& \dots & 0 & 0 \\
0 & 0 & 1 & \dots & 0 & 0 \\
0 & 0 & 0 & \dots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & \dots & 0 & 1 \\
-\frac{g^0}{g^p} & -\frac{g^1}{g^p} & -\frac{g^2}{g^p} & \dots & -\frac{g^{p-2}}{g^p} & -\frac{g^{p-1}}{g^p}
\end{pmatrix},
\quad
F = \begin{pmatrix}
0\\
0\\
0\\
\vdots\\
0\\
\frac{f}{g^p}
\end{pmatrix}.$$

In this paper we address the question of how this solution theory changes if the coefficient functions and the right hand side in are allowed to additionally depend on a real parameter $t$. Thus we shall be investigating ODEs of the form $$\label{eq:main_problem}
Pu \equiv \sum_{i=0}^p g^i(t,x) u_i = f(t,x),$$ where, analogously to the above, $u_i:={\partial}^i u/{\partial}x^i$, $i=0,\dots,p$, as well as systems of the form $$\label{eq:main_problem_system}
v_1 = A(t,x) v + F(t,x),$$ for $(t,x)$ varying in some open subset $\Omega$ of ${\mathbb R}^2$. Here $g^0,\dots,g^p,f\in {\rm C}(\Omega,\mathbb R^p)$, $g^p(t,x)\ne0$ for all $(t,x)\in\Omega$, $A\in {\rm C}(\Omega,\mathrm M_p(\mathbb R))$ and $F\in {\rm C}(\Omega,\mathbb R^p)$. We are, in particular, interested in determining the influence of the topology of $\Omega$ on the structure of the solution 