The differential equation of rectangular hyperbolas whose axes are asymptotes of the hyperbola $x^2-y^2=a^2$, is

$x \frac{d y}{d x}=-y$

The equation of the rectangular hyperbola whose axes are asymptotes of the hyperbola $x^2-y^2=a^2$ is

$x y=c^2$, where $c^2=a^2 / 2$

This is a one parameter family of curves. Differentiating with respect to $x$, we get

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