Attempts at a General Einstein Equation for an Arbitrary FLRW Cosmology

I tried to derive a general Einstein field equation for an arbitrary FLRW cosmology. That is, one that can handle any of the possible spatial curvatures: hyperbolic, spherical, or flat. Deriving the equation was easy, solving it was not! It ends up being a nonlinear, second-order ODE, with singularities at a=0, which turns out to be the Big Bang singularity, which obviously is of physical significance. Anyways, here’s a log of my notebook, showing the attempts. More to follow!