Page 287 - Schaum's Outline of Differential Equations

270 LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS [CHAP. 27

27.15. Use the power series method to find the general solution near x = 0 of

Using the results of Problem 27.14, we have that a 2 and a 3 are given by (4) and a n for (n = 4, 5, 6, ...) is given
by (3). It follows from this recurrence formula that

Thus,

The third series is the particular solution. The first and second series together represent the general solution of
2
the associated homogeneous equation (x +4)y" + xy = 0.
27.16. Find the recurrence formula for the power series solution around t=0 for the nonhomogeneous
2
2
differential equation (d yldt ) + ty = e t+1 .
Here P(t) = 0, Q(t) = t, and (f>(t) = e t+1 are analytic everywhere, so t=0 is an ordinary point. Substituting
Eqs. (27.5) through (27.7), with t replacing x, into the given equation, we find that

Recall that e t+1 has the Taylor expansion e' + = t" I n! about t = 0. Thus, the last equation can be rewritten as

Equating coefficients of like powers of f, we have

In general, (n + 2)(n+ l)a n + 2+ a n_ l= eln\ for n= 1, 2,..., or,

which is the recurrence formula for n= 1, 2, 3,.... Using the first equation in (_/), we can solve for a 2 = e/2.

27.17. Use the power series method to find the general solution near t = 0 for the differential equation given in
Problem 27.16.
Using the results of Problem 27.16, we have a 2 = e/2 and a recurrence formula given by Eq. (2). Using this
formula, we determine that