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

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

where $c^2=a^2/2$ This is a one parameter family of curves. Differentiating with respect to x, we get

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