176 Chapter 2 The Heat Equation
which is proved true if the left-hand side is zero:
∫r
l(
φn′′φm−φ′′mφn)
dx= 0.This integral is integrable by parts:
∫rl(
φn′′φm−φ′′mφn)
dx=
[
φ′n(x)φm(x)−φ′m(x)φn(x)]∣∣r
l−∫rl(
φ′nφm′−φ′mφ′n)
dx.Thelastintegralisobviouslyzero,sowehave
(
λ^2 m−λ^2 n)∫r
lφn(x)φm(x)dx=[
φn′(x)φm(x)−φm′(x)φn(x)]∣∣r
l.Bothφnandφmsatisfy the boundary condition atx=r,
β 1 φm(r)+β 2 φ′m(r)= 0 ,
β 1 φn(r)+β 2 φ′n(r)= 0.These two equations may be considered simultaneous equations inβ 1 andβ 2.
At least one of the numbersβ 1 andβ 2 is different from zero; otherwise, there
would be no boundary condition. Hence the determinant of the equations
must be zero:
φm(r)φ′n(r)−φn(r)φm′(r)= 0.A similar result holds atx=l.Thus
[
φn′(x)φm(x)−φ′m(x)φn(x)
]∣∣r
l=^0 ,and, therefore, we have proved the orthogonality relation
∫r
lφn(x)φm(x)dx= 0 , n=m,for the eigenfunctions of Eqs. (1)–(3).
We may make a much broader generalization about orthogonality of eigen-
functions with very little trouble. Consider the following model eigenvalue
problem, which might arise from separation of variables in a heat conduction
problem (see Section 9):
[
s(x)φ′(x)
]′
−q(x)φ(x)+λ^2 p(x)φ(x)= 0 , l<x<r,
α 1 φ(l)−α 2 φ′(l)= 0 ,
β 1 φ(r)+β 2 φ′(r)= 0.