Question

For the following partial differential equation, $$x \frac{\partial^2 f}{\partial x^2} + y \frac{\partial^2 f}{\partial y^2} = \frac{x^2 + y^2}{2}$$ which of the following option(s) is/are CORRECT?

• A. elliptic for $x>0$ and $y>0$
• B. parabolic for $x>0$ and $y>0$
• C. elliptic for $x = 0$ and $y>0$
• D. hyperbolic for $x<0$ and $y>0$

Hint:

To classify second-order PDEs, examine the coefficients of the highest-order terms. Elliptic equations have coefficients of the same sign, while hyperbolic equations have coefficients of opposite signs.

Answer 1

The Correct Option is A

Step 1: The given partial differential equation can be rewritten in the general form: $$a \frac{\partial^2 f}{\partial x^2} + b \frac{\partial^2 f}{\partial y^2} = f(x, y)$$ where: $$a = x, \quad b = y$$ Step 2: The classification of the PDE depends on the sign of the coefficients: If $a>0$ and $b>0$, the equation is elliptic. If $a<0$ and $b>0$, the equation is hyperbolic. If either $a$ or $b$ equals zero, the equation may degenerate to a different type. Step 3: Evaluating the given options: For $x>0$ and $y>0$, both coefficients are positive, so the equation is elliptic. Option (A) is correct. For $x<0$ and $y>0$, the coefficients have opposite signs, making the equation hyperbolic. Option (D) is correct. Conclusion: The correct answers are (A) and (D), confirming the classification of the given differential equation.