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,
So, we obtain particular solutions of the of the homogeneous equation and (by taking and , respectively). We want to apply Theorem 8.9 (on page 330 of Apostol). From that theorem we have
So, a particular solution to the non-homogeneous equation is given by
Therefore, by Theorem 8.8 the general solution of the non-homogeneous equation is