  We use Kovacic's algorithm to obtain all Liouvillian solutions, i.e.,
essentially all solutions in terms of quadratures, of the master equation which
governs the evolution of first order perturbations of the Schwarzschild
geometry. We show that all solutions in quadratures of this equation contain a
polynomial solution to an associated ordinary differential equation (ODE). This
ODE, apart from a few trivial cases, falls into the confluent Heun class. In
the case of the gravitational perturbations, for the Liouvillian solution $\chi
\int \frac {{\rm d}r_{\!\ast}}{\chi^{2}}$, we find in "closed form" the
polynomial solution P to the associated confluent Heun ODE. We prove that the
Liouvillian solution $\chi \int \frac {{\rm d}r_{\!\ast}}{\chi^{2}}$ is a
product of elementary functions, one of them being the polynomial P. We extend
previous results by Hautot and use the extended results we derive in order to
prove that P admits a finite expansion in terms of truncated confluent
hypergeometric functions of the first kind. We also prove, by using the
extended results we derive, that P admits also a finite expansion in terms of
associated Laguerre polynomials. We prove, save for two unresolved cases, that
the Liouvillian solutions $\chi$ and $\chi \int \frac {{\rm
d}r_{\!\ast}}{\chi^{2}}$, initially found by Chandrasekhar, are the only
Liouvillian solutions to the master equation. We improve previous results in
the literature on this problem and compare our results with theirs. Comments
are made for a more efficient implementation of Kovacic's algorithm to any
second order ODE with rational function coefficients. Our results set the stage
for deriving similar results in other black hole geometries 4-dim and higher.
