Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 14: Partial Derivatives - Section 14.3 - Partial Derivatives - Exercises 14.3 - Page 809: 81


The wave equation is satisfied.

Work Step by Step

We need to compute the wave equation. The wave equation can be defined as: $c^2 \dfrac{\partial^2 w}{\partial x^2}=-c^2 \sin (x+ct)$ In order to find the partial derivative, we will differentiate $w$ with respect to $t$, by keeping $x$ and $c$ as a constant, and vice versa: $\dfrac{\partial w}{\partial t}=c \cos (x+ct)$ and $\dfrac{\partial w}{\partial x}=\cos (x+ct)$ Now, $c^2 \dfrac{\partial^2 w}{\partial x^2}=c^2 \dfrac{\partial}{\partial x}(\dfrac{\partial w}{\partial x})=c^2 [-\sin (x+ct)]=-c^2 \sin (x+ct)$ Thus, the wave equation is satisfied.
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