$\Omega$ can be represented as $\varphi=\tilde\zeta\varphi^1$, where $\tilde\zeta:=\varphi(t,1)\in {\rm C}(\mathbb R)$. In this sense the function $\varphi^1$ constitutes a fundamental set of solutions of this equation on $\Omega$.

\[def:FundamentalSolutionSetOfODEWithParameter\] Given a linear ordinary differential equation $\mathcal P$: $Pu=0$ on an open subset $\Omega$ of the $(t,x)$-plane, where $P\in{{\rm DO}}(\Omega)$ with ${\mathop{\rm ord}\nolimits}P=p$ and $t$ plays the role of a parameter, we say that functions $\varphi^s\in {\rm C}^p_x(\Omega)$, $s=1,\dots,p$, satisfying this equation constitute

-   a *fundamental set of solutions* of $\mathcal P$ on $\Omega$ if any solution $u$ of $\mathcal P$ can uniquely be represented in the form $u=\zeta^s\varphi^s$ for certain functions $\zeta^s\in {\rm C}({\mathop{\rm pr}\nolimits}_t\Omega)$;

-   a *locally fundamental set of solutions* of $\mathcal P$ on $\Omega$ if there exists an open cover of the set $\Omega$ (by which we shall always understand a cover consisting of open subsets of $\Omega$) such that the restriction of any solution $u$ of $\mathcal P$ to any element $U$ of the cover, $u\big|_U$, can uniquely be represented in the form $u\big|_U=\zeta^s\,\varphi^s\big|_U$ for certain functions $\zeta^s\in {\rm C}({\mathop{\rm pr}\nolimits}_tU)$.

\[lem:OnWronskianOfFundamentalSolutionSetOfODEWithParameter\] Any solutions $\varphi^s\in {\rm C}^p_x(\Omega)$, $s=1,\dots,p$, of an equation $Pu=0$ with $P\in{{\rm DO}}(\Omega)$ and ${\mathop{\rm ord}\nolimits}P=p$ that satisfy the condition $\mathrm W(\varphi^1,\dots,\varphi^p)\ne0$ on $\Omega$ 