Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

By demanding that the coefficients of each power ofzvanish separately, we obtain the
three-term recurrence relation

(n+2)an+2− 2 nan+1+(n−2)an=0 forn≥ 0 ,

which determinesanforn≥2intermsofa 0 anda 1. Three-term (or more) recurrence
relations are a nuisance and, in general, can be difficult to solve. This particular recurrence
relation, however, has two straightforward solutions. One solution isan=a 0 for alln,in
which case (choosinga 0 = 1) we find

y 1 (z)=1+z+z^2 +z^3 +···=

1

1 −z

.

The other solution to the recurrence relation isa 1 =− 2 a 0 ,a 2 =a 0 andan=0forn>2,
so that (again choosinga 0 =1)weobtainapolynomialsolution to the ODE:

y 2 (z)=1− 2 z+z^2 =(1−z)^2.
The linear independence ofy 1 andy 2 is obvious but can be checked by computing the
Wronskian

W=y 1 y′ 2 −y 1 ′y 2 =

1

1 −z

[−2(1−z)]−

1

(1−z)^2

(1−z)^2 =− 3.

SinceW= 0, the two solutionsy 1 andy 2 are indeed linearly independent. The general
solution of the ODE is therefore

y(z)=

c 1

1 −z

+c 2 (1−z)^2.

We observe thaty 1 (and hence the general solution) is singular atz=1,whichisthe
singular point of the ODE nearest toz= 0, but the polynomial solution,y 2 , is valid for
all finitez.

The above example illustrates the possibility that, in some cases, we may find

that the recurrence relation leads toan=0forn>N, for one or both of the

two solutions; we then obtain apolynomialsolution to the equation. Polynomial

solutions are discussed more fully in section 16.5, but one obvious property of

such solutions is that they converge for all finitez. By contrast, as mentioned

above, for solutions in the form of an infinite series the circle of convergence

extends only as far as the singular point nearest to that about which the solution

is being obtained.

16.3 Series solutions about a regular singular point

From table 16.1 we see that several of the most important second-order linear

ODEs in physics and engineering have regular singular points in the finite complex

plane. We must extend our discussion, therefore, to obtaining series solutions to

ODEs about such points. In what follows we assume that the regular singular

point about which the solution is required is atz= 0, since, as we have seen, if

this is not already the case then a substitution of the formZ=z−z 0 will make

it so.

Ifz= 0 is a regular singular point of the equation

y′′+p(z)y′+q(z)y=0