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

$\left(x^2-y^2\right) \frac{d y}{d x}=2 x y$

We have,

$x^2+y^2-2 a y=0$            .....(i)

Differentiating w.r. to $x$, we get

$2 x+2 y \frac{d y}{d x}-2 a \frac{d y}{d x}=0 \Rightarrow a=\frac{x+y \frac{d y}{d x}}{\frac{d y}{d x}}$

Substituting this value of $a$ in (i), we get

$\left(x^2+y^2\right) \frac{d y}{d x}-2 y\left(x+y \frac{d y}{d x}\right)=0 \Rightarrow\left(x^2-y^2\right) \frac{d y}{d x}=2 x y$

This is the required differential equation.