First of all, don't use equals signs where things aren't equal. does NOT equal the rest of that stuff...
I would try .
Then
and
.
Substituting into your DE gives:
.
Therefore and
and .
This means .
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 .