Solution

From $Q=10^9$ and $\omega_0=10^4$ s$^{-1}$, we have $\gamma=\omega_0/(2Q)=5\times10^{-6}$ s$^{-1}$. The amplitude of the transient response decays according to the exponential $e^{-\gamma t}$, which becomes less than $5%$ for $t\approx3/\gamma$. Thus the forced oscillation is firmly established after roughly $6\times10^5$ s. An order-of-magnitude estimate is $\sim10^5-10^6$ s.