Page 294 - Schaum's Outline of Differential Equations

CHAP. 28] SERIES SOLUTIONS NEAR A REGULAR SINGULAR POINT 277

Solved Problems

28.1. Determine whether x = 0 is a regular singular point of the differential equation

As shown in Problem 27.1, x = 0 is an ordinary pont of this differential equation, so it cannot be a regular
singular point.

28.2. Determine whether x = 0 is a regular singular point of the differential equation

2
Dividing by 2x , we have

As shown in Problem 27.7, x = 0 is a singular point. Furthermore, both

are analytic everywhere: the first is a polynomial and the second a constant. Hence, both are analytic at x = 0, and
this point is a regular singular point.

28.3. Determine whether x = 0 is a regular singular point of the differential equation

3
Dividing by x , we have

Neither of these functions is defined at x = 0, so this point is a singular point. Here,

The first of these terms is analytic everywhere, but the second is undefined at x = 0 and not analytic there. Therefore,
x = 0 is not a regular singular point for the given differential equation.

28.4. Determine whether x = 0 is a regular singular point of the differential equation

2
Dividing by &c , we have

Neither of these functions is defined at x = 0, so this point is a singular point. Furthermore, both

are analytic everywhere: the first is a constant and the second a polynomial. Hence, both are analytic at x = 0, and
this point is a regular singular point.