Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

of the indicial equation areσ=σ 1 andσ=σ 2 then it follows that

Ly(z, σ)=a 0 (σ−σ 1 )(σ−σ 2 )zσ. (16.27)

Therefore, as in the previous section, we see that fory(z, σ)tobeasolutionof

the ODELy=0,σmust equalσ 1 orσ 2. For simplicity we shall seta 0 = 1 in the

following discussion.

Let us first consider the case in which the two roots of the indicial equation

are equal, i.e.σ 2 =σ 1. From (16.27) we then have

Ly(z, σ)=(σ−σ 1 )^2 zσ.

Differentiating this equation with respect toσwe obtain

∂

∂σ

[Ly(z, σ)]=(σ−σ 1 )^2 zσlnz+2(σ−σ 1 )zσ,

which equals zero ifσ=σ 1. But since∂/∂σandLare operators that differentiate

with respect to different variables, we can reverse their order, implying that

L

[

∂

∂σ

y(z, σ)

]

=0 atσ=σ 1.

Hence, the function in square brackets, evaluated atσ=σ 1 and denoted by
[
∂
∂σ

y(z, σ)

]

σ=σ 1

, (16.28)

is also a solution of the original ODELy= 0, and is in fact the second linearly

independent solution that we were looking for.

The case in which the roots of the indicial equation differ by an integer is

slightly more complicated but can be treated in a similar way. In (16.27), sinceL

differentiates with respect tozwe may multiply (16.27) by any function ofσ,say

σ−σ 2 , and take this function inside the operatorLon the LHS to obtain

L[(σ−σ 2 )y(z, σ)]=(σ−σ 1 )(σ−σ 2 )^2 zσ. (16.29)

Therefore the function

[(σ−σ 2 )y(z, σ)]σ=σ 2

is also a solution of the ODELy= 0. However, it can be proved§that this

function is a simple multiple of the first solutiony(z, σ 1 ), showing that it is not

linearly independent and that we must find another solution. To do this we

differentiate (16.29) with respect toσand find

∂

∂σ

{L[(σ−σ 2 )y(z, σ)]}=(σ−σ 2 )^2 zσ+2(σ−σ 1 )(σ−σ 2 )zσ

+(σ−σ 1 )(σ−σ 2 )^2 zσlnz,

§For a fuller discussion see, for example, K. F. Riley,Mathematical Methods for the Physical Sciences

(Cambridge: Cambridge University Press, 1974), pp. 158–9.