Chapter 52

Power series methods

of solving ordinary

differential equations

52.1 Introduction

Second order ordinary differential equations that can-
not be solved by analytical methods (as shown in
Chapters 50 and 51), i.e. those involvingvariable coeffi-
cients,can often besolved in theformof an infiniteseries
of powers of the variable. This chapter looks at some of
the methods that make this possible—by the Leibniz–
Maclaurin and Frobinius methods, involving Bessel’s
and Legendre’s equations, Bessel and gamma func-
tions and Legendre’s polynomials. Before introducing
Leibniz’s theorem, some trends with higher differential
coefficients are considered. To better understand this
chapter it is necessary to be able to:

(i) differentiate standard functions (as explained in

Chapters 27 and 32),

(ii) appreciate the binomial theorem (as explained in

Chapters 7), and

(iii) useMaclaurinstheorem(asexplainedinChapter8).

52.2 Higher order differential

coefficients as series

The following is an extension of successive differentia-
tion (see page 296), but looking for trends, or series,

as the differential coefficient of common functions

rises.

(i) Ify=eax,then

dy

dx

=aeax,

d^2 y

dx^2

=a^2 eax,andso

on.

If we abbreviate

dy

dx

asy′,

d^2 y

dx^2

asy′′,...and

dny

dxn

asy(n),theny′=aeax,y′′=a^2 eax,andthe

emerging pattern gives: y(n)=aneax (1)

For example, ify=3e^2 x,then

d^7 y

dx^7

=y(^7 )= 3 ( 27 )e^2 x=384e^2 x

(ii) Ify=sinax,

y′=acosax=asin

(

ax+

π

2

)

y′′=−a^2 sinax=a^2 sin(ax+π)

=a^2 sin

(

ax+

2 π

2

)

y′′′=−a^3 cosx

=a^3 sin

(

ax+

3 π

2

)

and so on.