the transient response

Strike a tuning fork and you hear an initial 'twang' that quickly fades, settling toward whatever steady tone the surroundings impose. That fading part — the system's own startup wiggle that dies away — is the transient response. It is how the system reacts to being switched on or disturbed, before it settles into its long-run behaviour.

Mathematically the transient is the complementary solution y_c = c1 y1 + c2 y2, the part carrying the arbitrary constants and depending on the initial conditions. For a stable system — for instance a damped mechanical oscillator or an RLC circuit with resistance — the homogeneous solutions contain decaying factors like e^(-at) with a > 0, so y_c tends to zero as time grows. After enough time only the particular solution remains, and the influence of how you started has been forgotten.

This split is the practical payoff of writing y = y_c + y_p. The transient tells you about the brief settling-in period — overshoot, ringing, how long until things calm down — while the steady state tells you the eventual driven behaviour. One honest caveat: 'transient' assumes the homogeneous solutions actually decay. If the unforced system is undamped (purely oscillatory y_c) or unstable (growing y_c), the complementary part does not fade, and calling it a transient would be misleading.