Question

Which partial differential equation represents the Laplace equation in two dimensions?

• A. \( \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0 \)
• B. \( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \)
• C. \( \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \)
• D. \( \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} \)

Hint:

Laplace equation = sum of second partial derivatives equals zero

Answer 1

The Correct Option is B

Concept: Laplace Equation
The Laplace equation is a second-order partial differential equation widely used in physics and engineering.
Step 1: Standard form in two dimensions
\[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \]
Step 2: Interpretation
It describes:

• Steady-state heat distribution
• Electrostatic potential
• Fluid flow

Step 3: Identify correct option
Option (B) matches the standard Laplace equation. Conclusion: \[ \text{Laplace equation in 2D is } \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \]