Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

Settingz= 0, all terms in the sum withn>0 vanish, implying that

[σ(σ−1) +s(0)σ+t(0)]a 0 =0,

which, since we requirea 0 = 0, yields theindicial equation

σ(σ−1) +s(0)σ+t(0) = 0. (16.16)

This equation is a quadratic inσand in general has two roots, the nature of

which determines the forms of possible series solutions.

The two roots of the indicial equation,σ 1 andσ 2 , are called theindicesof

the regular singular point. By substituting each of these roots into (16.15) in

turn and requiring that the coefficients of each power ofzvanish separately, we

obtain a recurrence relation (for each root) expressing eachanas a function of

the previousar(0≤r≤n−1). We will see that the larger root of the indicial

equation always yields a solution to the ODE in the form of a Frobenius series

(16.12). The form of the second solution depends, however, on the relationship

between the two indicesσ 1 andσ 2. There are three possible general cases: (i)

distinct roots not differing by an integer; (ii) repeated roots; (iii) distinct roots

differing by an integer (not equal to zero). Below, we discuss each of these in turn.

Before continuing, however, we note that, as was the case for solutions in

the form of a simple power series, it is always worth investigating whether a

Frobenius series found as a solution to a problem is summable in closed form

or expressible in terms of known functions. We illustrate this point below, but

the reader should avoid gaining the impression that this is always so or that, if

one worked hard enough, a closed-form solution could always be found without

using the series method. As mentioned earlier, this isnotthe case, and very often

an infinite series solution is the best one can do.

16.3.1 Distinct roots not differing by an integer

If the roots of the indicial equation,σ 1 andσ 2 , differ by an amount that is not

an integer then the recurrence relations corresponding to each root lead to two

linearly independent solutions of the ODE:

y 1 (z)=zσ^1

∑∞

n=0

anzn,y 2 (z)=zσ^2

∑∞

n=0

bnzn,

with both solutions taking the form of a Frobenius series. The linear independence

of these two solutions follows from the fact thaty 2 /y 1 is not a constant since

σ 1 −σ 2 is not an integer. Becausey 1 andy 2 are linearly independent, we may use

them to construct the general solutiony=c 1 y 1 +c 2 y 2.

We also note that this case includes complex conjugate roots whereσ 2 =σ 1 ∗,

sinceσ 1 −σ 2 =σ 1 −σ∗ 1 =2iImσ 1 cannot be equal to a real integer.