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A useful generalisation

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A useful generalisation
EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS
17.6 A useful generalisation
Sometimes we encounter inhomogeneous equations of a form slightly more general than (17.1), given by
Ly(x) − µρ(x)y(x) = f(x)
(17.53)
for some Hermitian operator L, with y subject to the appropriate boundary
conditions and µ a given (i.e. fixed) constant. To solve this equation we expand
y(x) and f(x) in terms of the eigenfunctions yn (x) of the operator L, which satisfy
Lyn (x) = λn ρ(x)yn (x).
Working in terms of the normalised eigenfunctions ŷn (x), we first expand f(x)
as follows:
b
∞
ŷn (x)
ŷn∗ (z)f(z)ρ(z) dz
f(x) =
n=0
a
b
∞
ρ(z)
=
a
ŷn (x)ŷn∗ (z)f(z) dz.
(17.54)
n=0
Using (17.29) this becomes
b
ρ(x)
f(x) =
a
∞
ŷn (x)ŷn∗ (z)f(z) dz
n=0
= ρ(x)
∞
b
ŷn (x)
a
n=0
ŷn∗ (z)f(z) dz.
(17.55)
Next, we expand y(x) as y = ∞
n=0 cn ŷn (x) and seek the coefficients cn . Substituting
this and (17.55) into (17.53) we have
b
∞
∞
ŷn (x)
ŷn∗ (z)f(z) dz,
(λn − µ)cn ŷn (x) = ρ(x)
ρ(x)
n=0
a
n=0
from which we find that
cn =
∞
n=0
b
a
ŷn∗ (z)f(z) dz
.
λn − µ
Hence the solution of (17.53) is given by
b
b
∞
∞
∞
ŷn (x)
ŷn (x)ŷn∗ (z)
y=
ŷn∗ (z)f(z) dz =
f(z) dz.
cn ŷn (x) =
λn − µ a
λn − µ
a
n=0
n=0
n=0
From this we may identify the Green’s function
G(x, z) =
∞
ŷn (x)ŷ ∗ (z)
n
n=0
572
λn − µ
.
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