# Ordinary Differential Equations/Homogenous 2

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## Mechanical Vibrations[edit]

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 where k is the spring constant. The force on the dampener is where c is the damping constant. And of course, the net force is . That gives us a system where

Remember that and . This gives us a differential equation of

In the case where c=0, we have just a mass on a spring. In this case, we have . Since k and m are both positive (by the laws of physics), the result is always a . This makes sense from a physical perspective- a spring moving back and forth forms a periodic wave of frequency