  Differential equations of the form $\ddot R=-kR^\gamma$, with a positive
constant $k$ and real parameter $\gamma$, are fundamental in describing
phenomena such as the spherical gravitational collapse ($\gamma=-2$), the
implosion of cavitation bubbles ($\gamma=-4$) and the orbital decay in binary
black holes ($\gamma=-7$). While explicit elemental solutions exist for select
integer values of $\gamma$, more comprehensive solutions encompassing larger
subsets of $\gamma$ have been independently developed in hydrostatics (see
Lane-Emden equation) and hydrodynamics (see Rayleigh-Plesset equation). I here
present a universal explicit solution for all real $\gamma$, invoking the beta
distribution. Although standard numerical ODE solvers can readily evaluate more
general second order differential equations, this explicit solution reveals a
hidden connection between collapse motions and probability theory that enables
further analytical manipulations, it conceptually unifies distinct fields, and
it offers insights into symmetry properties, thereby enhancing our
understanding of these pervasive differential equations.
