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MATH 251
Examination I
October 4, 2007
Name: Student Number: Section:
This exam has 12 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must be shown. Credit will not be given for an answer not supported by work. The point value for each question is in parentheses to the right of the question number.
You may not use a calculator on this exam. Please turn off and put away your cell phone.
Total:
Do not write in this box.
- (5 points) Which function below is the integrating factor μ(t) that could be used to solve the first order linear differential equation
t^2 y′^ − 4 ty = 0?
(a) t^4
(b) e^2 t
2
(c)
t^4 (d) e−^4 t
- (5 points) Consider the first order differential equation
y′^ = −t^3.
Which statement below is false?
(a) The equation is linear.
(b) The equation is separable.
(c) The equation is exact.
(d) The equation is autonomous.
- (5 points) Let y 1 (t) and y 2 (t) be any two solutions of the second order linear equation
(t^2 + 4)y′′^ + 2ty′^ − t^3 y = 0
In what general form must their Wronskian, W (y 1 , y 2 )(t), appear?
(a) C(t^2 + 4)
(b) C
t^2 + 4
(c)
C
(t^2 + 4)
(d) C(t^2 + 4)^2
- (5 points) A fish tank is initially filled with 400 liters of water containing 1 g/liter of dissolved oxygen. At t = 0, oxygenated water containing 10 g/liter of oxygen flows in at a rate of 4 liter/min. The well-mixed water is pumped out at a rate of 3 liter/min from the tank. Which of the initial value problems below describes, Q(t), the amount of dissolved oxygen in the tank at any time t > 0 (until the time when the tank, eventually, overflows)?
(a) Q′^ = 40 −
400 + t
Q, Q(0) = 400.
(b) Q′^ = 40 −
400 + t
Q, Q(0) = 1.
(c) Q′^ = 40 −
Q, Q(0) = 10.
(d) Q′^ = 40 −
400 − t
Q, Q(0) = 4000.
- (12 points) Consider the autonomous differential equation
y′^ = y(y − 5)(10 − y).
(a) (3 points) Find all equilibrium solutions.
(b) (5 points) Classify the stability of each equilibrium solution. Justify your answer.
(c) (2 points) If y(5000) = 6, what is (^) tlim→ ∞ y(t)?
(d) (2 points) If y(7) = 10, find y(21). (You do not need to solve the equation to find the answer.)
- (10 points) Consider the initial value problem
y′′^ − 7 y′^ + 6y = 0, y(0) = 5, y′(0) = 0.
(a) (8 points) Find the solution, y(t), of this initial value problem.
(b) (2 points) What is lim t→∞ y(t)?
- (12 points) Consider the nonhomogeneous second order linear equation of the form
y′′^ − 4 y′^ + 8y = g(t).
(a) (3 points) Find its complementary solution, yc(t).
For each of parts (b) through (d), write down the correct choice of the form of particular solution that you would use to solve the given equation using the Method of Undetermined Coefficients. DO NOT ATTEMPT TO SOLVE THE COEFFICIENTS.
(b) (3 points) y′′^ − 4 y′^ + 8y = 2e^2 t^ − 5 t^2 + sin 2t
(c) (3 points) y′′^ − 4 y′^ + 8y = −e^2 t^ sin 2t + 1
(d) (3 points) y′′^ − 4 y′^ + 8y = t^2 e−t^ cos 5t
- (12 points) Given that y 1 (t) = t^3 is a known solution of the second order linear differential equation t^2 y′′^ − ty′^ − 3 y = 0, t > 0.
Find the general solution of the equation.
