= Solution
Use <natural units> with $c=1$, write $H=\dot a/a$, and measure <cosmic time> from the <Big Bang>. The <cosmological continuity equation> follows by differentiating the <Friedmann equation> and using the <Friedmann acceleration equation>; for <pressureless matter> it gives $\dot\rho+3H\rho=0$, hence $\rho=\rho_0a^{-3}$ when the present <scale factor> is $a_0=1$. The present <critical density> and <cosmological density parameter> give
$$
\rho_{c0}=\frac{3H_0^2}{8\pi G},\qquad \rho_0=\Omega_0\rho_{c0},\qquad k=H_0^2(\Omega_0-1)>0.
$$
Consequently, with $C=H_0^2\Omega_0$, the <Friedmann equation> becomes $\dot a^2=C/a-k$.
Introduce <conformal time> through $dt=a\,d\tau$. A prime denotes a <derivative> with respect to $\tau$, so $a'=a\dot a$ and
$$
(a')^2=Ca-ka^2=k\left[\left(\frac C{2k}\right)^2-\left(a-\frac C{2k}\right)^2\right].
$$
Set $u=\sqrt{k}\tau$, with $u=0$ at the <Big Bang>. On the expanding branch the substitution $a=C(1-\cos u)/(2k)$ satisfies this equation. Integrating $dt=a\,du/\sqrt{k}$, with $t(0)=0$, gives the <closed matter-dominated Friedmann solution>:
$$
\boxed{a(\tau)=\frac{\Omega_0}{2(\Omega_0-1)}(1-\cos u),\qquad t(\tau)=\frac{\Omega_0}{2H_0(\Omega_0-1)^{3/2}}(u-\sin u),\qquad u=\sqrt{k}\tau.}
$$
The parametrization continues smoothly through the maximum <scale factor>, reached at $u=\pi$, onto the contracting branch. Its curve in the $(t,a)$ plane is a rescaled <cycloid>.
For a presently expanding universe, $H_0>0$ selects $0<u_0<\pi$. The normalization $a(u_0)=1$ gives
$$
\cos u_0=\frac2{\Omega_0}-1,\qquad \sin u_0=\frac{2\sqrt{\Omega_0-1}}{\Omega_0},\qquad u_0=\arccos\left(\frac2{\Omega_0}-1\right).
$$
Substitution yields the <age of a closed dust universe>:
$$
\boxed{t_0=\frac{\Omega_0}{2H_0(\Omega_0-1)^{3/2}}\left[\arccos\left(\frac2{\Omega_0}-1\right)-\frac{2\sqrt{\Omega_0-1}}{\Omega_0}\right].}
$$
The first later zero of the <scale factor> occurs at $u=2\pi$, so the <lifetime of a closed dust universe> gives the <Big Crunch> time
$$
\boxed{t_{\rm BC}=\frac{\pi\Omega_0}{H_0(\Omega_0-1)^{3/2}}.}
$$
This is the total lifetime measured from the <Big Bang>; the remaining <cosmic time> is $t_{\rm BC}-t_0$. As a check, the <limit> $\Omega_0\downarrow1$ gives $H_0t_0\to2/3$, while the <Big Crunch> recedes to infinite time, as expected for a flat <matter-dominated universe> with zero <cosmological constant>.
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