Solution
= Solution
Setting $a(t_0)=1$ gives
$$
t_0=\frac{2(\sqrt{\beta+\gamma}-\sqrt\beta)}\gamma
=\frac2{\sqrt{\beta+\gamma}+\sqrt\beta}.
$$
At the present epoch,
$$
H_0=\dot a(t_0)=\sqrt{\beta+\gamma},
$$
so
$$
t_0H_0=\frac{2\sqrt{\beta+\gamma}}
{\sqrt{\beta+\gamma}+\sqrt\beta}.
$$
For $\beta\gg\gamma$, this tends to $1$, giving $t_0\sim H_0^{-1}$. For $\gamma\gg\beta$, it tends to $2$, giving $t_0\sim2H_0^{-1}$.