## 13

## Series Solution of Ordinary Differential Equations and Special Functions

##### 13.0 INTRODUCTION

We have seen several methods of solving first order and second order differential equations. The solutions of these equations are expressed in closed form as a finite algebraic combination of elementary functions such as polynomial functions, trigonometric functions, exponential functions, logarithmic functions, and inverse trigonometric functions. However, the solutions of many differential equations arising from physical problems and important differential equations such as Bessel’s equation and Legendre’s equation cannot be expressed in terms of elementary functions in closed form.

In some such cases, it is easier to find an approximate solution ...