differential equation of circles

All circles of the plane form a three-parametric family


The parametres a,b,r may be eliminated by using successive differentiationsMathworldPlanetmath, when one gets

x-a+(y-b)y= 0,
1+y 2+(y-b)y′′=0,
3yy′′+(y-b)y′′′= 0.

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

(1+y 2)y′′′-3yy′′ 2= 0

It is of three, corresponding the number of parametres.

Title differential equation of circles
Canonical name DifferentialEquationOfCircles
Date of creation 2013-03-22 18:59:26
Last modified on 2013-03-22 18:59:26
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Example
Classification msc 34A34
Classification msc 51-00