thus simply connected; ([*f*]{}) disconnected non-$x$-simple set having a non-$x$-simple connected component. []{data-label="fig:VariationsOnXSimplicityAndConnectedness"}](Ris3){width=".9\linewidth"}

Fundamental sets of solutions of homogeneous linear ordinary differential equations depending on a parameter {#sec:FundamentalSolutionSetsOfLinODEsWithParameter}
============================================================================================================

Given a homogeneous linear $p$th order ordinary differential equation with the independent variable $x$ and the parameter $t$ and with continuous coefficients defined on an open set $\Omega\subseteq\mathbb R^2$ of $(t,x)$, the question is whether there exist $p$ continuous solutions[^2] of this equation with nonvanishing Wronskian on $\Omega$. In general, the answer is negative, as is illustrated by the following example.

\[ex:1stOrderHomLinODEWithVanishingWronskian\] Consider the linear homogeneous first-order ordinary differential equation $$\mathcal P\colon\quad u_x=\frac u{x^2+t^2}\quad\mbox{on}\quad \Omega=\mathbb R^2\setminus\{(0,0)\}.$$ For each fixed $t$, its general solution is $$u=C\exp\left(\frac1t\arctan\frac xt\right)\quad\mbox{if}\quad t\ne0, \qquad
u=C\exp\left(-\frac1x\right)\quad\mbox{if}\quad t=0,$$ where $C$ is an arbitrary constant. This solution is well defined on the entire $\Omega_t=\mathbb R$ if $t\ne0$, and should be considered separately on each $x$-semiaxis, $\mathbb R_{+}$ and $\mathbb R_{-}$, if $t=0$. The functions $$\begin{gathered}
u=\zeta^+(t)\exp\left(\frac1t\arctan\frac xt\right),\quad t>0,\ x\in\mathbb R,
\\[.5ex]
u=\zeta^-(t)\exp\left(\frac1t\arctan\frac xt\right),\quad t<0,\ x\in\mathbb R,\end{gathered}$$ where the parameter function $\zeta^+$ (resp. $\zeta^-$) runs through ${\rm C}(\mathbb R_{+})$ (resp. ${\rm C}(\mathbb R_{-})$), represent the general solutions of the equation $\mathcal P$ on the domains $\mathbb R_{+}\times\mathbb R$ and $\mathbb R_{-}\times\mathbb 