I'm learning about stochastic processes, and I want to solve
$$\mathop{dX_{t}} = \alpha X_{t} \mathop{dt} + \beta \mathop{dW_{t}}$$ where $X_{0} = x$.
I think that the solution uses Ito's Lemma; however, I am not too sure about how to get the answer. I also recognize this to be the Ornstein-Uhlenbeck process, and I think that the solution will be a diffusion process.
I would really appreciate some help.