 Solution

Choose the Big Bang at $\tau =t=0$. For $0<\Omega <1$, the expanding singular solution of the Riccati equation is $\mathcal{H}=\beta \coth (\beta \tau /2)$. Integrating $a^{\prime }/a=\mathcal{H}$ gives $a=C{\sinh }^2(\beta \tau /2)$. The Friedmann constraint fixes $C=\Omega /(1-\Omega )$, rather than leaving an independent amplitude. ThusNow $dt=ad\tau$. Integrating from the singular origin givesThis derives the flat matter-coasting-fluid Friedmann solution. For small $\tau$, $a\propto {\tau }^2$ and $t\propto {\tau }^3$, hence $a\propto t^{2/3}$. At late time $a\propto t$, as expected when the coasting fluid dominates.