## ABSTRACT

T ′′ + c2λT = 0, Θ′′ + γΘ = 0, R′′ + 1

r R′ +

( λ− γ

r2

) R = 0,

where T = T (t), Θ = Θ(θ) and R = R(r), and λ and γ are separation constants. The R-ODE is the eigenvalue version of Bessel’s equation.

c) Use Exercise 18 in Section 1.7 to conclude that γ = m2, where

equation in the standard form of Bessel’s equation of order α,

x2y′′ + xy′ + (x2 − α2)y = 0 (where, for the vibrating membrane, of course, α = m = 0, 1, 2, . . .).

e) If, instead, λ < 0, use the change of variable x = √−λ r to rewrite