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,

Then, let . This implies

Therefore,

Hence, which gives us the particular solution

Finally, the general solution is then

A particular solution to this excercise can also be found as the sum of the solutions of the previous two excercises (9 and 10 in section 8.17). The general solution follows from the theorem 8.8.

You can also assume that y1= p(x)e^x + q(x)e^2x.