# Regions where : $2u_{xx} + 4u_{xy} + 3u_{yy} - u = 0$ is hyperbolic, parabolic or elliptic

Exercise :

Describe the regions of the $(x,y)$-plane where the following equation is hyperbolic, parabolic or elliptic : $$2u_{xx} + 4u_{xy} + 3u_{yy} - u = 0$$

Attempt - Question :

In order to determine the regions, you have to work with the sign of the discriminant. My question though, is the following :

Should the discriminant be yielded from the initial differential equation or from an equation in canonical form (applying transformations) ?

If it should be yielded from the initial equation, then in the case of a PDE as :

$$au_{xx} + 2bu_{xy} + cu_{yy} + du_x + eu_y + fu +g = 0$$

we study the discriminant $\Delta = b^2 - 4ac$ and check its sign.

For my specific case, we have :

$$\Delta = 2^2 - 4\cdot 2\cdot 3=4-24<0$$

thus the equation is elliptic at every point $(x_0,y_0)$ in a domain $\Omega$ of $\mathbb R^2$ of the PDE.

$$\Delta = b^2 - ac = -2 < 0$$