Page 295 - Schaum's Outline of Differential Equations

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

28.5. Find a recurrence formula and the indicia! equation for an infinite series solution around x = 0 for the
differential equation given in Problem 28.4.
It follows from Problem 28.4 that x = 0 is a regular singular point of the differential equation, so Theorem 24.1
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 the coefficient of each power of x to zero, we find

and, for n > 1,

or,

Equation (2) is a recurrence formula for this differential equation.
From (1), either a 0 = 0 or

It is convenient to keep a 0 arbitrary; therefore, we must choose X to satisfy (3), which is the indicial equation.

1
28.6. Find the general solution near x = 0 of 8x y" + lOxy' + (x - l)y = 0.
The roots of the indicial equation given by (3) of Problem 28.5 are Xj = ^, and X 2 = — j. Since Xj - X 2 = |,
the solution is given by Eqs. (28.5) and (28.6). Substituting X = ^ into the recurrence formula (2) of Problem 28.5
and simplifying, we obtain

Thus,

and

Substituting X = -y into recurrence formula (2) of Problem 28.5 and simplifying, we obtain

Thus,

and