An exact solution could be found if only one were able to solve $$\int_0^te^{at'}\mathrm{d}W_{t'} $$ Using Ito's lemma, it's possible to convert the former to the following deterministic integral $$ \int_{0}^t e^{at'}W_{t'}\mathrm{d}t' $$ net of boundary terms. This procedure however shifts the problem but doesn't solve it. And simply calculating the mean and variance wouldn't solve it either, since for each given $\mu$ and $\sigma^2$ there are infinite different processes sharing these same numbers while being completely different from one another.
So? How should this be done?