## Thomas' Calculus 13th Edition

Published by Pearson

# Chapter 14: Partial Derivatives - Section 14.3 - Partial Derivatives - Exercises 14.3 - Page 809: 83

#### Answer

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: $\dfrac{\partial w}{\partial t}=c \cos (x+ct)-2 c\sin (2x+2ct)$ and $\dfrac{\partial w}{\partial x}= \cos (x+ct)-2 \sin (2x+2ct)$ Now, $\dfrac{\partial^2 w}{\partial t^2}= \dfrac{\partial}{\partial t}(c \cos (x+ct)-2 c\sin (2x+2ct))=-c^2 \sin (x+ct)-4c^2 \cos (2x+2ct)$ ...(1) and $c^2 \dfrac{\partial^2 w}{\partial x^2}=c^2 \dfrac{\partial}{\partial x}(\cos (x+ct)-2 \sin (2x+2ct))=-c^2 \sin (x+ct)-4c^2 \cos (2x+2ct)$ ...(2) Thus, equations (1) and (2) are equal, so the wave equation is satisfied.

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