# Ordinary Differential Equations/Homogenous 2

## Mechanical Vibrations

One place homogenous equations of constant coefficients are used is in mechanical vibrations. Lets imagine a mechanical system of a spring, a dampener, and a mass. The force on the string at any point is ${\displaystyle F=-kx}$ where k is the spring constant. The force on the dampener is ${\displaystyle F=-cv}$ where c is the damping constant. And of course, the net force is ${\displaystyle F=ma}$. That gives us a system where

${\displaystyle ma=-cv-kx}$

Remember that ${\displaystyle v=x'}$ and ${\displaystyle a=x''}$. This gives us a differential equation of

${\displaystyle mx''=-cx'-kx}$

${\displaystyle mx''+cx'+kx=0}$

In the case where c=0, we have just a mass on a spring. In this case, we have ${\displaystyle x''+{\frac {k}{m}}x=0}$. Since k and m are both positive (by the laws of physics), the result is always a ${\displaystyle y=c_{1}cos({\sqrt {\frac {k}{m}}}x)+c_{2}sin({\sqrt {\frac {k}{m}}}x)}$. This makes sense from a physical perspective- a spring moving back and forth forms a periodic wave of frequency ${\displaystyle {\frac {\sqrt {\frac {k}{m}}}{2\pi }}}$