Suppose a linear passive circuit is hooked up to an AC generator which outputs a sinusoidal voltage of angular frequency $\omega$.
How can we demonstrate rigorously that after a sufficient amount of time (after the transients have died out) the voltage across any of the individual circuit components will be a sinusoidal waveform of angular frequency $\omega$?
I think the idea is to show that the voltage across a circuit component satisfies a linear differential equation (*) with the constant term equal to the generator voltage $V=V_0\cos(\omega t)$. This is the "hard" part.
Then we know that there is a particular solution to (*) of the form $A\cos(\omega t + \phi)$. Next we should show that the homogenous solution either tends to zero or is a sinusoidal itself, and we'll be done. Now this is quite clear in the case of a second order LDE, but in general I don't know (can we even have circuits governed by LDE's of order higher than 2?).