You will need to solve the homogeneous problem
Then you will find a particular solution you can do this using the method of undetermined coefficients. Then you add the two solutions together.
Okay, I will go through the particular solution first. I will let denote the particular solution. Since the nonhomogeneous part is a trigonometric function we will let the particular solution be of the form . Now we must determine the coefficients . To do this we will plug back into the original DEQ.
and so
.
Thus we have
and
.
Therefore, we have
and
,
so
.
Finally, we see that our particular solution is
.
Deleted mistake see following post.
Sorry I made a mistake. The homogeneous solution is wrong. The solution of the characteristic equation is not . The solution is .
Therefore, the homogeneous solution is of the form
.
Finally, the general solution should be
.
To determine the constants we must have initial conditions.