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 $F=-kx$ where k is the spring constant. The force on the dampener is $F=-cv$ where c is the damping constant. And of course, the net force is $F=ma$. That gives us a system where

$ma=-cv-kx$

Remember that $v=x^{\prime }$ and $a=x^{″}$. This gives us a differential equation of

$m{x}^{″}=-c{x}^{\prime }-kx$

$m{x}^{″}+c{x}^{\prime }+kx=0$

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