Partial Differential Equation Classification

A general second-order linear partial differential equation (PDE) in two independent variables \(x\) and \(y\) has the form:

\[ A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + D \frac{\partial u}{\partial x} + E \frac{\partial u}{\partial y} + F u = G \] where \(A, B, C, D, E, F, G\) are functions of \(x\) and \(y\).

The classification of such a PDE (as elliptic, parabolic, or hyperbolic) depends on the coefficients of the highest-order derivatives (\(A, B, C\)) through the discriminant, given by \(B^2 - 4AC\).

• If \(B^2 - 4AC > 0\), the PDE is hyperbolic.
• If \(B^2 - 4AC = 0\), the PDE is parabolic.
• If \(B^2 - 4AC < 0\), the PDE is elliptic.

Coefficients of the PDE

The given partial differential equation is:

\[ x^2 \frac{\partial^2 u}{\partial x^2} - 2 x y \frac{\partial^2 u}{\partial x \partial y} - 3 y^2 \frac{\partial^2 u}{\partial y^2} + \frac{\partial u}{\partial x} - \frac{\partial u}{\partial y} = 0 \]

By comparing this equation with the general form, we can identify the coefficients \(A, B,\) and \(C\):

• \(A = x^2\)
• \(B = -2xy\)
• \(C = -3y^2\)

Discriminant Calculation

Now, we calculate the discriminant \(B^2 - 4AC\) using the coefficients we found:

\[ B^2 - 4AC = (-2xy)^2 - 4(x^2)(-3y^2) \]

First, square the term \(B\):

\[ (-2xy)^2 = (-2)^2 (x)^2 (y)^2 = 4x^2y^2 \]

Next, calculate \(4AC\):

\[ 4AC = 4(x^2)(-3y^2) = -12x^2y^2 \]

Now, subtract \(4AC\) from \(B^2\):

\[ B^2 - 4AC = 4x^2y^2 - (-12x^2y^2) = 4x^2y^2 + 12x^2y^2 = 16x^2y^2 \]

So, the discriminant is \(16x^2y^2\).

Analyzing the Discriminant

The type of the PDE depends on the sign of the discriminant \(16x^2y^2\). Let's analyze its sign for different values of \(x\) and \(y\):

The term \(x^2\) is always non-negative (\(\ge 0\)). The term \(y^2\) is always non-negative (\(\ge 0\)). Therefore, \(x^2y^2\) is always non-negative (\(\ge 0\)).

So, the discriminant \(16x^2y^2\) is always non-negative (\(\ge 0\)) because 16 is a positive constant.

• If \(x \ne 0\) and \(y \ne 0\), then \(x^2 > 0\) and \(y^2 > 0\). In this case, \(x^2y^2 > 0\), and the discriminant \(16x^2y^2 > 0\).
• If \(x = 0\) or \(y = 0\) (or both), then \(xy = 0\). In this case, \(x^2y^2 = 0\), and the discriminant \(16x^2y^2 = 0\).

Classifying the PDE Type

Based on the analysis of the discriminant \(16x^2y^2\):

• The PDE is hyperbolic in the region where \(xy \ne 0\). This region includes all points \((x, y)\) in \(\reals^2\) except those lying on the x-axis (\(y=0\)) or the y-axis (\(x=0\)).
• The PDE is parabolic on the lines where \(xy = 0\), which are the x-axis and the y-axis.
• The PDE is never elliptic because the discriminant is never negative.

We are looking for a description of the PDE's type in a specific region as given in the options. The condition \(xy \ne 0\) means the region where both \(x \ne 0\) and \(y \ne 0\).

One of the options states that the PDE is hyperbolic in the region \(\{(x, y) \in \reals^2 : xy \ne 0\}\). This matches our conclusion from the discriminant analysis.