= Solution
For $0<\Omega_M<1$, set $x=a^{3/2}$ and choose the expanding branch of the <Friedmann equation>. Then
$$
\dot x=\frac32H_0\sqrt{\Omega_M+(1-\Omega_M)x^2}.
$$
Separating variables and choosing <cosmic time> $t=0$ at the <Big Bang> gives
$$
\frac{\operatorname{arsinh}\!\left(x\sqrt{(1-\Omega_M)/\Omega_M}\right)}{\sqrt{1-\Omega_M}}=\frac32H_0t.
$$
Thus the <flat dust-and-vacuum expansion> is
$$
\boxed{a(t)=\left(\frac{\Omega_M}{1-\Omega_M}\right)^{1/3}\sinh^{2/3}\!\left(\frac32H_0\sqrt{1-\Omega_M}\,t\right).}
$$
Setting $a(t_0)=1$ gives the <age of a flat matter-Lambda universe>,
$$
t_0=\frac{2}{3H_0\sqrt{1-\Omega_M}}\operatorname{arsinh}\sqrt{\frac{1-\Omega_M}{\Omega_M}}.
$$
The two <matter-vacuum age endpoint limits> follow below.
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