Existence of Solution for a Differential Matrix Equation I have some problems proving that a Differential Matrix Equation has a solution. I apologize if the question is too elementary, but I've found this theorem stated everywhere on the web without any reference or clue about how to prove it.
What I exactly mean with a Differential Matrix Equation is:
$X'=AX+B$. Where $A$ is a matrix of size $n\times n$ and $B$ is a column vector. Both $A$ and $B$ have coefficients which are holomorfic functions in a convex open set $\Omega$ and continuous on the closure $\bar\Omega$. We also have an initial condition $X(z_0)=U_0$, where $U_0$ is a column vector of complex coefficients.
So far I know how to prove it when $A$ has constant coefficients. However the proof cannot be applied as it uses the trick of $(e^{Az})'=Ae^{Az}$ to find an explicit solution and in general $(e^A)'\neq Ae^A$ if $A$ is nonconstant.
I've also read about Magnus Series, but I don't fully understand them. Also I'm only interested in the existence, so I'd prefer an easier way to prove that there are solutions.  
 A: This is more a comment than a solution, but I do not have the power. Have you checked in 
Daletskii-Krein: Stability of Solutions of Differential Equations in Banach Space, Chapter 6?
A: This really sounds like homework. Anyway, try this. We work under the assumption that $A,B$ are entire functions, otherwise everything works the same but the results are local.
If $X(z)=u+iv$ is holomorphic, its derivative can be written as $X'(z)=u_x+iv_x$. Now separate real part and imaginary part of your system, what you get is a nice linear system of $2n$ equations for $2n$ unknown functions, in the variable $x$, with $y$ as a parameter. Standard advanced calculus results give you the existence of a smooth global solution, smoothly dependent on the parameter $y$ (don't ask for references, just differentiate the system w.r.to $y$ and repeat). Now do the same trick writing $X'(z)=v_y-iu_y$: you get another set of $2n$ smooth functions. But of course $X'(z)=AX+B=u_x+iv_x=v_y-iu_y$ which means that $u,v$ satisfy the Cauchy-Riemann conditions and $X$ is holomorphic.
