Advanced Engineering Mathematics, 7th Edition

5.2 Solutions about Singular Points

INTRODUCTION

The two differential equations y″ − xy = 0 and xy″ + y = 0 are similar only in that they are both examples of simple linear second-order DEs with variable coefficients. That is all they have in common. Since x = 0 is an ordinary point of the first equation, we saw in the preceding section that there was no problem in finding two distinct power series solutions centered at that point. In contrast, because x = 0 is a singular point of the second DE, finding two infinite series solutions—notice we did not say “power series solutions”—of the equation about that point becomes a more difficult task.

A Definition

A singular point x = x₀ of a linear differential equation

a₂(x)y″ + a₁(x)y′ + a₀(x)