Question Bank • QB_EM_DE_012 MCQ • 2 Marks

Section 1: Engineering Mathematics

Differential Equations

Subtopic: Partial Differential Equations – Laplace, heat and wave equations

Question Statement:

The two-dimensional steady-state heat conduction equation (Laplace's equation) $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ is classified as:

Options:

  • (A) Elliptic partial differential equation
  • (B) Parabolic partial differential equation
  • (C) Hyperbolic partial differential equation
  • (D) Nonlinear dispersive partial differential equation

Answer Key:

A

Step-by-Step Solution & Explanation

A second-order PDE $A u_{xx} + B u_{xy} + C u_{yy} + \dots = 0$ is classified by the discriminant $\Delta = B^2 - 4AC$: For Laplace's equation: $A = 1, B = 0, C = 1$. $\Delta = B^2 - 4AC = 0^2 - 4(1)(1) = -4 < 0$ Since $\Delta < 0$, the equation is strictly elliptic.
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