Find the general solution of the second-order differential equation

If the solution is not valid everywhere, describe the interval on which it is valid.

The general solution of the homogeneous equation

is given by Theorem 8.7 with and . This gives us ; hence, . Thus,

To find a particular solution of let . Then,

Therefore,

Setting the coefficients of like powers of to be equal and solving for the constants we get

By Theorem 8.8 (page 330 of Apostol) the general solution of the given differential equation is then