# Parameter-dependent linear ordinary differential equations and topology of domains

## TL;DR

This paper explores how the solvability of parameter-dependent linear ODEs is influenced by the topology of the domain, providing a complete characterization and extending the analysis to Schwartz distributions.

## Contribution

It offers a comprehensive topological characterization of the solvability of parameter-dependent linear ODEs and extends the theory to Schwartz distributions.

## Key findings

- Solvability depends on topological properties of the domain.
- Complete characterization of solution existence for parameter-dependent equations.
- Extension of results to Schwartz distributions.

## Abstract

The well-known solution theory for (systems of) linear ordinary differential equations undergoes significant changes when introducing an additional real parameter. Properties like the existence of fundamental sets of solutions or characterizations of such sets via nonvanishing Wronskians are sensitive to the topological properties of the underlying domain of the independent variable and the parameter. We give a complete characterization of the solvability of such parameter-dependent equations and systems in terms of topological properties of the domain. In addition, we also investigate this problem in the setting of Schwartz distributions.