= Solution
For $0<\Omega<1$, choose the <Big Bang> to occur at $\tau=t=0$ and retain the expanding solution. The first integral found above is
$$
a'^2=H_0^2(\Omega a+ba^2).
$$
Use $a=(\Omega/b)\sinh^2u$. Substitution gives $u'=H_0\sqrt b/2=\alpha/2$, so $u=\alpha\tau/2$. This gives the <flat matter-coasting-fluid Friedmann solution>:
$$
\boxed{a(\tau)=\frac{\Omega}{b}\sinh^2\frac{\alpha\tau}{2}=\frac{\Omega}{2b}\,[\cosh(\alpha\tau)-1].}
$$
Integrate $dt/d\tau=a$, with the same zero of time:
$$
\boxed{t(\tau)=\frac{\Omega}{2b\alpha}[\sinh(\alpha\tau)-\alpha\tau]=\frac{H_0^{-1}\Omega}{2b^{3/2}}[\sinh(\alpha\tau)-\alpha\tau].}
$$
As a check, $a'=(\Omega\alpha/2b)\sinh(\alpha\tau)$, and the identity $\sinh^2v=(\cosh v-1)(\cosh v+1)$ verifies $a'^2=H_0^2(\Omega a+ba^2)$. Also $t'=a$ exactly. At early times $a\simeq\Omega H_0^2\tau^2/4$ and $t\simeq\Omega H_0^2\tau^3/12$, reproducing the matter-dominated relation $a\propto t^{2/3}$.
The endpoint $\Omega=1$ is obtained by taking the smooth limit: $a=H_0^2\tau^2/4$ and $t=H_0^2\tau^3/12$. At $\Omega=0$ the <coasting cosmic-string universe> instead has $a=H_0t$ and $\mathcal H=H_0$. Its <Big Bang> is at $\tau=-\infty$, so a finite conformal-time origin at the bang is no longer available; choosing $a=1$ at $\tau=0$ gives $a=e^{H_0\tau}$ and $t=H_0^{-1}e^{H_0\tau}$. The physical-age limit remains regular.
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