336 Chapter 5 Higher Dimensions and Other Coordinates
of variables:
1
ρ^2{
∂
∂ρ(
ρ^2 ∂u
∂ρ)
+^1
sin(φ)∂
∂φ(
sin(φ)∂u
∂φ)}
= 0 ,
0 <ρ<c, 0 <φ<π, (1)
u(c,φ)=f(φ), 0 <φ<π. (2)
From the assumptionu(ρ, φ)=R(ρ)(φ), it follows that
(ρ^2 R′(ρ))′
R(ρ)+(sin(φ)′(φ))′
sin(φ)(φ)= 0.
Both terms are constant, and the second is negative,−μ^2 , because the bound-
ary condition atρ=cwill have to be satisfied by a linear combination of func-
tions ofφ. The separated equations are
(
ρ^2 R′
)′
−μ^2 R= 0 , 0 <ρ<c,( 3 )
(
sin(φ)′)′
+μ^2 sin(φ)= 0 , 0 <φ<π. ( 4 )Neither equation has a boundary condition. However,ρ=0 is a singular point
of the first equation, and bothφ=0andρ=πare singular points of the sec-
ond equation. (At these points, the coefficient of the highest-order derivative
is zero, while some other coefficient is nonzero.) At each of the singular points,
we impose a boundedness condition:
R( 0 )bounded,( 0 ) and (π)bounded.
Equation (4) can be simplified by the change of variables x=cos(φ),
(φ)=y(x).(Ofcourse,xisnotthe Cartesian coordinate.) By the chain rule,
the relevant derivatives are
d
dφ=−sin(φ)dy
dx,
d
dφ(
sin(φ)d
dφ)
=sin^3 (φ)d(^2) y
dx^2
−2sin(φ)cos(φ)dy
dx
.
The differential equation becomes
sin^2 (φ)d^2 y
dx^2 −2cos(φ)dy
dx+μ(^2) y= 0 ,
or, in terms ofxalone,
(
1 −x^2
)
y′′− 2 xy′+μ^2 y= 0 , − 1 <x< 1. (5)In addition, we require thaty(x)be bounded atx=±1.