Math 381 Fall 2006 Exam 1
Instructor: Sabin Cautis
Wednesday, October 11, 2006
Instructions: This is a closed book, closed notes exam. You have fifty minutes.
Do all the problems but notice there is a bonus problem which you should try
only after finishing all the other questions. Please do all your work on the paper
provided – using the extra page if you need extra space. You must show your
work to receive full credit on a problem.
Please print you name clearly here.
Print name:
Upon finishing please sign the pledge below:
On my honor I have neither given nor received any aid on this exam.
Grader’s use only:
1a.
/45
1b.
/10
2a.
/30
2b.
/15
Bonus.
/5
1. Consider the following partial differential equation
∂2u
∂u
=−
2
∂x
∂t
where u = u(x, t) is a function of x and t.
(a)[45 points] Use the separation of variables method to find the general
solution to this PDE on the interval [0, L] subject to the boundary conditions u(0, t) = 0 and u(L, t) = 0. Please show all your work including all
three cases (λ < 0, λ = 0, λ > 0) for the possible eigenvalues.
(b)[10 points] What is the general solution if the boundary conditions are
changed to u(0, t) = 0 and u(L, t) = 1? (you may assume the general
solution from part (a) is given).
2. Let f (x) = 1.
(a) [30 points] Find the sine Fourier series of f (x) on the interval [−2, 2]
(please show your work and simplify your answer as much as possible).
(b) [15 points] To what function does the sine Fourier series converge on
the interval [−4, 4] (draw a precise graph). Very briefly explain why this
is the case.
3. [5 points BONUS]
State the divergence theorem! (either in two or three dimensions) Hint: if
you don’t remember it, think of the fundamental theorem of calculus and
guess a generalization to two or three dimensions.