I have a question regarding the Ornstein -Uhlenbeck process. We have a simplified version with Stochastic Integral Equation: $X_t=-a\int^t_0 X_s\,ds +B_t$. B is the Brownian motion.
And its analytic solution is $X_t=e^{-at}\int^t_0 e^{as}\,dB_s$.
How do I prove that this is the case? I know that $e^{-at}=-a\int^t_0 e^{-as}\,ds$ and that I'm somehow supposed to make use of this result but I am still unable to get from the analytic solution to the integral equation.
Thanks!