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

$$x-a+(y-b)y^{\prime }=\mathrm{\hspace{0.33em}0},$$

$$1+y^{\prime \mathrm{\hspace{0.17em}2}}+(y-b)y^{\prime \prime }=0,$$

$$3y^{\prime }{y}^{\prime \prime }+(y-b)y^{\prime \prime \prime }=\mathrm{\hspace{0.33em}0}.$$

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

$$(1+y^{\prime \mathrm{\hspace{0.17em}2}})y^{\prime \prime \prime }-3y^{\prime }{y}^{\prime \prime \mathrm{\hspace{0.17em}2}}=\mathrm{\hspace{0.33em}0}$$

It is of three, corresponding the number of parametres.