Question

obtain the differential equation of all circles of radius r .

Answer 1

From the implicit equation of the circle (x−u)2+(y−v)2=a2(x−u)2+(y−v)2=a2, you get

x′(x−u)+y′(y−v)=0x′(x−u)+y′(y−v)=0

by implicit differentiation. Add the initial condition

x(0)=u+a,y(0)=vx(0)=u+a,y(0)=v

You can write the differential equations as

x′=−y+v,y′=x−ux′=−y+v,y′=x−u

which is especially

Answer 2

Answer:

From the implicit equation of the circle

Step-by-step explanation:

(x-u)^2 + (y-u) ^2=a^2

We get

x'(x-u) +y'(y-v) =0

By implicit differentiation add the initial condition

x(0)=u+a, y(0)=v

We can write the different equation as

x'= - y+v

y'=x-u

Which is especially nice for circle centered at the origin