# Find the differential equation of all circles of radius a

Can someone please post a detailed step-by-step procedure. Given the circle with a radius a, what is the differential equation of the circle.

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quoting you, "what is the differential equation of the circle"? –  Ilya Dec 3 '11 at 18:40
Does this other account belong to you as well? If so, we can flag the moderators to merge these two accounts; it'll make it convenient for you to keep track of your questions. Thanks, –  Srivatsan Dec 3 '11 at 23:45
Nope, that account does not belong to me Srivatsan. –  Nikhil Mulley Dec 4 '11 at 5:12

From the implicit equation of the circle $(x-u)^2+(y-v)^2=a^2$, you get $$x'(x-u)+y'(y-v)=0$$ by implicit differentiation. Add the initial condition $$x(0)=u+a, \quad y(0)=v$$
You can write the differential equations as $$x'=-y+v, \quad y' = x-u$$ which is especially nice for circles centered at the origin.
Circle equation $$(x-C_1)^2+(y-C_2)^2=a^2\quad (0)$$ Differentiate twice by $x$ $$(x-C_1)+y'(y-C_2)=0\quad (1)$$ $$1+(y-C_2)y''+(y')^2=0\quad (2)$$ From $(2)$ we obtain $$C_2=y+\frac{(y')^2+1}{y''}$$ Then substitute in $(1)$ and $(0)$ $$(x-C_1)-y'\frac{(y')^2+1}{y''}=0\quad(3)$$ $$(x-C_1)^2+\left(\frac{(y')^2+1}{y''}\right)^2=a^2\quad(4)$$ From $(3)$ we obtain $$x-C_1=y'\frac{(y')^2+1}{y''}$$ Then substitute in $(4)$ $$\left(y'\frac{(y')^2+1}{y''}\right)^2+\left(\frac{(y')^2+1}{y''}\right)^2=a^2$$ After some simplifications we get $$((y')^2+1)^3=(ay'')^2$$