this problem by zero to $\Omega$ gives a solution of $\mathcal P$ on $\Omega$ as required.

Suppose that there is another $x$-simple piece $\tilde U$ of $\Omega$ such that ${\mathop{\rm pr}\nolimits}_t U\cap{\mathop{\rm pr}\nolimits}_t\tilde U\ne\varnothing$. We choose a value $t_0\in{\mathop{\rm pr}\nolimits}_t U\cap{\mathop{\rm pr}\nolimits}_t\tilde U$ and additionally set the condition $\chi^1(t_0)\ne0$ in the above construction, which results in a solution $\psi\in {\rm C}^p_x(\Omega)$ of $\mathcal P$ with ${\mathop{\rm supp}\nolimits}\psi\subset U$ and $\psi\big(t_0,\theta(t_0)\big)\ne0$. If the equation $\mathcal P$ admitted a fundamental set $\{\varphi^1,\dots,\varphi^p\}$ of solutions on $\Omega$, where $p={\mathop{\rm ord}\nolimits}P$, then the restrictions of these solutions to $U$ and to $\tilde U$ would form fundamental sets of solutions of $\mathcal P$ on $U$ and on $\tilde U$, respectively. Thus, Corollary \[cor:WronkianOfFSS\] would imply that the Wronskian of these solutions does not vanish on $U\cup\tilde U$. Let us analyze the expansion $\psi=\zeta^s\varphi^s$, where $\zeta^s\in {\rm C}({\mathop{\rm pr}\nolimits}_t\Omega)$. Since $\psi=0$ on $\tilde U$, the functions $\zeta^s$ would vanish on ${\mathop{\rm pr}\nolimits}_t\tilde U$, which contradicts the condition $\psi\big(t_0,\theta(t_0)\big)\ne0$.

If an open set $\Omega$ contains no $x$-simple pieces, then there may exist a differential equation $Pu=0$ with $P\in{{\rm DO}}(\Omega)$, possessing only the zero solution on $\Omega$:

\[ex:1stOrderODEWithNoNonzeroSolutions\] On the “infinitely punctured open square” $\Omega$ presented in Example 