Let $a$, $b$, $c$, and $e$ be constants, $W_1$ and $W_2$ be Brownian motions with correlation $\rho$, and $f(t)$ and $g(t)$ be deterministic functions of time. Let $X$ satisfy $$d(X(t))=(aX(t)+ef(t)g(t))dt+f(t)X(t)dW_1(t)+g(t)X(t)dW_2(t).$$ Compute the expected value of $X(T)^2$ given $X(t)$ for some $0\le t\le T$.
If $e=0$, we can use Ito's rule to write $d(\log X)$ as an expression independent of $X$. Integrating gives that $X(T)|X(t)$ is log-normal. If $e\neq 0$, $d(\log X)$ is no longer independent of $X$. I can't think of a way around this issue.