  We derive a Sturm-Liouville system of equations for the exact calculation of
the survival probability in first-passage time problems. This system is the one
associated with the Wiener-Hopf integral equation obtained from the theory of
random walks. The derived approach is an alternative to the existing literature
and we tested it against direct calculations from both discrete- and
continuous-time random walks in a manageable, but meaningful, example. Within
this framework, the Sparre Andersen theorem results to be a boundary condition
for the system.
