I need to find the region in the $xy$-plane where the following PDE is hyperbolic, parabolic and elliptic depending on the value of $a$ (real constant):

$$yu_{xx}+2au_{xy} + x^2u_{yy}+(x^2-a^2)u_x+(y^2-a^2)u_y=0$$

This is what I did:

For the PDE to be hyperbolic, we need: $a^2-yx^2>0$ and since $a^2>0$ we have the following conditions on $x$ and $y$:

$$x \in ]- \infty ;+ \infty [$$ $$y>0$$

However, for the rest, I am a bit lost, I don't know if my method is correct. This case by case thing is not my strong point.

Thanks !


1 Answer 1


The discriminant is $D(x, y) = a^2 - yx^2$. The classification is that, at a given point $(x, y)$, the PDE is elliptic if $D(x, y) > 0$, parabolic if $D(x, y) = 0$, and hyperbolic if $D(x, y) < 0$.

I'll analyse the case where $a = 0$ and let you attempt the case for $a\neq 0$, since the idea is similar. For $a = 0$, the discriminant reduces to $D(x, y) = -yx^2$ and the PDE is

  • elliptic on $\{(x, y)\colon x\neq 0, y < 0\}$;
  • parabolic on both the $x$- and $y$-axis;
  • hyperbolic on $\{(x, y)\colon x\neq 0, y > 0\}$.
  • $\begingroup$ I forgot a term hehe... $\endgroup$
    – bsaoptima
    Dec 27, 2022 at 18:01

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