I was trying to solve the following problem:
The partial differential equation $y^{3}u_{xx}-(x^{2}-1)u_{yy}=0$ is
(a) parabolic in $\{(x,y):x<0\}$,
(b) hyperbolic in $\{(x,y):y>0\}$,
(c) elliptic in $\Bbb R^{2}$,
(d) parabolic in $\{(x,y):x>0\}$.
I have to determine which of the given options is correct. I know that a partial differential equation of the form $Au_{xx}+Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+F=0$ is parabolic, hyperbolic or elliptic according as $B^{2}-4AC=0$, $>0$ or $<0$ respectively. Here, I see $B^{2}-4AC=y^{3}(x^2-1)$. From hereon, I could not progress. Please help.