f5 t
«
f6 ett
f f 3 4f 4 3f 6 e 2Ÿt"3 cos 5Ÿt " 3 !Ÿt " 3 4e t 3e t t
If you want me to verify your answer to any part of the following problem, call me
over. If you are wrong, I will give you the correct answer and you will lose part or
all of the indicated points but you will be able to proceed to subsequent parts.
2. (40 points) Consider the 2 mass, 3 spring system shown below.
The masses and spring constants are
gm
gm
m 1 m 2 1 gm, k 1 k 3 4
, k2 6
2
sec
sec 2
The walls are at x 0 and x 80 cm. The rest positions of the masses are at x 1r 20
cm and x 2r 60 cm. Let x 1 Ÿt and x 2 Ÿt be the positions of the masses at time t in sec.
The initial conditions are
cm ,
cm
x 1 Ÿ0 20 cm,
x 2 Ÿ0 0 sec
x 2 Ÿ0 62 cm,
x 1 Ÿ0 16 sec
U
U
a. (3 pts) Write out the second order differential equations satisfied by x 1 Ÿt and x 2 Ÿt .
2
m 1 d x21 "k 1 Ÿx 1 " 20 " k 2 Ÿx 1 " 20 k 2 Ÿx 2 " 60 dt
d2x
m 2 22 "k 2 Ÿx 2 " 60 k 2 Ÿx 1 " 20 " k 3 Ÿx 2 " 60 dt
d 2 x 1 "10 x " 20 6
Ÿ 1
Ÿx 2 " 60 dt 2
d 2 x 2 6 x " 20 " 10 x " 60
Ÿ 2
Ÿ 1
dt 2
b. (4 pts) To simplify the equations make the change of variables from x 1 Ÿt and x 2 Ÿt to
and
y 2 Ÿt x 2 Ÿt " 60
y 1 Ÿt x 1 Ÿt " 20
which measure the distance of each mass from its rest position. Write out the
second order differential equations and the initial conditions satisfied by y 1 Ÿt and
y 2 Ÿt .
d2y1
"10y 1 6y 2
dt 2
d2y2
6y 1 " 10y 2
dt 2
y 1 Ÿ0 0
y 1 Ÿ0 16
y 2 Ÿ0 2
y 2 Ÿ0 0
U
U
c. (10 pts) Find the Laplace transforms Y 1 Ÿs and Y 2 Ÿs of the solutions y 1 Ÿt and
y 2 Ÿt . DO NOT simplify or invert the transforms.
s 2 Y 1 " sy 1 Ÿ0 " y 1 Ÿ0 "10Y 1 6Y 2
s 2 Y 2 " sy 2 Ÿ0 " y 2 Ÿ0 6Y 1 " 10Y 2
s 2 Y 1 " 16 "10Y 1 6Y 2
s 2 Y 2 " 2s 6Y 1 " 10Y 2
10 Y 1 " 6Y 2 16
"6Y 1 Ÿs 2 10 Y 2 2s
U
Ÿs
2
U
Cross multiply and add:
Ÿs
2
10 2 " 36 Y 1 16Ÿs 2 10 12s
16Ÿs 2 10 12s
2
2
Ÿs 10 " 36
2
Y 1 16s4 12s2 160
s 20s 64
Y1 Ÿs
2
10 2 " 36 Y 2 2sŸs 2 10 96
2sŸs 2 10 96
2
2
Ÿs 10 " 36
3
Y 2 2s4 20s2 96
s 20s 64
Y2 dy 1
dy 2
and v 2 Ÿt . Write out a system of first order
dt
dt
differential equations for y 1 Ÿt , y 2 Ÿt , v 1 Ÿt and v 2 Ÿt . (Please list the equations and
variables in this order.)
d. (3 pts) Now let v 1 Ÿt dy 1
dt
dy 2
dt
dv 1
dt
dv 2
dt
v1
v2
"10y 1 6y 2
6y 1 " 10y 2
e. (10 pts) Identify the coefficient matrix of the first order system.
(Please list the variables in the order y 1 , y 2 , v 1 and v 2 .
HINT: How many rows and columns are there in the coefficient matrix?)
Find its eigenvalues. Find the eigenvector for one eigenvalue.
(DO NOT find the other three eigenvectors.)
A
0
0
1
0
0
0
0
1
"10
6
0
0
6
"10
0
0
"5
0
detŸA " 51 "5
A " 51 0
1
6
0
1
0
0
"5
0
1
"10
6
"5
0
6
"10
0
"5
"5
1
6
0
1 "10
"5 0
"10 0 "5
6
"5
"10 "5
"5Ÿ"5 3 " 105 1Ÿ100 105 2 " 36 5 4 205 2 64 0
5 2 "20 o 400 " 256 "10 o 6 "4, "16
2
5 o "4 ,
o "16 2i, "2i, 4i, "4i
Look at 5 2i:
"2i
0
1
0
u1
0
"2i
0
1
u2
"10
6
"2i
0
u3
6
"10
0
"2i
u4
1
u 2i u 1
1
2i
2i
0
0
0
®
0
1
u "2i u 1
1
"2i
"2i
"2iu 1 u 3
"2iu 2 u 4
"10u 1 6u 2 " 2iu 3
6u 1 " 10u 2 " 2iu 4
1
u 4i u 1
"1
4i
"4i
0
0
0
0
1
u "4i u 1
"1
"4i
4i
f. (10 pts) Now suppose an external force F 35 cosŸ3t is applied to mass 1 only.
Write out the modified second order differential equations for y 1 Ÿt and y 2 Ÿt . Using
the method of undetermined coefficients, find a particular solution for y 1 Ÿt and
y 2 Ÿt . (In other words, guess the form of y 1 Ÿt and y 2 Ÿt based on the form of the
external force, plug into the equations and solve for the coefficients.)
d2y1
"10y 1 6y 2 35 cos 3t
dt 2
d2y2
6y 1 " 10y 2
dt 2
y 1p A sin 3t B cos 3t
y 1p "9A sin 3t " 9B cos 3t
y 2p C sin 3t D cos 3t
y 2p "9C sin 3t " 9D cos 3t
UU
UU
"9A sin 3t " 9B cos 3t "10ŸA sin 3t B cos 3t 6ŸC sin 3t D cos 3t 35 cos 3t
"9C sin 3t " 9D cos 3t 6ŸA sin 3t B cos 3t " 10ŸC sin 3t D cos 3t Coefficient of sin 3t:
"9A "10A 6C
"9C 6A " 10C
®
A 6C
®
C 6A
®
A0
C0
Coefficient of cos 3t:
"9B "10B 6D 35
"9D 6B " 10D
®
B " 6D 35
®
D 6B
Particular solution:
y 1p " cos 3t
y 2p "6 cos 3t
®
B "1
D "6
3. (15 points) There are two tanks of water. Tank A initially holds 20 gal of sugar water
with a concentration of 0.3 lb per gal. Tank B initially holds 30 gal of pure water. Pure
water flows into tank A at the rate of 3 gal/hr and sugar water from tank A flows into
tank B at the rate of 4 gal/hr. Sugar water from tank B is pumped back into tank A at
the rate of 1 gal/hr and sugar water is drained from tank B at the rate of 3 gal /hr.
Set up the differential equations and initial conditions for the amount of sugar in
each tank. Be sure to say what your variables mean. DO NOT solve it. Draw a rough
qualitative graph of the amount of sugar in each tank as a function of time.
Let AŸt be the lbs of sugar in tank A. Let BŸt be the lbs of sugar in tank B.
Differential Equations:
dA 1gal
dt
hr
B lb
" 4gal
30gal
hr
A lb
20gal
" 1A 1 B
5
30
out
in
dB 4gal
dt
hr
A lb
" 1gal
20gal
hr
in
B lb
" 3gal
30gal
hr
out
B lb
1A" 2 B
5
15
30gal
out
Initial Conditions:
AŸ0 20gal 0.3 lb 6lb
gal
BŸ0 0
6
5
4
3
2
1
0
AŸt BŸt MAPLE COMPUTATIONS
4. (30 points) Again consider the spring-mass system of problem 2. In part 2(b), you
should have obtained the differential equations
d2y1
"10y 1 6y 2
dt 2
d2y2
6y 1 " 10y 2
dt 2
and the initial conditions
cm ,
y 2 Ÿ0 2 cm,
y 1 Ÿ0 16 sec
y 1 Ÿ0 0,
y 2 Ÿ0 0.
U
U
a. In part 2(c), you should have found the Laplace transforms of the solutions are
2
3
Y 1 Ÿs 16s4 12s2 160 and Y 2 Ÿs 2s4 20s2 96
s 20s 64
s 20s 64
i. (3 pts) Use the inverse Laplace transform to find the solutions y 1 Ÿt and y 2 Ÿt .
ii. (3 pts) Plot y 1 Ÿt and y 2 Ÿt for 0 t t t 4=.
iii. (2 pts) What is the period of oscillation? Answer in a sentence.
b. In part 2(e), you should have found the eigenvalues 5 2i, "2i, 4i, "4i.
i. (2 pts) How are these reflected in the solutions found in part 4(a)?
Answer in a sentence.
c. In part 2(f), a force was applied to mass 1 and you should have obtained the
differential equations
d2y1
"10y 1 6y 2 35 cosŸ3t dt 2
d2y2
6y 1 " 10y 2
dt 2
i. (3 pts) Using the simplest way possible in Maple, find the solution satisfying
the initial conditions
cm ,
y 2 Ÿ0 2 cm,
y 1 Ÿ0 16 sec
y 1 Ÿ0 0,
y 2 Ÿ0 0.
U
U
ii. (3 pts) Plot y 1 Ÿt and y 2 Ÿt for 0 t t t 4=.
iii. (2 pts) What is the period of oscillation? Answer in a sentence.
iv. (2 pts) From the form of the solution, not the plot, why is the period not =?
Answer in a sentence.
d. Now suppose the force on mass 1 is changed to F 24 cosŸ2t .
i. (3 pts) Using the simplest way possible in Maple, find the solution satisfying
the initial conditions
y 1 Ÿ0 0,
cm ,
y 1 Ÿ0 16 sec
y 2 Ÿ0 0.
ii. (3 pts) Plot y 1 Ÿt and y 2 Ÿt for 0 t t t 4=.
iii. (4 pts) What is happening to the solution? Why is the solution not periodic?
Why is this solution unphysical? What would need to be changed in the
differential equations to make them more physical. Answer in sentences.
y 2 Ÿ0 2 cm,
U
U
To Turn in Your Maple Computations:
1. Save your Maple file as lastname_final.mws
2. Print your file as follows:
a. Click on FILE, PRINT and Printer Command.
lpr -J ” Yasskin Maple Final”
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:
To
yasskin
Attachment
lastname_final.mws (or the exact name of your Maple file)
Subject
Last Name
Final
d. Call Dr. Yasskin over to check your email.
^X
and
Y.
e. Mail the letter by typing