Page 302 - Schaum's Outline of Differential Equations

CHAP. 28] SERIES SOLUTIONS NEAR A REGULAR SINGULAR POINT 285

and

Then

This is in the form of (28.9) with d -^ = d a = a a, di = 0, d 2 = a 0, di = 0, d 4 = •a 0,....The general solution is
y = c 1y 1(x) + c 2y 2(x).

2
2
28.18. Use the method of Frobenius to find one solution near x = 0 of x y" + (x + 2x)y' - 2y = 0.
Here
and

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

Dividing by x^, factoring the coefficient of a n, and equating to zero the coefficient of each power of x, we obtain

and, for n > 1,

which is equivalent to

2
From (_/), the indicial equation is X + X - 2 = 0, which has roots Xj = 1 and ^ = -2. Since Xj - ^ = 3, a positive
integer, the solution is given by Eqs. (28.5) and (28.9). Substituting X = 1 into (2), we obtain a n = [-11 (n + 3)]a n _ 1;
which in turn yields