damped harmonic oscillator
Pluck a guitar string, jolt a car on its suspension, or let a pendulum swing in air, and the motion oscillates but fades — energy bleeds away to friction or resistance until the system comes to rest. The damped harmonic oscillator is the single equation that captures all of these, and it is the canonical worked example of the whole linear-ODE theory.
Its equation is m x'' + c x' + k x = 0: mass times acceleration, plus a damping force proportional to velocity, plus a restoring force proportional to displacement. The characteristic equation m r^2 + c r + k = 0 sorts behavior by the discriminant, conventionally summarized by the damping ratio. Underdamped (light damping) gives a decaying oscillation, e^(-gamma t) times a sinusoid — it rings and slowly dies. Critically damped is the borderline that returns to rest fastest without overshoot. Overdamped (heavy damping) returns slowly with no oscillation at all, a sum of two decaying exponentials.
This trichotomy is engineering bread and butter. A car's shock absorbers are tuned near critical damping so the body settles quickly without bouncing; a door closer is overdamped so it never slams; a bell is deliberately underdamped so it rings. The identical equation, with charge in place of displacement, describes the current in an RLC circuit — resistance is the damping, inductance the inertia, the inverse of capacitance the stiffness — which is why the same mathematics serves both mechanical and electrical design.
x'' + 2x' + 5x = 0 gives roots r = -1 plus-or-minus 2i, so x = e^(-t)(c_1 cos 2t + c_2 sin 2t): underdamped, oscillating at frequency 2 inside a decaying envelope e^(-t).
Underdamped motion: a sinusoid trapped inside a shrinking exponential envelope.
Damping lowers the observed oscillation frequency slightly below the undamped natural frequency; the system does not ring at sqrt(k/m) once damping is present, a subtlety often overlooked.