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MATH 251
Examination I
July 3, 2012
FORM A
Name: Student Number: Section:
This exam has 10 questions for a total of 100 points. Show all you your work! 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. For other problems, points might be deducted, at the sole discretion of the instructor, for an answer not supported by a reasonable amount of 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.
- (8 points) Consider the differential equation
(t^2 − 9)y′^ −
t t − 1
y =
9 sin t t
Answer the following questions. Briefly explain your answers.
(a) (2 points) Is the equation linear?
(b) (2 points) Is the equation separable?
(c) (4 points) Given the initial condition y(2) = −2, without solving the equation, state the largest interval in which a unique solution is guaranteed to exist.
- (5 points) True or false: The function μ(t) = t^2 e−^3 t^ is a suitable integrating factor that can be used to solve the equation below. Justify your answer by finding the correct μ(t).
t^2 y′^ − (3t^2 − 2 t)y = e^4 t
- (12 points) A tank of 200-liter capacity initially contains 300 grams of salt dissolved in 100 liters of water. Starting at t = 0 additional solution with a concentration of 3 grams/liter begins to flow into the tank at the rate of 4 liter/min. The thoroughly mixed mixture is allowed to flow out of the bottom of the tank at the same rate.
(a) (4 points) Write down an initial value problems that describes the amount of salt, Q(t), in grams, that would be in the tank at any time t
(b) (6 points) Determine the quantity of salt that will be in the tank after 2 mins.
(c) (2 points) To what value would the concentration of salt in the solution in the tank ap- proach eventually (as t → ∞)?
- (10 points) Consider the equation
(2λx^5 y^3 −
x^2
) + (3x^6 y^2 − 4 λ)y′^ = 0.
(a) (4 points) Find the value of λ such that the equation becomes an exact equation.
(b) (6 points) Find its general solution. You may leave your answer in implict form.
- (13 points) Suppose y 1 (t) = 15t^3 and y 2 (t) = − 4 t^3 ln(t) are two solutions of a certain second order homogeneous linear equation
y′′^ + p(t) y′^ + q(t) y = 0.
(a) (4 points) Find the Wronskian W (y 1 , y 2 )(t).
(b) (2 points) True or false: y 1 and y 2 form a set of fundamental solutions of this equation. Why or why not?
(c) (3 points) Write down a general solution of the differential equation.
(d) (2 points) True or false: y 3 = 9t^6 ln(t) is also a solution of this equation. Why or why not?
(e) (2 points) True or false: y 4 = 0 is also a solution of this equation. Why or why not?
- (12 points) Consider the second order linear equation
y′′^ + 6y′^ + 10y = 0.
(a) (3 points) Find its general solution.
(b) (7 points) Find the solution satisfying the initial conditions y(5031) = 6, and y′(5031) = −1.
(c) (2 points) Let y(t) be the solution found in (b). Evaluate lim t→∞
y(t).
