constitute a locally fundamental set of solutions of this equation.

It suffices to consider a covering of $\Omega$ by balls $U_j:=B_{\varepsilon_j}(z_j)\subseteq\Omega$, $j\in J$, where $J$ is some index set, and $z_j=(t_j,x_j)\in\Omega$. For an arbitrary solution $u$ of the equation $Pu=0$ and for each of these balls, we have the representation , where the functions are defined, for each $t\in(t_j-\varepsilon_j,t_j+\varepsilon_j)$, as solutions of the system $\zeta^{js}(t)\varphi^s_{s'-1}(t,x_j)=u_{s'-1}(t,x_j)$, $s'=1,\dots,p$.

\[thm:OnFundSolutionSetOfLinODEsWithParameter\] Given an open subset $\Omega$ of the $(t,x)$-plane, the following are equivalent:

-   Any homogeneous linear ordinary differential equation $Pu=0$ with $P\in{{\rm DO}}(\Omega)$ admits a fundamental set of solutions on $\Omega$ with Wronskian nonvanishing on the entire $\Omega$.

-   $\Omega$ is an $x$-simple region.

[([*ii*]{})]{}${}\Rightarrow{}$[([*i*]{})]{}: Consider an arbitrary $P\in{{\rm DO}}(\Omega)$. In view of Lemma \[lem:OnOpenX-SimpleRegion\], there exists a function $\theta\in {\rm C}^\infty(I)$ with $I={\mathop{\rm pr}\nolimits}_t\Omega$ such that its graph is contained in $\Omega$. For each $t\in I$ and $s\in\{1,\dots,p\}$, we consider the initial value problem for the equation $Pu=0$ on $\Omega_t$ with the initial conditions $u_{s'-1}=\delta_{ss'}$, $s'=1,\dots,p$, at $x=\theta(t)$ and then vary $t$ through $I$. Here $\delta_{ss'}$ is the Kronecker delta. The collection of the solutions $\varphi^s\colon\Omega\to\mathbb R$ of the above problems then satisfies the required properties.

[([*i*]{})]{}${}\Rightarrow{}$[([*ii*]{})]{}: Supposing that the open set 