The Chemistry Maths Book, Second Edition

13.2 The power-series method 369

13.2 The power-series method

Many important second-order linear differential equations

y′′ 1 + 1 p(x)y′ 1 + 1 q(x)y 1 = 1 r(x) (13.1)

have at least one particular solution that can be expressed as a power series,

(13.2)

The series is substituted into the differential equation to determine the numbers

a

0

, a

1

, a

2

, =, and the particular solutions for a physical system are obtained by the

application of the appropriate boundary or initial conditions.

The power-series method can be used whenp(x),q(x)andr(x)are polynomials

or if they can be expanded as power series in x. It can therefore be used to solve the

Legendre, associated Legendre and Hermite equations (the first two are transformed

into the standard form (13.1) by division by ( 11 − 1 x

2

)).

We demonstrate the power-series method in Examples 13.1 and 13.2 by solving

two equations for which we already know the solutions; a first-order equation in

Example 13.1 and a second-order equation in Example 13.2.

EXAMPLE 13.1Use the power-series method to solve the equation

By equation (13.2), we express the solution as the power series

Then

and, substituting in the differential equation,

= 1 (a

1

1 + 1 a

0

) 1 + 1 (2a

2

1 + 1 a

1

)x 1 + 1 (3a

3

1 + 1 a

2

)x

2

1 +1-

dy

dx

+= +ya axax()+ + + + +()aaxax+

12 3

2

01 2

2

23 

dy

dx

aaxax max

m

m

m

=+ + +=

=

−

∑

12 3

2

1

1

23 

∞

ya axax ax ax

m

m

m

=+ + + +=

=

∑

01 2

2

3

3

0



∞

dy

dx

+=y 0

yx a ax ax ax ax

m

m

m

()=+ + + +=

=

∑

01 2

2

3

3

0



∞