= Solution
For $\Omega_0>1$, introduce <conformal time> by $dt/d\tau=a$. Since $a'=a\dot a$, the constraint becomes
$$
(a')^2=H_0^2\Omega_0(1-\beta a^2).
$$
On the initial expanding branch, separate variables and choose $a=t=\tau=0$ at the <Big Bang>. With $\alpha=H_0\sqrt{\Omega_0-1}$,
$$
\boxed{a(\tau)=\frac{\sin(\alpha\tau)}{\sqrt\beta},\qquad
t(\tau)=\frac{1-\cos(\alpha\tau)}{\alpha\sqrt\beta}.}
$$
The second formula follows by integrating $dt/d\tau=a$. The model reaches maximum size at $\alpha\tau=\pi/2$ and recollapses at $\alpha\tau=\pi$.
Today $a=1$ and $H_0>0$ selects the expanding branch, so
$$
\tau_0=\alpha^{-1}\arcsin\sqrt\beta,
\qquad\cos(\alpha\tau_0)=\sqrt{1-\beta}=\Omega_0^{-1/2}.
$$
Substitution yields the requested age
$$
\boxed{t_0=H_0^{-1}\frac{\sqrt{\Omega_0}-1}{\Omega_0-1}
=\frac1{H_0(\sqrt{\Omega_0}+1)}.}
$$
In particular the numerator is $\sqrt{\Omega_0}-1$, not $\sqrt{\Omega_0-1}$; the PDF displays the former. The flat-model limit is $1/(2H_0)$.
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