in Section \[sec:ExistenceOfSolutionsOfInhomLinODEsWithParameter\], where we derive necessary and sufficient conditions for solvability in terms of $x$-simplicity and also quantify the ‘degree of non-solvability’ in case some connected component of $\Omega$ fails to be $x$-simple. In Section \[sec:DistrSolutionsOfLinODEsWithParameter\] we then turn to the distributional setting, singling out the relevant case of ${\rm C}^0$-semiregular distributions. The final Section \[sec:LinSystemsOfODEsWithParameter\] is devoted to the study of systems of parameter-dependent linear ODEs. In Appendix \[sec:DistributionsWithVanishingPartialDerivatives\] we prove a structure theorem for distributions with vanishing partial derivatives on domains that are simple with respect to a variable. This is required for deriving the general form of distributional solutions to parameter-dependent (systems of) linear ODEs on such domains in Sections \[sec:DistrSolutionsOfLinODEsWithParameter\] and \[sec:LinSystemsOfODEsWithParameter\].

Sets simple with respect to a variable {#sec:xSimpleSets}
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Given a subset $U$ of the $(t,x)$-plane, by ${\rm lb}_U$ and ${\rm ub}_U$ we denote the *lower* and *upper bounds of $U$ in $x$*, which are functions from ${\mathop{\rm pr}\nolimits}_tU$ to $\mathbb R\cup\{-\infty,+\infty\}$ defined by $${\rm lb}_U(t):=\inf U_t,\quad {\rm ub}_U(t):=\sup U_t,\quad t\in{\mathop{\rm pr}\nolimits}_tU.$$ In view of the inequality ${\rm lb}_U(t)\leqslant x\leqslant{\rm ub}_U(t)$ for any $t\in{\mathop{\rm pr}\nolimits}_tU$ and any $x\in U_t$, the functions ${\rm lb}_U$ and ${\rm ub}_U$ may attain values only from $\mathbb R\cup\{-\infty\}$ 