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 assume is a solution. Then,

Therefore,

Now, let , then we have

Thus, $y_1 = \frac{1}{4} x e^{2x}. \] Hence, the general solution is given by