Page 296 - Schaum's Outline of Differential Equations

CHAP. 28] SERIES SOLUTIONS NEAR A REGULAR SINGULAR POINT 279

The general solution is

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

28.7. Find a recurrence formula and the indicia! equation for an infinite series solution around x = 0 for the
differential equation

It follows from Problem 28.2 that x = 0 is a regular singular point of the differential equation, so Theorem 28.1
holds. Substituting Eqs. (28.2) through (28.4) into the left side of the given differential equation and combining
coefficients of like powers of x, we obtain

Dividing by x^" and simplifying, we have

Factoring the coefficient of a n and equating each coefficient to zero, we find

and, for n > 1,

or,

Equation (2) is a recurrence formula for this differential equation.
From (_/), either a 0 = 0 or

It is convenient to keep a 0 arbitrary; therefore, We require A, to satisfy the indicial equation (3).

2
28.8. Find the general solution near x = 0 of 2x y" + 7x(x + I)/ - 3y = 0.
The roots of the indicial equation given by (3) of Problem 28.7 are A,j = -| and A, 2 = -3. Since A,j — A, 2 = |, the
solution is given by Eqs. (28.5) and (28.6). Substituting A, = £ into (2) of Problem 28.7 and simplifying,
we obtain

Thus,

and