T ′′ + c2λT = 0, Θ′′ + γΘ = 0, R′′ + 1
r R′ +
( λ− γ
) 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