  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.
