Question

Consider the second order Partial Differential Equation (PDE)
\[ 4x^2 \frac{\partial^2 u}{\partial x^2} + 4(x + y) \frac{\partial^2 u}{\partial x \partial y} + (x^2 + y^2) \frac{\partial^2 u}{\partial y^2} - u = 0. \] Then which one of the following statements is correct?

• A. The PDE is hyperbolic in the region \(\{ (x,y) \in \mathbb{R}^2 : -1<x<0, \, y<0 \}\)
• B. The PDE is hyperbolic in the region \(\{ (x,y) \in \mathbb{R}^2 : 1<x<\infty, \, y<0 \}\)
• C. The PDE is elliptic in the region \(\{ (x,y) \in \mathbb{R}^2 : 0<x<1, \, y>0 \}\)
• D. The PDE is parabolic in the region \(\{ (x,y) \in \mathbb{R}^2 : 1<x<\infty, \, y>0 \}\)

Hint:

To classify second-order partial differential equations, calculate the discriminant \( \Delta = B^2 - AC \), where \(A\), \(B\), and \(C\) are the coefficients of the second derivatives. Based on \( \Delta \), you can classify the PDE as hyperbolic, elliptic, or parabolic.

Answer 1

The Correct Option is A

We are given a second-order partial differential equation (PDE). To classify the PDE, we need to examine the discriminant of the associated quadratic form, which is based on the coefficients of the second-order partial derivatives.
The general form of a second-order PDE is: \[ A \frac{\partial^2 u}{\partial x^2} + 2B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + \cdots = 0 \] where \(A\), \(B\), and \(C\) are the coefficients of the second-order partial derivatives.
For the given equation, we identify:
- \( A = 4x^2 \)
- \( B = 2(x + y) \)
- \( C = x^2 + y^2 \)
The discriminant \( \Delta \) is given by: \[ \Delta = B^2 - AC \] Substituting the values: \[ \Delta = [2(x + y)]^2 - 4x^2 (x^2 + y^2) \] \[ \Delta = 4(x + y)^2 - 4x^2 (x^2 + y^2) \] Now, for classification:
- If \( \Delta>0 \), the PDE is hyperbolic.
- If \( \Delta = 0 \), the PDE is parabolic.
- If \( \Delta<0 \), the PDE is elliptic.
By examining the discriminant in the given region, we find that the PDE is hyperbolic in the region \(\{ (x,y) \in \mathbb{R}^2 : -1<x<0, \, y<0 \}\).
Thus, the correct option is (A).