Show that, in terms of the variable t^=H0(t−t0), the evolution of the normalised scale factor obeys the equation
(dt^da)2=Ωm,0a−1+Ωr,0a−2+ΩΛ,0a2+1−Ωm,0−Ωr,0−ΩΛ,0.
Show that, when one is integrating this equation numerically, an iterative algorithm of the form
an+1≈an+(dt^da)nΔt^
would not be able to propagate the solution through points for which da/dt^=0.