Page 297 - Schaum's Outline of Differential Equations

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

Substituting X = -3 into (2) of Problem 28.7 and simplifying, we obtain

Thus,

and, since a 4 = 0, a n = 0 for n > 4. Thus,

The general solution is

where ki = Cia 0 and k 2 = c 2a 0.

2
28.9. Find the general solution near x = 0 of 3.x )/' - xy' + y = 0.
2
Here P(x) = -l/(3x) and Q(x) = l/(3x ); hence, x = 0 is a regular singular point and the method of Frobenius is
applicable. Substituting Eqs. (28.2) through (28.4) into the differential equation and simplifying, we have
2
2
2
+
jt^SA, - 4A, + l]a 0 + ^ 1 + 1 [3X + 2X]a 1 + ••• +^ "[3(X + n) -4(X + n) + l]a n + ••• = 0
Dividing by x and equating all coefficients to zero, we find
and
From (1), we conclude that the indicial equation is 3A, 2 — 4A, + 1 = 0, which has roots A,j = 1 and A, 2 = i.

Since A,j — A, 2 = |, the solution is given by Eqs. (28.5) and (28.6). Note that for either value of A,, (2) is satisfied by
simply choosing a n = 0, n > 1. Thus,

and the general solution is

where k± = c^ and k 2 = c 2a Q.

28.10. Use the method of Frobenius to find one solution near x = 0 of x y" + xy' + x y = 0.
z
2
Here P(x) = 1/x and Q(x) = 1, 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,

2
and, for n > 2, (X, + n) a n + a n _ 2 = 0, or,