Subtopic: Partial Differential Equations – Laplace, heat and wave equations
Question Statement:
The one-dimensional wave equation $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ is classified as which type of partial differential equation?
Options:
(A) Hyperbolic
(B) Parabolic
(C) Elliptic
(D) Non-linear
Answer Key:
A
Step-by-Step Solution & Explanation
For $A u_{xx} + B u_{xt} + C u_{tt} = 0$, here $A = c^2$, $B = 0$, $C = -1$. The discriminant is $B^2 - 4AC = 0 - 4(c^2)(-1) = 4c^2 > 0$, which classifies the equation as hyperbolic.