Wronskians nonvanishing on $\Omega$ and no fundamental set of solutions on $\Omega$.

\[cor:WronkianOfFSS\] Given an open $x$-simple subset $\Omega$ of the $(t,x)$-plane, a solution set $\{\varphi^1,\dots,\varphi^p\}$ of a $p$th order linear ordinary differential equation $\mathcal P$: $Pu=0$ with $P\in{{\rm DO}}(\Omega)$ is fundamental on $\Omega$ if and only if the Wronskian of these solutions does not vanish on $\Omega$.

\[cor:OnOpenSetsWithXSimplePieces\] 1. If an open set $\Omega$ has an $x$-simple piece, then any differential equation $Pu=0$ with $P\in{{\rm DO}}(\Omega)$, possesses a solution that is not identically zero on $\Omega$.

2\. If there are $x$-simple pieces of $\Omega$ with overlapping projections to the $t$-axis, then any equation of the above form admits no fundamental set of solutions on $\Omega$.

Fix $P\in{{\rm DO}}(\Omega)$ with ${\mathop{\rm ord}\nolimits}P=p$ and let $U$ be an $x$-simple piece of $\Omega$. In view of Lemma \[lem:OnOpenX-SimpleRegion\], there exists a function $\theta\in {\rm C}^\infty({\mathop{\rm pr}\nolimits}_tU)$ whose graph is contained in $U$. For each $t\in{\mathop{\rm pr}\nolimits}_tU$, we consider the initial value problem for the equation $\mathcal P$: $Pu=0$ on $U_t$ with the initial conditions $u_{s-1}=\chi^s(t)$, $s=1,\dots,p$, at $x=\theta(t)$ and then vary $t$ through ${\mathop{\rm pr}\nolimits}_tU$. Here $\chi^1$, …, $\chi^p$ are (smooth) bump functions on ${\mathop{\rm pr}\nolimits}_tU$. The continuation of the solution of 