The equation is hyperbolic at a point when $\boxed{b^2-ac>0}$. The principal form has one positive and one negative <eigenvalue> and two distinct real <characteristic directions>. In the given equation $a=x^2$, $b=0$, $c=-y^2$, so $b^2-ac=x^2y^2>0$ throughout the <positive quadrant>, consistent with the two canonical characteristic families used above.

Solution

Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2009 / Differential equations / 8 / iii / Solution

 Solution (source code)