Page 298 - Schaum's Outline of Differential Equations

CHAP. 28] SERIES SOLUTIONS NEAR A REGULAR SINGULAR POINT 281

The stipulation n > 2 is required in (3) because a n _ 2 is not defined for n = 0 or n = 1. From (1), the indicial equation
2
is X = 0, which has roots, A,j = A^ = 0. Thus, we will obtain only one solution of the form of (28.5); the second solution,
y 2(x), will have the form of (28.7).
2
Substituting X = 0 into (2) and (3), we find that aj = 0 and a n = -(l/w )a n _ 2 - Since «i = 0, it follows that
Q = a 3 = a 5 = a 7= •••. Furthermore,

and, in general, , (k= 1, 2,3, ...). Thus,

28.11. Find the general solution near x = 0 to the differential equation given in Problem 28.10.
One solution is given by (4) in Problem 28.10. Because the roots of the indicial equation are equal, we use Eq. (28.8)
to generate a second linearly independent solution. The recurrence formula is (3) of Problem 28.10, augmented by (2)
of Problem 28.10 for the special case n = 1. From (2), aj = 0, which implies that 0 = a 3 = a s = a 7 = • • •. Then, from (3),

Substituting these values into Eq. (28.2), we have

Recall that In x. (When differentiating with respect to X, x can be thought of as a constant.) Thus,

and

which is the form claimed in Eq. (28.7). The general solution is y = c^^x) + C 2y 2(x).