spaces of  and of . Basic examples show that solvability may completely break down already for very simple sets $\Omega$ (e.g. for $\Omega$ the punctured plane, cf. Example \[ex:1stOrderInhomLinODEWithoutSolutions\] below). Conversely, for nice enough domains, e.g. for rectangles, the parameter-dependent theory is practically the same as in the single-variable case. We want to find out which properties of the domain determine the solvability of linear parameter-dependent ODEs. Indeed, we will completely characterize the solvability of  and of  in terms of a topological property of $\Omega$, namely the so-called $x$-simplicity of $\Omega$, a notion well-known from elementary integration theory (cf., e.g., ). In addition, we characterize the existence of fundamental sets of solutions of  or of fundamental matrices of , and the nonvanishing of the corresponding Wronskians, again in terms of the $x$-simplicity of $\Omega$ (or its connected components).

As may also be viewed as a specific kind of linear partial differential equation, the question of existence of solutions in terms of properties of the underlying domain bears some resemblance to notions like Hörmander’s concept of $P$-convexity [@Hoermander1976; @Hoermander1983_2]. In particular, it is of interest to address the problems stated above also within the framework of Schwartz distributions.

In the remainder 