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MATH 251
Final Exam
December 20, 2007
Name: Student Number: Section:
This exam has 15 questions for a total of 150 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.
- (6 points) What is the form of the general solution of the equation
y′′^ − y′^ − 2 y = 4t^2 e−t^?
(a) y(t) = At^3 e−t^ + Bt^2 e−t^ + Cte−t
(b) y(t) = t^2 (A cos t + B sin t)
(c) y(t) = At^3 e−t^ + Bt^2 e−t^ + Cte−t^ + De−t^ + Ee^2 t
(d) y(t) = t^2 e−t(A cos t + B sin t)
- (6 points) Find the inverse Laplace transform of the function
F (s) =
2(s − 1)e−^2 s s^2 − 2 s + 2
(a) 2u(t − 1)et^ cos t
(b) 2u(t − 2)et−^2 cos(t − 2)
(c) 2u(t − 2)et−^2 sin(t − 2)
(d) 2u(t − 2)et^ sin t
- (6 points) Which of the following second order homogeneous linear equations has y 1 (t) = e^2 t and y 2 (t) = te^2 t^ as two of its solutions?
(a) y′′^ + 2y′^ + y = 0
(b) y′′^ − 4 y′^ + 4y = 0
(c) y′′^ − 2 y = 0
(d) y′′^ + 2ty′^ = 0
- (6 points) The explicit solution of the initial value problem
dy dt
= y^2 cos(t) , y(0) = 1
is given by:
(a) y = t
(b) y = (^1) −sin(^1 t)
(c) y = 2 3
1 sin(t)+
(d) y =
1 2 (sin
(^2) (t) − 1)
- (6 points) Which of the following functions below is a solution of the wave equation
utt = 4uxx?
(a) e−^4 π^2 t^ sin(πx)
(b) sin(x − 2 t)
(c) x^2 + t^2
(d) 1 + 4 cos(t) + x^2
- (6 points) A mass-spring system with damping is described by the initial value problem
u′′^ + 6u′^ + 9u = 0, u(0) = 1, u′(0) = 0,
where u is the displacement of the mass from its equilibrium position. Then the motion of the mass is:
(a) Periodic.
(b) Oscillatory but not periodic.
(c) Critically damped and u(t) is a positive decreasing function of t for t > 0.
(d) Overdamped.
- (15 points) A college student borrows $5000 to buy a car. The lender charges interest at an annual rate of 10%. Assume the interest is compounded continuously and that the student makes payments continuously at a constant annual rate k. (a) Let S(t) denote the amount (in dollars) which the student owes at time t. Set up the differential equation for S(t).
(b) Solve the equation in (a) for S(t) using the given value of S(0).
(c) Determine the payment rate k that is required to pay off the loan in 5 years.
- (20 points) Find the eigenvalues and eigenfunctions of the boundary value problem
X′′^ + λX = 0, X(0) = 0, X(π/2) = 0.
(Show your work in all three cases: λ = 0, λ < 0 and λ > 0 .)
- (20 points) Let f be the periodic function with period 2π such that
f (x) =
2 , −π ≤ x < 0 , − 2 , 0 ≤ x < π.
(a) Find the Fourier series of the function f.
(b) To what value does the Fourier series in (a) converge at x = π?
- (20 points) The temperature distribution u(x, t) of a metal rod insulated at both ends is gov- erned by the initial-boundary value problem
ut = 5uxx, 0 < x < 3 , t > 0 , ux(0, t) = ux(3, t) = 0, u(x, 0) = 10 + 4 cos(2πx/3) − 2 cos(4πx/3).
(a) Solve the above initial-boundary value problem for u(x, t).
(b) What is the steady state temperature distribution?
