Ordinary Differential Equations/Homogenous 2

Mechanical VibrationsEdit

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