I've been brushing up again on my differential equations. In reading through the derivations for the use and application of them to model physical systems I have hit a road bump in my intuitive understanding.

The 'general' form of a first order system, after some rearranging to put focus on the time constant, is:

(1/Tau)*dy(t)/dt + y(t) = f(t), where f(t) is some forcing function.

Now if we set f(t) = 0 and provide some y(0) condition, we can solve the 'homogenous' form and get an exponential decay (assuming it is stable).

If we now bring f(t) back in and assume it is not zero, my text states the following for the 'general solution':

y(t) = yh(t) + yp(t), where yh(t) is the exponentially decaying homogeneous form mentioned above, an yp(t) is a 'particular' solution to the forcing function.

This isn't obvious or intuitive to me. Can anyone help me understand how the general form can be proven to provide the general solution?

Regards,

James