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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.

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up vote 2 down vote accepted

None of the options are correct.

The discriminant $\Delta = B^2 - 4AC$ satisfies

  • parabolic: $\Delta = 0$ iff $x = \pm 1$ or $y = 0$
  • hyperbolic: $\Delta > 0$ iff either ($y > 0$ and $|x| > 1$) or ($y < 0$ and $|x| < 1$)
  • elliptic: $\Delta < 0$ iff either ($y < 0$ and $|x| > 1$) or ($y > 0$ and $|x| < 1$)
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