Page 285 - Schaum's Outline of Differential Equations

268 LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS [CHAP. 27

27.10. Find the general solution near t = 0 for the differential equation given in Problem 27.9.
We have from Problem 23.9 that

Then evaluating recurrence formula (2) in Problem 27.9 for successive integer values of n beginning with n = 1, we
find that

Substituting these values into Eq. (27.5) with x replaced by t, we obtain as the general solution to the given differ-
ential equation

27.11. Determine whether x = 0 or x= lisan ordinary point of the differential equation

for any positive integer n.
2
We first transform the differential equation into the form of Eq. (27.2) by dividing by x - 1. Then

Both of these functions have Taylor series expansions around x = 0, so both are analytic there and x = 0 is an
ordinary point. In contrast, the denominators of both functions are zero at x = 1, so neither function is defined there
and, therefore, neither function is analytic there. Consequently, x= 1 is a singular point.

27.12. Find a recurrence formula for the power series solution around x = 0 for the differential equation given
in Problem 27.11.
To avoid fractions, we work with the differential equation in its current form. Substituting Eqs. (27.5) through
(27.7), with the dummy index n replaced by k, into the left side of this equation, we have that

Combining terms that contain like powers of x, we obtain