The differential equation for the family of curves $x^2+y^2-2ay=0,$ where $a$ is an arbitrary constant is

Given the equation $x^2+y^2-2ay=0$.
Differentiating with respect to $x$:
$2x+2y{y}^{\prime }-2a{y}^{\prime }=0$.
We need to eliminate the arbitrary constant $a.$
From the derivative: $2a{y}^{\prime }=2x+2y{y}^{\prime }\Rightarrow a=\frac{2x+2y{y}^{\prime }}{2y^{\prime }}=\frac{x+y{y}^{\prime }}{y^{\prime }}$.
Substitute this value of $a$ back into the original equation:
$x^2+y^2-2(\frac{x+y{y}^{\prime }}{y^{\prime }})y=0$
$x^2+y^2-\frac{2xy+2y^2{y}^{\prime }}{y^{\prime }}=0$
Multiply by $y^{\prime }$:
$x^2{y}^{\prime }+y^2{y}^{\prime }-2xy-2y^2{y}^{\prime }=0$
$x^2{y}^{\prime }-y^2{y}^{\prime }=2xy$
$(x^2-y^2)y^{\prime }=2xy$.