1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

2.7 Sturm–Liouville Problems 177
Let us carry out the procedure used in the preceding with this problem. The
eigenfunctions satisfy the differential equations
(
sφ′n

)′

−qφn+λ^2 npφn= 0 ,

(

sφm′

)′

−qφm+λ^2 mpφm= 0.

Multiply the first byφmand the second byφn, subtract (the terms containing
q(x)cancel), and move the term containingpφnφmto the other side:
(
sφ′n

)′

φm−

(

sφ′m

)′

φn=

(

λ^2 m−λ^2 n

)

pφnφm. (4)

Integrate both sides fromltor, and apply integration by parts to the left-hand
side:
∫r

l

[(

sφn′

)′

φm−

(

sφm′

)′

φn

]

dx

=

[

sφn′φm−sφm′φn

]∣∣r

l−

∫r

l

(

sφn′φm′ −sφn′φ′m

)

dx.

The second integral is zero. From the boundary conditions we find that

φn′(r)φm(r)−φm′(r)φn(r)= 0 ,

φn′(l)φm(l)−φm′(l)φn(l)= 0

by the same reasoning as before. Hence, we discover the orthogonality relation
∫r

l

p(x)φn(x)φm(x)dx= 0 ,λ^2 n=λ^2 m

for the eigenfunctions of the problem stated.
During these operations, we have made some tacit assumptions about in-
tegrability of functions after Eq. (4). In individual cases, where the coefficient
functionss,q,andpand the eigenfunctions themselves are known, one can
easily check the validity of the steps taken. In general, however, we would like
to guarantee the existence of eigenfunctions and the legitimacy of computa-
tionsafterEq.(4).Todoso,weneedthefollowing.

Definition

The problem

(sφ′)′−qφ+λ^2 pφ= 0 , l<x<r, (5)

α 1 φ(l)−α 2 φ′(l)= 0 , (6)

β 1 φ(r)+β 2 φ′(r)=0(7)