Chapter 6 - Power Series Solutions of Ordinary Differential Equations

Abstract

This chapter provides an overview of power series solutions of ordinary differential equations. It reviews the basic properties of power series and proofs of the major theorems. If the power series converges absolutely for all values of x, then the power series has infinite radius of convergence. A power series may be differentiated and integrated term-by-term on its interval of convergence. The chapter discusses Taylor's theorem. If two power series are equal, it means that their corresponding coefficients are equal. LogicalExpand is used to equate the coefficients. The chapter illustrates power series solutions using ordinary points, and defines ordinary point and singular point. It also shows power series solutions using regular singular points with various examples. If x0 is a singular point of y “(x) + p(x)(x)y’(x) + q(x)y (x)=0, x0 is a regular singular point that means that both (x - x0) p(x) and (x - x0)2 q(x) are analytic at x = x0. If x0 is not a regular singular point, x is called an irregular singular point.