Consider a general linear SDE, $$ dX_t = (a(t)X_t+b(t))dt+(g(t)X_t+h(t))dW_t $$ an explicit solution to this has been given in detail here: Solution to General Linear SDE The solution is written as: $$ X_t = Y_t \left( X_0 + \int_0^t (b(s)-h(s)g(s)) Y^{-1}_s ds + \int_0^t h(s) Y^{-1}_s dW_s \right) $$ Here $Y_t$ is the solution to the homogeneous case, $$ Y_t = \exp \left( \int_0^t (a(s)-g^2(s)/2) ds + \int_0^t g(s) dW_s \right) $$
What I would like to understand is exactly how to get the probability distribution of $X_t$. In particular, I can't see how to solve the integrals that contain $Y^{-1}_s$
