Parabolic Partial Differential Equation

A parabolic partial differential equation is a type of second-order partial differential equation that involves both second derivatives with respect to time and space variables. It is characterized by having a unique time variable and multiple space variables.

Given Equation:
The given partial differential equation is:
x^2uxx - 2xyuxy + y^2uyy + xux + yuy = 0

Classifying the Equation:

To classify the given equation, we need to determine the nature of its characteristic curves. This can be done by considering the discriminant of the equation, which is given by:

D = (-2xy)^2 - 4(x^2)(y^2) = 4(x^2)(y^2) - 4(x^2)(y^2) = 0

Discriminant D = 0:
When the discriminant D is zero, the equation is parabolic.

Explanation:
In this case, the discriminant D is equal to zero, indicating that the given partial differential equation is parabolic. This means that the equation represents a parabolic type of second-order partial differential equation.

The presence of the term x^2uxx in the equation indicates that it is second-order in the spatial variable x. The terms -2xyuxy and y^2uyy involve both spatial variables x and y, while the terms xux and yuy are first-order in the spatial variables.

The fact that the equation involves both second derivatives with respect to the spatial variables and first derivatives with respect to the spatial variables is a characteristic of a parabolic equation.

Therefore, the correct answer is option 'A' - Parabolic.

Overall, the given partial differential equation is a parabolic equation due to its discriminant being zero. It represents a specific class of second-order partial differential equations that have unique time variables and multiple space variables.