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,

We then find a particular solution of the non-homogeneous equation by assuming is a solution. This has derivatives

Therefore,

Integrating twice we can solve for ,

(We can ignore the constants of integration here since we only need to satisfy the equation . Any constants of integration are not going to matter when we take derivatives of .)

Therefore the particular solution is

and the general solution (by Theorem 8.8) is

2y’ + Y = 0; Y=e^-x/2

Sorry, what is theorem 8.7? Where can I find it?