# differential equation of circles

All circles of the plane form a three-parametric family

 $(x-a)^{2}+(y-b)^{2}\;=\;r^{2}.$

The parametres $a,\,b,\,r$ may be eliminated by using successive differentiations, when one gets

 $x-a+(y-b)y^{\prime}\;=\;0,$
 $1+y^{\prime\,2}+(y-b)y^{\prime\prime}=0,$
 $3y^{\prime}y^{\prime\prime}+(y-b)y^{\prime\prime\prime}\;=\;0.$

The two last equations allow to eliminate also $b$, yielding the differential equation of all circles of the plane:

 $(1+y^{\prime\,2})y^{\prime\prime\prime}-3y^{\prime}y^{\prime\prime\,2}\;=\;0$

It is of three, corresponding the number of parametres.

Title differential equation of circles DifferentialEquationOfCircles 2013-03-22 18:59:26 2013-03-22 18:59:26 pahio (2872) pahio (2872) 5 pahio (2872) Example msc 34A34 msc 51-00