I recently studied a problem which involved two particles joined by a harmonic spring 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 ($A'$, $\phi'$) in terms of the function $f(t)$?

I would think that some sort of perturbation theory or multiscale analysis might work here.

As a very simple example, one can consider the function

$f(t)= \begin{cases} 1 &\quad 0 < t < 2\pi\\ 0 &\quad\text{otherwise} \\ \end{cases}$

in the case $\omega=1$. This case can be solved exactly by matching the solutions in the three regions $t<0$, $0<t<2\pi$ and $t>2\pi$, which gives the conditions $A\cos\phi=A'\cos\phi'$ and $A(\sin\phi + \cos\phi) = A'\sin\phi'$. Thence, one can obtain relations for the phase shift, for example:

$\phi' = \tan^{-1}(1+\tan\phi) \approx \frac \pi 4 + \frac 1 2 \phi$ when $\phi\approx 0$.

However, this sort of analysis is clearly not possible for a general $f(t)$, so some sort of approximate approach is needed.