the case, by the Liouville–Ostrogradski formula we obtain$\mathrm W(\varphi^1,\dots,\varphi^p)(t,\tilde x_2+\varepsilon)
=\mathrm W(\varphi^1,\dots,\varphi^p)(t,\tilde x_1-\varepsilon)e^{-G(t)}\to\infty$, $t\to\tilde t_0{}^-$, which contradicts the continuity of $\varphi^1$, …, $\varphi^p$ at $(\tilde t_0,\tilde x_2+\varepsilon)$.

If a connected component of an open set of $\Omega$ is not an $x$-simple region, then for each $p\in\mathbb N$ there exists an infinite-parameter family of equations of the form $Pu=0$ with $P\in{{\rm DO}}^\omega_1(\Omega)$ of order $p$ such that the Wronskian of any $p$ solutions of any of them vanishes on the same line segment $\{t_0\}\times[x_1,x_2]$ contained in $\Omega$.

We follow the proof of Theorem \[thm:OnFundSolutionSetOfLinODEsWithParameter\] and consider an operator $P\in{{\rm DO}}^\omega_1(\Omega)$ of the form $P=\sum_{q=0}^pg^q{\partial}_x^q$, where $g^p(t,x)=1$, $g^{p-1}(t,x)=-f(t,x)\big((x-\tilde x_1)^2+(t-\tilde t_0)^2\big)^{-1}$ for $(t,x)\in\Omega$, $g^q$, $q=0,\dots,p-2$, are arbitrary elements of ${\rm C}^\omega(\Omega)$, and $f$ is an arbitrary positive function in ${\rm C}^\omega(\Omega)$ that is separated from zero on the intersection of a neighborhood of $(\tilde t_0,\tilde x_1)$ with $\Omega$. The coefficients of the Taylor expansions of the functions $f$ and $g^q$, $q=0,\dots,p-2$, can serve as parameters of the family of equations $Pu=0$, which obviously has the required properties.

If each connected component of an open non-$x$-simple set $\Omega$ is $x$-simple, then any equation $Pu=0$ with $P\in{{\rm DO}}(\Omega)$ admits sets of ${\mathop{\rm ord}\nolimits}P$ solutions with 