Page 282 - Schaum's Outline of Differential Equations

CHAP. 27] LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS 265

which is the recurrence formula for this problem.

27.3. Find the general solution near x = 0 of /' - xy' + 2y = 0.
Successively evaluating the recurrence formula obtained in Problem 27.2 for n = Q, 1,2, ... , we
calculate

Note that since a 4 = 0, it follows from the recurrence formula that all the even coefficients beyond a 4 are also zero.
Substituting (_/) into Eq. (27.5) we have

If we define

then the general solution (2) can be rewritten as y = a 0yi(x) + aiy 2(x).

21 A. Determine whether x = 0 is an ordinary point of the differential equation

Here P(x) = 0 and Q(x) = 1 are both constants; hence they are analytic everywhere. Therefore, every value of
x, in particular x = 0, is an ordinary point.

27.5. Find a recurrence formula for the power series solution around x = 0 for the differential equation given
in Problem 27.4.
It follows from Problem 27.4 that x = 0 is an ordinary point of the given equation, so Theorem 27.1 holds.
Substituting Eqs. (27.5) through (27.7) into the left side of the differential equation, we find