I am used to thinking of the resonance phenomenon in terms of a small driving force building up a large vibration amplitude. But when you solve the equation for a damped, driven oscillator, you get a transient part that decays with time, and a steady-state part that is, well, steady. How does this account for the slow buildup from small amplitude to large amplitude? I would have thought there would be a term with exponentially increasing amplitude.
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In the case of the forced harmonic oscillator, the steady solution is the one showing resonance. It has to do, in this case, with how far is the frequency of the external force from the natural frequency of the system. If they are alike, we have maximal amplitude, and so, resonance. But again, it appears in the steady solution not in the decaying part.
I think your confusion is done by considering resonance as an effect of a small driven force in relation with big amplitudes, when in fact, it has to do with proximity of frequencies in a system (like the aforementioned harmonic oscillator).
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1$\begingroup$ This is all correct, but you've missed something: a 'slow buildup' happens starting from a resting of nearly resting configuration. So take the general solution in the case of $x_0 = \dot{x}_0 = 0$. The system starts with zero energy and the driving force adds a little each cycle. The maximum size is still governed by the closeness of the driving and natural frequencies and the overall strength of the damping. $\endgroup$ Commented Jan 5, 2018 at 7:51