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

Question

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.

Answer 1

Your formula is off. It's

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

But in any case, the equation is elliptic everywhere

Answer 2

Short answer to your question is it does not matter whether you are working with the original PDE or in canonical form. It should give you the same answer.

Note that the person who previously answered the question has a 4 missing in the discriminant formula.