16

Series solutions of ordinary

differential equations

In the previous chapter the solution of both homogeneous and non-homogeneous

linear ordinary differential equations (ODEs) of order≥2 was discussed. In par-

ticular we developed methods for solving some equations in which the coefficients

were not constant but functions of the independent variablex. In each case we

were able to write the solutions to such equations in terms of elementary func-

tions, or as integrals. In general, however, the solutions of equations with variable

coefficients cannot be written in this way, and we must consider alternative

approaches.

In this chapter we discuss a method for obtaining solutions to linear ODEs

in the form of convergent series. Such series can be evaluated numerically, and

those occurring most commonly are named and tabulated. There is in fact no

distinct borderline between this and the previous chapter, since solutions in terms

of elementary functions may equally well be written as convergent series (i.e. the

relevant Taylor series). Indeed, it is partly because some series occur so frequently

that they are given special names such as sinx,cosxor expx.

Since we shall be concerned principally with second-order linear ODEs in this

chapter, we begin with a discussion of these equations, and obtain some general

results that will prove useful when we come to discuss series solutions.

16.1 Second-order linear ordinary differential equations

Any homogeneous second-order linear ODE can be written in the form

y′′+p(x)y′+q(x)y=0, (16.1)

wherey′=dy/dxandp(x)andq(x) are given functions ofx. From the previous

chapter, we recall that the most general form of the solution to (16.1) is

y(x)=c 1 y 1 (x)+c 2 y 2 (x), (16.2)