Differential Equations

6

Integration in Series: Legendre, Bessel and Chebyshev Functions

6.1 LEGENDRE FUNCTIONS

6.1.1 Introduction

If a homogeneous linear differential equation of the form L(y) = (Σa_rD^r)y = 0 has constant coefficients a_r, it can be solved by elementary methods, and the functions involved in solutions are elementary functions such as xⁿ, e^x, log x, sin x etc. However, if such an equation has variable coefficients and functions of the independent variable x, it has to be solved by other methods. Legendre's equation and Bessel's equation are two very important equations of this type. We take up below, solution of these equations by application of two standard methods of solution:

1. The power series method and
2. The Frobenius¹ method, which is an extension ...