Page 300 - Schaum's Outline of Differential Equations

CHAP. 28] SERIES SOLUTIONS NEAR A REGULAR SINGULAR POINT 283

from which we obtain

and, in general, a k = Thus,

28.15. Find the general solution near x = 0 to the differential equation given in Problem 28.14.
One solution is given by (3) in Problem 28.14 for the indicial. root A,j = 2. If we try the method of Frobenius
with the indicial root X 2 = 1, recurrence formula (2) of Problem 28.14 becomes

which leaves a 1; undefined because the denominator is zero when n=\. Instead, we must use (28.10) to generate a
second linearly independent solution. Using the recurrence formula (2) of Problem 28.14 to solve sequentially for
a n (n = 1, 2, 3, ...) in terms of X, we find

Substituting these values into Eq. (28.2) we obtain

and, since X - X 2 = X - 1,

Then

and

This is the form claimed in Eq. (28.9), with d_i = -l, d 0 = a Q, di = 0, d 3 =|a 0,.... The general solution is
y = crf^x) + C 2y 2(x).