I recently studied a problem which involved a harmonic oscillator moving in a potential and through some manipulation, I obtained the equation

$x''(t) = -\omega^2x + f(t)x$,

where $f(t)$ is a function such that $\lim_{t\to \pm \infty} f(t) = 0$.

If $f(t)=0$, this is just harmonic motion and the solutions are $x(t)=A\sin(\omega t + \phi)$.

If $f(t)$ is non-zero, however, $x(t)$ is still harmonic asymptotically (in the limit $t\to\pm\infty$); however, the two limits may now be different -- the sine waves may have a different amplitude and phase.

If we say that $\lim_{t\to -\infty} x(t) = A\sin(\omega t + \phi)$ and $\lim_{t\to \infty} x(t) = A'\sin(\omega t + \phi')$, is it possible to obtain some relationship between ($A$, $\phi$) and ($A'$, $\phi'$)?