# Classifying 2nd order PDE

Classify the following PDEs according to being elliptic, hyperbolic or parabolic. And for each one sketch the regions of ellipticity, parabolicity, and hyperbolicity

1. Tricomi equation $$y\partial_{xx} u + \partial_{yy} u = 0,\qquad u = u(x, y);$$
2. $$x^2\partial_{xx} u + 2xy\partial_{xy} u + y^2\partial_{yy} u + \left(\partial_{x} u\right)^2 − e^u = 0$$

So I know that the first one is hyperbolic and the second is elliptic. but I don't quite know how to sketch them.

• Are you sure that they are hyperbolic/elliptic for all $x$ and $y$? If not, then for what values of $x$ and $y$ is this true? Where it is not true, how can they be classified? Feb 17, 2020 at 18:57
• That's what my friend and I ended up with, but if it is wrong then I would love to know the correct answer. Feb 17, 2020 at 19:13

We use the notation $$Au_{xx}+Bu_{xy}+Cu_{yy}$$. We then have ellipticity when $$\Delta := B^2-4AC<0$$, parabolicity when $$\Delta=0$$, and hyperbolicity when $$\Delta>0$$.
1. For $$y>0$$, we have $$\Delta = -4y<0$$, so the equation is hyperbolic in this region. Similarly, we have a parabolic equation when $$y=0$$ and hyperbolic equation when $$y<0$$.
2. We have $$\Delta = 4x^2y^2-4x^2y^2 = 0$$ for all $$x$$ and $$y$$, so the equation is parabolic everywhere.