= Solution
Put $D=\Omega^2-\omega^2$, $E=2\gamma\omega$, and $N=D^2+E^2$. Substituting a sine-and-cosine trial solution for the <forced harmonic oscillator> gives
$$
\boxed{x(t)=\frac A N\left[D\sin\omega t-E\cos\omega t\right],\qquad t\leq0.}
$$
The complementary solution is $e^{-\gamma t}(C\cos\nu t+S\sin\nu t)$, where $\nu=\sqrt{\Omega^2-\gamma^2}$. Every nonzero such solution is unbounded as $t\to-\infty$, so boundedness on the whole negative time axis forces $C=S=0$. This fixes the phase as well as the amplitude. In particular,
$$
x_0=x(0)=-\frac{AE}{N},\qquad v_0=\dot x(0)=\frac{AD\omega}{N}.
$$
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