Higher Engineering Mathematics

I

Differential equations

52

Power series methods of solving

ordinary differential equations

52.1 Introduction

Second order ordinary differential equations that
cannot be solved by analytical methods (as shown
in Chapters 50 and 51), i.e. those involving vari-
able coefficients, can often be solved in the form
of an infinite series 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 equa-
tions, Bessel and gamma functions 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) use Maclaurins theorem (as explained in

Chapter 8).

52.2 Higher order differential

coefficients as series

The following is an extension of successive dif-
ferentiation (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, and so

on.

If we abbreviate

dy

dx

asy′,

d^2 y

dx^2

asy′′, ... and

dny

dxn

asy(n), theny′=aeax,y′′=a^2 eax, and the

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

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

d^7 y

dx^7

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

(ii) Ify=sinax,

y′=a cosax=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.

In general, y(n)=ansin

(

ax+

nπ

2

)

(2)

For example, if

y=sin 3x, then

d^5 y

dx^5

=y(5)

= 35 sin

(

3 x+

5 π

2

)

= 35 sin

(

3 x+

π

2

)

=243 cos 3x

(iii) Ify=cosax,

y′ =−asinax=acos

(

ax+

π

2

)

y′′ =−a^2 cosax=a^2 cos

(

ax+

2 π

2

)

y′′′ =a^3 sinax=a^3 cos

(

ax+

3 π

2

)

and so on.

In general, y(n)=ancos

(

ax+

nπ

2

)

(3)