Let us start with the equation for
q
(
θ
).
The second boundary condition in (12.50)
requires that
q
(
θ
) be 2
π
periodic.
Therefore, the required solutions are the elementary
trigonometric functions
q
(
θ
) = cos
mθ
or
sin
mθ,
where
μ
=
m
2
,
(12
.
53)
with
m
= 0
,
1
,
2
, . . .
a nonnegative integer.
Substituting the formula for the separation constant,
μ
=
m
2
, the differential equation
for
p
(
r
) takes the form
r
2
d
2
p
dr
2
+
r
dp
dr
+ (
λ r
2
−
m
2
)
p
= 0
,
0
≤
r
≤
1
.
(12
.
54)
Ordinarily, one imposes two boundary conditions in order to pin down a solution to such a
second order ordinary differential equation. But our Dirichlet condition, namely
p
(1) = 0,
only specifies its value at one of the endpoints. The other endpoint is a
singular point
for
the ordinary differential equation, because the coefficient of the highest order derivative,
namely
r
2
, vanishes at
r
= 0. This situation might remind you of our solution to the Euler
differential equation (4.100) in the context of separable solutions to the Laplace equation
on the disk. As there, we only require the solution to be bounded at
r
= 0, and so seek
eigensolutions that satisfy the boundary conditions

p
(0)

<
∞
,
p
(1) = 0
.
(12
.
55)
While (12.54) appears in a variety of applications, it is more challenging than any
ordinary differential equation we have encountered so far. Indeed, most solutions cannot
be written in terms of the elementary functions (rational functions, trigonometric functions,
exponentials, logarithms, etc.) you see in first year calculus. Nevertheless, owing to their
ubiquity in physical applications, its solutions have been extensively studied and tabulated,
and so are, in a sense, wellknown, [
105
,
142
].
To simplify the analysis, we make a preliminary rescaling of the independent variable,
replacing
r
by
z
=
√
λ r.
Note that, by the chain rule,
dp
dr
=
√
λ
dp
dz
,
d
2
p
dr
2
=
λ
d
2
p
dz
2
,
and hence
r
dp
dr
=
z
dp
dz
,
r
2
d
2
p
dr
2
=
z
2
d
2
p
dz
2
.