the steady-state response
Wait long enough after switching a stable driven system on and the startup wiggles die away, leaving a clean, persistent pattern that just tracks the input. A loudspeaker fed a steady tone eventually vibrates at that tone; a driven circuit settles into an output at the driving frequency. That lasting, input-following part is the steady-state response.
Mathematically the steady state is the particular solution y_p of the forced equation y'' + p y' + q y = g(x) — specifically the piece that remains after the transient y_c has decayed to zero. For sinusoidal forcing g = F cos(wt), the steady state is itself a sinusoid at the SAME frequency w but generally with a different amplitude and a phase shift: y_ss = R cos(wt - phi). The amplitude R and phase lag phi depend on the driving frequency and the system's parameters, which is exactly the information a frequency-response analysis extracts.
The steady state matters because it is what you observe in the long run and what most engineering design targets: the eventual output level, the phase lag, whether the response is large (near resonance) or small. One honesty point: a true steady state requires the transient to actually decay, i.e. a stable (typically damped) system. With no damping and resonant forcing, there is no steady state at all — the response grows without bound — so 'steady state' is a concept that lives where the system settles.
For y'' + 2y' + 2y = 10 cos(t), the steady state is y_ss = 2 cos(t) + 4 sin(t) = 2√5 cos(t - phi) with phi = arctan(2): same frequency as the forcing, shifted amplitude and phase. Whatever the initial conditions, the solution approaches this as t grows.
The steady state is the lasting particular response that tracks the forcing.
Steady state does not mean 'constant' — under a sinusoidal drive the steady state keeps oscillating forever; it means the part that no longer depends on initial conditions, after the transient has gone.