Page 301 - Schaum's Outline of Differential Equations

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

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2
28.16. Use the method of Frobenius to find one solution near x = 0 of Jt /' + xy' + (x - \)y = 0.
Here

so x = 0 is a regular singular point and the method of Frobenius is applicable. Substituting Eqs. (28.2) through (28.4)
into the left side of the differential equation, as given, and combining coefficients of like powers of x, we obtain

Thus,

and, for n > 2, [(A + nf - l]a n + a n _ 2 = 0, or,

2
From (_/), the indicial equation is X - 1 = 0, which has roots A^ = 1 and A 2 = —1. Since AJ - A^ = 2, a positive integer,
the solution is given by (28.5) and (28.9). Substituting X = 1 into (2) and (3), we obtain aj = 0 and

Since «i = 0, it follows that 0 = a 3 = a 5 = a 7= •••. Furthermore,

and, in general,

Thus,

28.17. Find the general solution near x = 0 to the differential equation given in Problem 28.16.
One solution is given by (4) in Problem 28.16 for the indicial root A : = 1. If we try the method of Frobenius
with the indicial root A 2 =—1, recurrence formula (3) of Problem 28.16 becomes

which fails to define a 2 because the denominator is zero when n = 2. Instead, we must use Eq. (28.10) to generate
a second linearly independent solution. Using Eqs. (2) and (3) of Problem 28.16 to solve sequentially for
a n(n= 1, 2, 3, ...) in terms of A, we find 0 = a l = a 3 = a 5 = ••• and

Thus,

Since A-A 2 = A + 1,