EXAM
Exam 2
Math 3350–D01, Fall 2013
Oct. 23, 2013
• 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 5 problems. There are 340 points
total.
Good luck!
90 pts.
Problem 1. In each part, find the general solution of the differential equation,
or solve the initial value problem.
A.
y 00 + 5y 0 + 6y = 0,
y(0) = 1,
y 0 (0) = 0.
B.
y 00 − 2y 0 + y = 0
C.
y 00 − 4y 0 + 5y = 0.
D. An Euler-Cauchy equation
x2 y 00 + 2xy 0 − 2y = 0
40 pts.
Problem 2. Find the general solution. In the case of complex roots, find the
general real-valued solution.
A. y (4) − 8y (3) + 24y 00 − 32y 0 + 16y = 0.
The characteristic polynomial is
p(λ) = λ4 − 8λ3 + 24λ2 − 32λ = (λ − 2)4 .
B. y (4) − 8y 3 + 42y 00 − 104y 0 + 169y = 0
The characteristic polynomial is
p(λ) = λ4 − 8λ3 + 42λ2 − 104λ + 169 = (λ − (2 + 3i))2 (λ − (2 − 3i))2
1
80 pts.
Problem 3. Use the method of Undetermined Coefficients (either version) to find the general solution
A.
y 00 − 3y 0 + 2y = 2x2 + 1.
B.
y 00 − y 0 − 2y = 4xex
C.
y 00 − y 0 − 2y = e−x
D.
y 00 − y 0 − 2y = sin(x).
60 pts.
Problem 4. Find the general solution by the method of variation of parameters. No credit for doing it by a different method.
y 00 + y = sec(x).
70 pts.
Problem 5. A tank contains 10 gallons of water. Five gallons of brine per
minute flow into the tank, each gallon of brine containing 1 pound
of salt. Five gallons of brine flow out of the tank per minute. Assume that the
tank is kept well stirred.
Find a differential equation for the number of pounds of salt in the tank (call
it y, say).
Assuming the tank intially contains 1 pound of salt, solve this differential
equation.
How much salt is in the tank after 5 minutes?
At what time will there be 9 lbs of salt in the tank? Give a numerical answer
accurate to two decimal places.
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