[Math] Find the differential equation of all circles of radius 1 centered on the y-axis

Question

I need to find the differential equation of all circles of the form:

$$ x^2 + (y -C_1)^2 = 1$$

Differentiating w.r.t $x$ once yields:
$$ x + (y-C_1) y' =0 $$

Twice:

$$ 1+ (y-C_1) y'' +(y')^2 =0 $$

How would I Then find the resulting equation

Answer 1

A hint:

Eliminate $C_1$ from the first and the second equation.

Note that exactly two circles pass through each point in the strip $-1<x<1$. Therefore we should expect a differential equation of the form $y'^2=\ldots\ $. A second order differential equation has infinitely many solution curves through each point $(x_0,y_0)$ in its domain.