Question

Form the differential equation of the family of circle touching the y-axis at origin

Answer 1

Circle touches x axis
$x^2+(y-r{)}^2=r^2$
$x^2+y^2=2yr\Rightarrow r=\left(\frac{x^2+y^2}{2y}\right)$
$\frac{dr}{dx}=0=\frac{2x+2y\cdot \frac{dy}{dx}}{2y}-\frac{x^2+y^2}{2y^2}\frac{dy}{dx}$

Rearrange this

$\frac{dy}{dx}=\frac{2xy}{x^2-y^2}$