# How to classify a linear 2nd order PDE?

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 !

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 $$\{(x, y)\colon x\neq 0, y > 0\}$$.