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: 84


The wave equation is satisfied.

Work Step by Step

We need to compute the wave equation. The wave equation can be defined as: $c^2 \dfrac{\partial^2 w}{\partial x^2}= \dfrac{\partial^2 w}{\partial t^2}$ 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: $\dfrac{\partial w}{\partial t}=\dfrac{c}{x+ct}$ and $\dfrac{\partial w}{\partial x}= \dfrac{1}{x+ct}$ Now, $ \dfrac{\partial^2 w}{\partial t^2}= \dfrac{\partial}{\partial t}( \dfrac{1}{x+ct})= \dfrac{0-(c) (c)}{(x+ct)^2}=\dfrac{-c^2}{(x+ct)^2}$ ...(1) and $c^2 \dfrac{\partial^2 w}{\partial x^2}=c^2 \dfrac{\partial}{\partial x}(\dfrac{1}{x+ct})=\dfrac{-c^2}{(x+ct)^2}$ ...(2) Thus, equations (1) and (2) are equal and so the wave equation is satisfied.
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