EXAM
Final Exam
Math 3350, Summer 2009
August 6, 2009
• 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 10 problems. There are 480 points
total.
Good luck!
60 pts.
Problem 1. In each part, find the general solution of the differential equation,
or solve the given initial value problem. You must show the steps
in solving the equation by one of the methods given in class, you can’t just write
down the answer.
A.
B.
C.
50 pts.
dy
4y
=
,
dx
1 + 2x
y(0) = 5.
dy
− 5y = x3 e5x
dx
dy
− 4y = ex y −1/2
dx
Problem 2. Newton’s law of cooling says that the time rate of change dT /dt
of the temperature T of a body is proportional to the difference
between T and the temperature M of the surrounding medium (the temperature
of the surrounding medium is assumed to stay constant).
A soda can at a temperature of 35◦ is placed in a room that is at 75◦ . After
5 minutes the can has warmed to a temperature of 45◦ .
A. Find the differential equation for the temperature T of the can and solve it
to find T as a function of time.
B. What is the temperature of the can after 15 minutes? Give a numerical
answer accurate to two decimal places.
C. At what time will the temperature of the coffee be 74◦ ? Give an numerical
answer that is accurate to two decimal places.
1
90 pts.
Problem 3. In each part, find the general solution of the differential equation,
or solve the initial value problem.
A.
y 00 − 4y 0 + 3y = 0,
y(0) = 2,
y 0 (0) = 1.
B.
y 00 − 4y 0 + 4y = 0
C.
y 00 + 4y 0 + 20y = 0.
D.
x2 y 00 + 2xy 0 − 6y = 0.
40 pts.
Problem 4. Use the method of Undetermined Coefficients to find the
general solution
A.
y 00 − 3y 0 + 2y = x
B.
y 00 + 2y 0 + y = e−x
40 pts.
Problem 5. Find the general solution by the method of variation of parameters. No credit for doing it by a different method.
y 00 − 4y 0 + 4y = x2 e2x .
40 pts.
Problem 6. In each part, find the inverse Laplace Transform. Use a calculator
to do the partial fractions!
A.
F (s) =
6 s3 − 3 s − 2 + s5
2
s3 (s + 1) (s − 2)
B.
F (s) =
s
s2 − 4s + 13
2
40 pts.
40 pts.
40 pts.
Problem 7. Find the Laplace Transform of the function
(
t2 0 < t < 2
f (t) =
0, 2 < t < ∞
Problem 8. Find the Inverse Laplace Transform of the function
2
s
−2s
F (s) = e
+ 2
(s − 2)3
s +4
Problem 9. Solve the initial value problem using Laplace transforms. (No
credit for using a different method.) Use a calculator to do the
partial fractions!
y 00 + 3y 0 + 2y = te−t ,
40 pts.
y(0) = 0,
y 0 (0) = 1
Problem 10. Consider the function
f (x) = x − 1,
0 < x < 1.
Find the Fourier sine series of f (x) on [0, 1]. This is also known as the odd halfrange expansion. You can certainly use your calculator to find the integrals!
3