= 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 $a'/a=\mathcal H$ gives $a=C\sinh^2(\beta\tau/2)$. The Friedmann constraint fixes $C=\Omega/(1-\Omega)$, rather than leaving an independent amplitude. Thus
$$
\boxed{a(\tau)=\frac\Omega{2(1-\Omega)}[\cosh(\beta\tau)-1]}.
$$
Now $dt=a\,d\tau$. Integrating from the singular origin gives
$$
\boxed{t(\tau)=\frac{H_0^{-1}\Omega}{2(1-\Omega)^{3/2}}[\sinh(\beta\tau)-\beta\tau]}.
$$
This 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.
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