# Relating Behaviour of a Driven System to the Undriven Modes

1. May 1, 2015

### PGPhys

Is it generally true that if you have a system of particles with some arbitrary interactions due to their relative positions which you can solve in some way (maybe numerically, maybe analytically), you can relate the long term behaviour to the undriven case if it is being driven by a time-varying force? Like in the example of the driven harmonic oscillator, you have a transient term and a steady state term for any given driving frequency - is this a general fact? It seems like it since the differential equation governing the behaviour would simply have some driving term added:

So if you know behaviour of:
$$f(x,\frac{dx}{dt},\frac{d^2x}{dt^2}) = 0$$

Are there any immediately obvious characteristics of:
$$f(x,\frac{dx}{dt},\frac{d^2x}{dt^2}) = G(t)$$
?

I don't know if the whole business of complementary function and particular integral is specific to certain types of differential equation - my interactions involve many non-trivial terms and potentials. I'm not well grounded in the theory of differential equations and am not sure where to start. Thanks!

2. May 1, 2015