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 $$v=x'$$ and $$a=x''$$. This gives us a differential equation of

$$mx''=-cx'-kx$$

$$mx''+cx'+kx=0$$

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 $$y=c_1cos(\sqrt{\frac{k}{m}}x)+c_2sin(\sqrt{\frac{k}{m}}x)$$. This makes sense from a physical perspective- a spring moving back and forth forms a periodic wave of frequency $$\frac{\sqrt{\frac{k}{m}}}{2 \pi}$$