Given a differential matrix equation, ie $X'=A(z)X+B(z)$ where both $A$ and $B$ are matrix of size $n\times n$ with coefficients that are holomorfic functions in a convex open set $\Omega$ and continuous on the closure $\bar \Omega$, and an initial data: $X(z_0)=u$, I know there exists a solution.
However, I haven't been able to find on the internet a proof of the existence. So the question is how to prove it.
I already know it when $A(z)$ has constant coefficients, but it cannot be extended to this case. Also I've read about Magnus Series. Although I don't fully understand them, I'd prefer a easier proof of the existence, as I'm not really interested in a generic formula.