Mass-Spring ODE

An ideal spring described by Hooke's law

$$f(t) \;=\;k\, x(t)$$

where $k$ denotes the spring constant, $x(t)$ denotes the spring displacement from rest at time $t$ , and $f(t)$ is the force required for displacement $x(t)$

If the force on a mass is due to a spring then, as discussed later, we may write the ODE as

$$k\, x(t) + m\, \ddot x(t) \;=\;0$$

   (Spring Force + Mass Inertial Force = 0)