I am following this ARTICLE on solving the differential equations. Take a closer look at page 580, chapter "11.7 Nonhomogeneous Linear Systems" - Theorem 23.
Follwing this work, brought me to the following equation $$x_1(t)=e^{At}x_{1.0}+e^{At}\int_{t_0}^te^{-Ar}F_1(r)dr$$ Now my problem here is that $F_1(r)$ is in fact a function of $x(t)$ for example $$F(r)=F_0\sin(x(r))$$ where $x(r)$ under the integral is the same as $x(t)$ - is the function that I am looking for!
I assume there is no way around that? Any other ideas how to solve a system
$$\frac{\mathrm{d} }{\mathrm{d} t}\begin{pmatrix} x\\ y\\ \dot x\\ \dot y \end{pmatrix}=\begin{pmatrix} O &I \\ M^-1 K&M^-1B \end{pmatrix}\begin{pmatrix} x\\ y\\ \dot x\\ \dot y \end{pmatrix}+\binom{\vec 0}{M^-1\vec F(t)}$$ where you can assume that all the matrices M, K, B are known.