Derivation of the Diff Eq

Assume that there are two forces only:

The restoring force exerted by the spring: \( -ky \);
The damping force: \( -\gamma\frac{dy}{dt} \).

Here, \(k\) is the spring constant and \(\gamma\) is the damping coefficient.

The Derivation of the Diff Equation

\( \begin{array}{c} \begin{array}{rcl} \mbox{(mass)(acceleration)} &=&\mbox{(the net force on the mass),} \\ \displaystyle m\frac{d^2 y}{dt^2} &=& \displaystyle -ky-\gamma\frac{dy}{dt}, \end{array} \\ \boxed{m\frac{d^2 y}{dt^2} +\gamma\frac{dy}{dt} +ky =0} \end{array}\)