 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 givesThe 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,