Damped and Forced Oscillations — What Happens to a Mass on a Spring a Few Seconds Later?

A mass m attached to a spring of spring constant k moves while the air resistance $-\gamma \frac{dx}{dt}$ acts on it. Once you fix the initial position x(0) = L and the initial velocity $v_0$, then the differential equation decides the whole future. We plot the answer straight from the formula, and see that there are only three ways to come to rest, and look at where the resonance occurs when the mass is shaken from outside.

Mochizuki Research Group, Tokyo University of Science | How to solve second-order linear ODEs (characteristic equation, differential operator method, Wronskian, variation of parameters), and how forced oscillation leads to Green’s functions and linear response, are collected on the derivation page (in Japanese).