1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

2.8 Expansion in Series of Eigenfunctions 181

8. Showthat0isaneigenvalueoftheproblem

(sφ′)′+λ^2 pφ= 0 , l<x<r,

φ′(l)= 0 ,φ′(r)= 0 ,

wheresandpsatisfy the conditions of a regular Sturm–Liouville problem.

9. Find all values of the parameterμfor which there is a nonzero solution of
   this problem:

φ′′+μφ= 0 ,

φ( 0 )+φ′( 0 )= 0 ,φ(a)+φ′(a)= 0.

One solution is negative. Does this contradict Theorem 2?

2.8 Expansion in Series of Eigenfunctions

We have seen that the eigenfunctions that arise from a regular Sturm–Liouville
problem

(sφ′)′−qφ+λ^2 pφ= 0 , l<x<r, (1)

α 1 φ(l)−α 2 φ′(l)= 0 , (2)

β 1 φ(r)+β 2 φ′(r)=0(3)

are orthogonal with weight functionp(x):
∫r

l

p(x)φn(x)φm(x)dx= 0 , n=m, (4)

and it should be clear, from the way in which the question of orthogonality
arose, that we are interested in expressing functions in terms of eigenfunction
series.
Suppose that a functionf(x)is given in the intervall<x<rand that we
wish to expressf(x)in terms of the eigenfunctionsφn(x)of Eqs. (1)–(3). That
is, we wish to have

f(x)=

∑∞

n= 1

cnφn(x), l<x<r. (5)

The orthogonality relation Eq. (4) clearly tells us how to compute the coef-
ficients. Multiplying both sides of the proposed Eq. (5) byφm(x)p(x)(where