Here is matrix differential equation: $$ \mu \frac {dX}{dt}=AX+XB+C(t) $$ $$ X(0) = X_0 $$
Here $\mu$ is real diagonal matrix, $X$ is $m$ by $n$ matrix. $A$, $B$ are real square matrices of constant coefficients.
Without $\mu$ it would be standard Sylvestor Matrix ODE. I'm looking for a solution of it or maybe a method to try.