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Second order partial differential equation: $$(x-y)^2/4 u_{xx} + (x-y) \sin(x^2+y^2) u_{xy} + \cos(x^2+y^2) u_{yy} +\dots=0$$ is

a. Elliptic in $\{(x,y): x≠y, x^2+y^2<\frac{\pi}{6}\}$

b. Hyperbolic in $\{(x,y): x≠y, \frac{\pi}{4}<x^2+y^2<\frac{3\pi}{4}\}$

c. Elliptic in $\{(x,y): x≠y, \frac{\pi}{4}<x^2+y^2<\frac{3\pi}{4}\}$

d. Hyperbolic in $\{(x,y): x≠y, x^2+y^2<\frac{\pi}{4}\}$

I am stuck on this problem . Can anyone help me please......

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What are the definitions of an elliptic or hyperbolic PDE? And in what regions are the coefficients to $u_{xx}$, $u_{xy}$, $u_{yy}$ positive or negative, and what do they have to do with the definition of an elliptic/hyperbolic PDE? – Mario Carneiro Dec 18 '12 at 9:15

compute the expression $S^2-4RT$ . here $S=(x-y)\sin(x^2+y^2)$ , $R=(x-y)^2/4$ , $T=\cos^2(x^2+y^2)$. after calculating we have
$S^2-4RT=(x-y)^2\cos 2(x^2+y^2)$

so the option ($3$) is correct

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