Perfect Fluid Cosmological Universes: One equation of state and the most general solution

Considering a homogeneous and isotropic universe characterized by the Friedmann-Lema\^itre-Robertson-Walker (FLRW) line element, in this work, we have prescribed a general formalism for the cosmological solutions when the equation of state of the cosmic substance follows a general structure $\phi (p, \rho) = 0$, where $p$, $\rho$ are respectively the pressure and the energy density of the cosmic substance. Using the general formalism we recover some well known solutions, namely, when the cosmic substance obeys the linear equation of state, a Chaplygin type equation of state, or a nonlinear equation of state. Thus, the current work offers a new technique to solve the cosmological solutions without any prior relation between $p$ and $\rho$.