The differential equation of the family of circles touching y-axis at the origin is

As required circle touches y - axis at the origin.
$\therefore$ Let Center of the circle is $d(a,0)$ and radius is a
$\therefore$ Equation of circle will be,
${\left(x-a\right)}^2+{\left(y-0\right)}^2=a^2$
$\Rightarrow x^2-2ax+a^2+y^2=a^2$
$\Rightarrow x^2+y^2-2ax=0$ .... (i)
By differentiating above equation w.r.t x, we get
$2x+2y\frac{dy}{dx}-2a=0$
$\therefore 2a=2x+2y\frac{dy}{dx}$ .... (ii)
From (i) and (ii),
$x^2{+ y}^2-\left(2x+2y\frac{dy}{dx}\right)x=0$
$\Rightarrow x^2+y^2-2x^2-2xy\frac{dy}{dx}=0$
$\Rightarrow x^2-y^2+2xy\frac{dy}{dx}=0$