Let $X_0$ be a standard normal random variable and suppose that $$dX_t=-\frac{1}{2}X_tdt+dB_t.$$ $X_0$ is independent of the Brownian motion. Find the distribution of $X_t$ for $t\geq0$ and find $\text{cov}(X_t,X_s)$ for all $t,s$.
So far, I have attempted to solve the SDE using Ito's lemma, but this has seemed to imply that there exists no solution of the form $X_t=f(t,B_t)$. So I am beginning to think that maybe it is possible to describe the distribution without explicitly solving the SDE. But I'm not sure how to do this - any advice would be greatly appreciated!