). Use the eigenfunction method to solve the following PDE
on the square 0
≤
x
≤
1, 0
≤
y
≤
1, with boundary conditions
u
(
x,
0) =
u
(
x,
1) =
u
(0
, y
) =
u
(1
, y
) = 0. (Hint: Now you’ll have an inFnite sum.)
∂
2
u
∂x
2
+
∂
2
u
∂y
2
=
xy
6) (10 points) ²ind all eigenvalues and corresponding eigenfunctions for the regular Sturm
Liouville problem
y
′′
+
λy
= 0 with boundary conditions
y
′
(0) = 0 and
y
′
(1) = 0. (Hint: Not
all of the eigenvalues are positive.)
7) (10 points) ²ind the solution to the wave equation in polar coordinates:
∂
2
u
∂t
2
=
∇
2
u
where
u
(
r, t
) is radially symmetric, with
u
(1
, t
) = 0 for all
t
≥
0,
u
(
r,
0) =
J
0
(
α
2
r
) and
u
t
(
r,
0) =
J
0
(
α
4
r
) for all 0
< r <
1. Your answer will NOT be an inFnite sum.
8) (10 points) If the functions
f
(
r, θ
) and
g
(
r, θ
) are independent of
θ
, show that the solution
to the wave equation in polar coordinates (page 213) reduces to the solution found for the
symmetric case (page 202). In particular, show how
a
mn
,
b
mn
,
a
∗
mn
, and
b
∗
mn
reduce to
A
n
and
B
n
, and show that the double summation turns into a single sum.
9) (10 points) What is your favorite PDE, and why? Answers like “They all are my favorite!”
and “I don’t like any of them” are not acceptable.
10) (10 points) It’s your 10 free points! Hooray!!
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 Spring '11
 NormanKatz
 Eigenvalue, eigenvector and eigenspace, wave equation, Eigenfunction, infinite sum, Vibrations of a circular drum

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