In chapter 2, various theorems were discussed which guaranteed the existence and uniqueness of solutions to a single or a system of ordinary differential equations (ODEs). The method used was a constructive one, because it proved the existence of the solution by exhibiting the solution itself. This method of obtaining solutions is not particularly useful in general, and it is our purpose here to discuss the solution of differential equations in series form. We will limit the discussion to linear differential equations of the second order. The methods for higher-order differential equations are similar, but the analysis becomes tedious and complicated. In this chapter we will consider the solution of linear differential equations in which the coefficients are analytic or have at most a few singularities. Most of the special functions which occur in the solution of engineering problems arise as solutions of differential equations. In this way we are led to the functions of Bessel, Legendre, Laguerre, Hermite, and to the hypergeometric function.

4.1 SOME DEFINITIONS

A power series is an infinite series of the form

where a is the center and c _j are the coefficients of the power series.

is called the nth partial sum of the power series. If the nth partial sum is subtracted from the power series, the resulting expression is called the remainder,

If at a point x = x ₀, lim _{n ? ?} S _n( x ₀) exists and...