The differential equation corresponding to the family of circles in the plane touching the Y-axis at the origin, is

$\begin{array}{l}Equationofcircle\\ x^2+y^2+2gx=0--\left(1\right)\\ diffw.r.tox\\ \Rightarrow 2x+2y\frac{dy}{dx}+2g=0\\ multiplyingwith\text{'}x\text{'}\\ \Rightarrow 2x^2+2xy\frac{dy}{dx}+2gx=0--\left(2\right)\\ \left(2\right)-\left(1\right)\\ x^2-y^2+2xy\frac{dy}{dx}=0\\ \frac{dy}{dx}=\frac{y^2-x^2}{2xy}\end{array}$