I am having problem trying to find of the equation:
i tried but this doesn't work, because i am still left with a constant of .
Well you're obviously not going to get the same answer, because a sum of an exponential with a trigonometric function is not the same as a product of them.
I don't know why you're overcomplicating this. The easiest way is to use the same "family" of functions as you are given.
In this case, your "family" is the product of an exponential and the trigonometric functions.
The only time you would have to do differently is if this "family" appears in your homogenous solution. Then you have to multiply it by .