Given a matrix $X(t)=e^{tA}$, we know that $X(t)$ is the solution of the following matrix differential equation: $ \frac{dX(t)}{dt} =X(t) \cdot A .$
Now could anyone help to construct a matrix differential equation in terms of $Y(t)$, such that $Y(t)=e^{tA} \cdot e^{tB}$ is its solution?
(NOTE: the matrices $A$ and $B$ do not commute, meaning that $e^{A+B} \neq e^A \cdot e^B.$ )