Page 292 - Schaum's Outline of Differential Equations

CHAPTER 28

Series Solutions

Near a Regular

Singular Point

REGULAR SINGULAR POINTS

The point x 0 is a regular singular point of the second-order homogeneous linear differential equation

2
if XQ is not an ordinary point (see Chapter 27) but both (x — x 0)P(x) and (x — x 0) Q(x) are analytic at XQ. We only
consider regular singular points at XQ = 0; if this is not the case, then the change of variables t = x — XQ will translate
XQ to the origin.

METHOD OF FROBENIUS
Theorem 28.1. If x = 0 is a regular singular point of (28.1), then the equation has at least one solution of
the form

where A, and a n (« = 0, 1, 2, ...) are constants. This solution is valid in an interval
0 < x < R for some real number R.
To evaluate the coefficients a n and A, in Theorem 28.1, one proceeds as in the power series method of
Chapter 27. The infinite series

with its derivatives

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