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 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}