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2.7 Sturm–Liouville Problems 179

this section. In particular, the problem

φ′′+λ^2 φ= 0 , 0 <x<a,

φ( 0 )= 0 , hφ(a)+κφ′(a)= 0

is a regular Sturm–Liouville problem, in which

s(x)=p(x)= 1 , q(x)= 0 ,α 1 = 1 ,α 2 = 0 ,

β 1 =h,β 2 =κ.

All conditions of the definition are met.

2. A less trivial example is

(xφ′)′+λ^2

( 1

x

)

φ= 0 , 1 <x< 2 ,φ( 1 )= 0 ,φ( 2 )= 0.

We i d e n t i f ys(x)=x,p(x)= 1 /x,q(x)=0. This is a regular Sturm–

Liouville problem. The orthogonality relation is

∫ 2

1

φn(x)φm(x)^1

x

dx= 0 , n=m.

The conclusions of Theorems 1 and 2 hold for both examples. 

EXERCISES

1. The general solution of the differential equation in Example 2 is

φ(x)=c 1 cos

(

λln(x)

)

+c 2 sin

(

λln(x)

)

.

Find the eigenvalues and eigenfunctions, and verify the orthogonality rela-

tion directly by integration.

2. Check the results of Theorem 2 for the problem consisting of

φ′′+λ^2 φ= 0 , 0 <x<a,

with boundary conditions

a.φ( 0 )=0, φ(a)=0; b. φ′( 0 )=0, φ′(a)=0.

In caseb,λ^21 =0.

3. Find the eigenvalues and eigenfunctions, and sketch the first few eigenfunc-
   tions of the problem

φ′′+λ^2 φ= 0 , 0 <x<a,