The Chemistry Maths Book, Second Edition

372 Chapter 13Second-order differential equations. Some special functions

in whichb(x)andc(x)are polynomials or can be expanded as power series in x. It

includes the simple power-series method discussed in the previous section as a special

case and can be used for all the equations in Table 13.1, and for many of the linear

differential equations in the sciences.

Every differential equation of the form (13.3) has at least one solution that can be

expressed as

(13.4)

where r(which may be zero) is an indicial parameterto be determined, anda

0

1 ≠ 10.

We consider first the case ofb(x) 1 = 1 b

0

, c(x) 1 = 1 c

0

, whereb

0

andc

0

are constants:

x

2

y′′ 1 + 1 b

0

xy′ 1 + 1 c

0

y 1 = 1 0 (Euler–Cauchy equation) (13.5)

(for convenience, the equation has been multiplied byx

2

). We have

y 1 = 1 x

r

[a

0

1 + 1 a

1

x 1 +1-]

y′ 1 = 1 x

r− 1

[ra

0

1 + 1 (r 1 + 1 1)a

1

x 1 +1-] (13.6)

y′′ 1 = 1 x

r− 2

[r(r 1 − 1 1)a

0

1 + 1 (r 1 + 1 1)ra

1

x 1 +1-]

and substitution into (13.5) gives

x

r

[r(r 1 − 1 1)a

0

1 + 1 (r 1 + 1 1)ra

1

x 1 +1-]

• 1 b

0

x

r

[ra

0

1 + 1 (r 1 + 1 1)a

1

x 1 +1-] 1 + 1 c

0

x

r

[a

0

1 + 1 a

1

x 1 +1-] 1 = 10

or, in full,

(13.7)

The coefficient of each power of xin this equation must be zero so that, for the

coefficient ofx

r

(m 1 = 10 ),

r

2

1 + 1 (b

0

1 − 1 1)r 1 + 1 c

0

1 = 10 (13.8)

This is called the indicial equation, and the roots are the possible values of the

parameter rin (13.4). In general,b(x)andc(x)in (13.3) are polynomials or can be

expanded as power series in x:

b(x) 1 = 1 b

0

1 + 1 b

1

x 1 + 1 b

2

x

2

1 + 1 b

3

x

3

1 +1-, c(x) 1 = 1 c

0

1 + 1 c

1

x 1 + 1 c

2

x

2

1 + 1 c

3

x

3

1 +1-

but the indicial equation (13.8) remains valid.

m

m

mr

rm b rm cax

=

+

∑

++−++













=

0

2

00

10

∞

()()()

yx x a ax ax ax x ax

rr

m

m

m

() (=++++=)

=

∑

01 2

2

3

3

0



∞