  The scope of this paper is to compare two different approaches for solving
the Wheeler-DeWitt (WDW) equation in the presence of homogeneous matter
(inflaton) and perturbations around it. The standard Born-Oppenheimer (BO)
decomposition, which consists of factorizing out the gravitational wave
function and then defining the flow of the time through it, and a more general
BO decomposition where the whole minisuperspace wave function is factorized
out. The two approaches are compared, for simplicity, in the case of a
minimally coupled inflaton with a flat potential. The consistency of the latter
decomposition is checked against the former by comparing the resulting
perturbation (Mukhanov-Sasaki) equations. Finally a few solutions to the
homogeneous WDW not suitable for the traditional BO treatment are presented and
the corresponding Mukhanov-Sasaki equations are evaluated.
