Advanced Engineering Mathematics, 7th Edition

5.1 Solutions about Ordinary Points

INTRODUCTION

In Section 3.3 we saw that solving a homogeneous linear DE with constant coefficients was essentially a problem in algebra. By finding the roots of the auxiliary equation we could write a general solution of the DE as a linear combination of the elementary functions x^k, x^ke^αx, x^ke^αxcos βx, and x^ke^αxsin βx, k a nonnegative integer. But as pointed out in the introduction to Section 3.6, most linear higher-order DEs with variable coefficients cannot be solved in terms of elementary functions. A usual course of action for equations of this sort is to assume a solution in the form of infinite series and proceed in a manner similar to the method of undetermined coefficients (Section 3.4). In this section ...