Page 305 - Schaum's Outline of Differential Equations

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

28.24. Find the general solution near x = 0 of the hypergeometric equation

where A and B are any real numbers, and C is any real nonintegral number.
Since x = 0 is a regular singular point, the method of Frobenius is applicable. Substituting, Eqs. (28.2) through
(28.4) into the differential equation, simplifying and equating the coefficient of each power of x to zero, we obtain

as the indicial equation and

as the recurrence formula. The roots of (1) are A^ = 0 and A^ = 1 - C; hence, A : - A^ = C - 1. Since C is not an integer,
the solution of the hypergeometric equation is given by Eqs. (28.5) and (28.6).
Substituting A, = 0 into (2), we have

which is equivalent to

Thus

and y>i(x) = a QF(A, B; C; x), where

The series F(A, B; C; x) is known as the hypergeometric series; it can be shown that this series converges for -1 < x < 1.
It is customary to assign the arbitrary constant ag the value 1. Then y\(x) = F(A, B; C; x) and the hypergeometric series
is a solution of the hypergeometric equation.
To find y 2(x), we substitute A, = 1 - C into (2) and obtain

or

Solving for a n in terms of a 0, and again setting a 0 = 1, it follows that

The general solution is y = c^^x) + C 2y 2(x).