Page 303 - Schaum's Outline of Differential Equations

286 SERIES SOLUTIONS NEAR A REGULAR SINGULAR POINT [CHAP. 28

and, in general,

Hence,

which can be simplified to

28.19. Find the general solution near x = 0 to the differential equation given in Problem 28.18.
One solution is given by (3) in Problem 28.18 for the indicial root A,j = 1. If we try the method of Frobenius
with the indicial root X 2 = -2, recurrence formula (2) of Problem 28.18 becomes

which does define all a n(n > 1). Solving sequentially, we obtain

k
and, in general, a k = (-I) a 0lk\. Therefore,

This is precisely in the form of (28.9), with rf_j = 0 and d n = (-I)"a 0/n\. The general solution is

28.20. Find a general expression for the indicial equation of (28.1).
2
Since x = 0 is a regular singular point; xP(x) and x Q(x) are analytic near the origin and can be expanded in
Taylor series there. Thus,

2
Dividing by x and x , respectively, we have

Substituting these two results with Eqs. (28.2) through (28.4) into (28.1) and combining, we obtain

which can hold only if

Since a 0^0 (a 0 is an arbitrary constant, hence can be chosen nonzero), the indicial equation is