Mathematical Methods for Physics and Engineering - Matematica.NET

16Series solutions of ordinarydifferential equationsIn the previous chapter the solution of both homogeneous and non-homogeneouslinear ordinary differential equations (ODEs) of order ≥ 2 was discussed. In particularwe developed methods for solving some equations in which the coefficientswere not constant but functions of the independent variable x. In each case wewere able to write the solutions to such equations in terms of elementary functions,or as integrals. In general, however, the solutions of equations with variablecoefficients cannot be written in this way, and we must consider alternativeapproaches.In this chapter we discuss a method for obtaining solutions to linear ODEsin the form of convergent series. Such series can be evaluated numerically, andthose occurring most commonly are named and tabulated. There is in fact nodistinct borderline between this and the previous chapter, since solutions in termsof elementary functions may equally well be written as convergent series (i.e. therelevant Taylor series). Indeed, it is partly because some series occur so frequentlythat they are given special names such as sin x, cosx or exp x.Since we shall be concerned principally with second-order linear ODEs in thischapter, we begin with a discussion of these equations, and obtain some generalresults that will prove useful when we come to discuss series solutions.16.1 Second-order linear ordinary differential equationsAny homogeneous second-order linear ODE can be written in the formy ′′ + p(x)y ′ + q(x)y =0, (16.1)where y ′ = dy/dx and p(x) andq(x) are given functions of x. From the previouschapter, we recall that the most general form of the solution to (16.1) isy(x) =c 1 y 1 (x)+c 2 y 2 (x), (16.2)531