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 $e^{A+B} \neq e^A \cdot e^B.$ )
Notice that $$\begin{eqnarray*} \frac{d}{dt} Y &=& \left(\frac{d}{dt} e^{t A}\right) e^{t B} + e^{t A} \left(\frac{d}{dt}e^{t B}\right) \\ &=& A e^{t A} e^{t B} + e^{t A}e^{t B} B \\ &=& A Y + Y B. \end{eqnarray*}$$ Since $A$ commutes with itself, it also commutes with $e^{t A}$, so $\frac{d}{dt} e^{t A} = A e^{t A} = e^{t A}A$. If you are unfamiliar with matrix calculus, here is not a bad place to start.