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

$y\frac{dy}{dx}=-y$

The equation of the rectangular hyperbola whose axes are asymptotes of the hyperbola $x^2-y^2=a^2$ is $xy=c^2$ where $c^2=\frac{a^2}{2}$. On differentiating, $x\frac{dy}{dx}+y=0$ is the required D.E.