Suppose we are given the following matrix differential equation in $X \in \Bbb R^{n \times n}$.
$$ \dot{X} (t) = AX(t) + X(t)A^{T} + M(t) $$
where $A \in \Bbb R^{n \times n}$ is given and the initial condition for is $X(t_0) =: X_0$. Find a solution $X(t)$.
How can I approach this problem? Any suggestions are appreciated.
If there were no $M(t)$ term, then the solution would be
$$X(t) = \Phi(t,t_0)X_0\Phi(t,t_0)^{T}$$
Indeed, this solution satisfies the both differential equation and the initial condition. I could not understand how $M(t)$ affects the solution?
For example, if I could write the solution like
$$ X(t) = \Phi(t,t_0)X $$
then solution for $\dot{X} = AX(t) + M(t)$ would be
$$ X(t) = \Phi(t,t_0)X_0 + \int_{t_0}^{t}\Phi(t,\tau)M(\tau)d\tau $$