Mathematical Methods for Physics and Engineering : A Comprehensive Guide

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS

17.6 A useful generalisation

Sometimes we encounter inhomogeneous equations of a form slightly more gen-

eral than (17.1), given by

Ly(x)−μρ(x)y(x)=f(x) (17.53)

for some Hermitian operatorL, withysubject to the appropriate boundary

conditions andμa given (i.e.fixed) constant. To solve this equation we expand

y(x)andf(x) in terms of the eigenfunctionsyn(x) of the operatorL, which satisfy

Lyn(x)=λnρ(x)yn(x).

Working in terms of the normalised eigenfunctionsyˆn(x), we first expandf(x)

as follows:

f(x)=

∑∞

n=0

yˆn(x)

∫b

a

yˆ∗n(z)f(z)ρ(z)dz

=

∫b

a

ρ(z)

∑∞

n=0

yˆn(x)yˆ∗n(z)f(z)dz. (17.54)

Using (17.29) this becomes

f(x)=

∫b

a

ρ(x)

∑∞

n=0

ˆyn(x)yˆn∗(z)f(z)dz

=ρ(x)

∑∞

n=0

yˆn(x)

∫b

a

yˆ∗n(z)f(z)dz. (17.55)

Next, we expandy(x)asy=

∑∞
n=0cnyˆn(x) and seek the coefficientscn. Substituting
this and (17.55) into (17.53) we have

ρ(x)

∑∞

n=0

(λn−μ)cnˆyn(x)=ρ(x)

∑∞

n=0

yˆn(x)

∫b

a

yˆ∗n(z)f(z)dz,

from which we find that

cn=

∑∞

n=0

∫b

aˆy

∗

n(z)f(z)dz

λn−μ

.

Hence the solution of (17.53) is given by

y=

∑∞

n=0

cnyˆn(x)=

∑∞

n=0

yˆn(x)

λn−μ

∫b

a

ˆy∗n(z)f(z)dz=

∫b

a

∑∞

n=0

yˆn(x)yˆn∗(z)

λn−μ

f(z)dz.

From this we may identify the Green’s function

G(x, z)=

∑∞

n=0

yˆn(x)yˆ∗n(z)

λn−μ

.