c) Using the ±ourier series itself, prove that
∞
s
k
=0
1
(2
k
+ 1)
2
=
π
2
8
8)
(10 points)
a) Let
a
=
b
= 1,
f
1
(
x
) = 0,
f
2
(
x
) =
x
,
g
1
(
y
) = 0, and
g
2
(
y
) =
y
. Solve the Laplace
equation with boundary conditions
u
(
x,
0) =
f
1
(
x
),
u
(
x,
1) =
f
2
(
x
),
u
(0
, y
) =
g
1
(
y
), and
u
(1
, y
) =
g
2
(
y
).
b) Of course, your answer in part a) is really, really ugly. But there is a nice, pretty answer
for
u
(
x, y
). Can you Fnd a really simple expression of
u
(
x, y
)?
9)
(10 points) Consider the PDE
∂
2
u
∂x
2
+
∂
2
u
∂x∂y
+
∂
2
u
∂y
2
= 0
a) Verify that this PDE is elliptic.
b) Perform a change of variables on this PDE:
α
=
ax
+
by
β
=
cx
+
dy
Choose the constants
a
,
b
,
c
, and
d
in such a way to make
u
αα
+
u
ββ
= 0. In other words,
this PDE is just a change of basis away from Laplace’s equation.
c) The cubic polynomial
u
(
x, y
) =
x
3

3
xy
2
is a solution to Laplace’s equation. Determine
a cubic polynomial that is a solution to
u
xx
+
u
xy
+
u
yy
= 0.
10
(10 points) Eh, nothing is inspiring me. Ten free points for everyone!!!
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 Spring '11
 NormanKatz
 Critical Point, Partial differential equation, inﬁnite sum identity

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