free damped vibration
Push a swing once and walk away: it keeps going on its own, but the air and the friction at the pivot steal a little energy each pass, so the swings get smaller until it stops. That is free damped vibration — 'free' because nobody is pushing it anymore, 'damped' because something is draining its energy. Almost every real system, once disturbed and left alone, behaves this way.
Set the forcing to zero in the spring equation to get m x'' + c x' + k x = 0. The behaviour depends entirely on the tug-of-war between the damping c (which kills motion) and the spring k (which keeps motion going). Mathematically you look at the discriminant c^2 - 4 m k of the characteristic equation. Three outcomes are possible: if damping is weak, the system oscillates while shrinking (underdamped); if damping is just strong enough, it returns to rest as fast as possible without overshooting (critically damped); if damping is very strong, it oozes back slowly with no oscillation at all (overdamped).
Free damped vibration is the 'transient' part of any system's behaviour — the part that fades. Because it always decays to rest, its long-term value is zero, which is why, when you add a steady driving force later, only the forced steady-state response survives at long times. Understanding which of the three regimes you are in tells you whether a door slams or eases shut, whether a car ride is bouncy or harsh, and how fast a circuit settles after a switch flips.
For m x'' + c x' + k x = 0 with m = 1, k = 4: c = 1 gives oscillating decay (underdamped, since 1 < 16), c = 4 gives the fastest non-oscillating return (critically damped), and c = 10 gives a slow creep (overdamped).
Same spring and mass, three damping values, three completely different return shapes.
All three regimes eventually reach rest; the difference is only how. A common myth is that more damping always settles faster — past the critical value, extra damping actually slows the return, because the heavy resistance also resists the motion back toward centre.