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

Furthermore,

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

You can also assume that y1= Ax + B + p(x)e^x.

Or find a particular solution for both 2xe^x and x separately and add them up