Subtopic: Partial Differential Equations – Laplace, heat and wave equations
Question Statement:
The general two-dimensional Laplace equation $\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ is classified as:
Options:
(A) Elliptic partial differential equation
(B) Hyperbolic partial differential equation
(C) Parabolic partial differential equation
(D) Ordinary differential equation
Answer Key:
A
Step-by-Step Solution & Explanation
For $A u_{xx} + B u_{xy} + C u_{yy} = 0$, here $A = 1, B = 0, C = 1$. The discriminant $B^2 - 4AC = 0 - 4(1)(1) = -4 < 0$, which defines an elliptic PDE.