(a) For this part, let
v
=
u

1, then
v
satisfies homogenous boundary condition, and
∇
2
v
=
r
cos(2
θ
), do
eigenfunction expansion for
v
,
v
(
a, θ
) = 0
→
f
(
a
) =
J
m
‡
√
λa
·
= 0
(6)
holds for
λ
6
= 0 (there is no nontrivial solution for
λ
= 0). Thus,
λ
=
λ
mn
=
j
m,n
a
¶
2
,
m
= 0
,
1
,
2
, . . . , n
= 1
,
2
,
3
, . . .
,
(7)
where
j
m,n
is the
n
th zero of
J
m
(
z
)
, z >
0
. So, the function
v
mn
=
A
mn
J
m
‡
p
λ
mn
r
·
cos (
mθ
) +
B
mn
J
m
‡
p
λ
mn
r
·
sin (
mθ
)
(8)
solves the PDE system
∇
2
v
mn
+
λ
mn
v
mn
= 0
(9)
with
v
mn
(
a, θ
) = 0. We therefore write
v
as the series
v
(
r, θ
) =
∞
X
m
=0
∞
X
n
=1
A
mn
J
m
‡
p
λ
mn
r
·
cos (
mθ
) +
∞
X
m
=1
∞
X
n
=1
B
mn
J
m
‡
p
λ
mn
r
·
sin (
mθ
)
,
(10)
which implies that
∇
2
v
=

∞
X
m
=0
∞
X
n
=1
λ
mn
A
mn
J
m
‡
p
λ
mn
r
·
cos (
mθ
)

∞
X
m
=1
∞
X
n
=1
λ
mn
B
mn
J
m
‡
p
λ
mn
r
·
sin (
mθ
)
.
(11)
We also write a series expansion for
r
cos(2
θ
):
r
cos(2
θ
) =
∞
X
m
=0
∞
X
n
=1
C
mn
J
m
‡
p
λ
mn
r
·
cos (
mθ
) +
∞
X
m
=1
∞
X
n
=1
D
mn
J
m
‡
p
λ
mn
r
·
sin (
mθ
)
,
(12)
where (by orthogonality of the eigenfunctions)
C
mn
=
R
a
0
r
2
J
2
(
√
λ
2
n
r
)
dr
R
a
0
J
2
2
(
√
λ
2
n
r
)
rdr
m
= 2
0
m
6
= 2
(13)
and
D
mn
= 0
(14)
So,
∇
2
u
=
r
cos(2
θ
)
(15)
2