## Thomas' Calculus 13th Edition

Published by Pearson

# Chapter 14: Partial Derivatives - Section 14.3 - Partial Derivatives - Exercises 14.3 - Page 809: 86

#### 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: $\dfrac{\partial w}{\partial t}=\dfrac{\partial (5 \cos (3x+3ct))}{\partial t}+\dfrac{\partial (e^{x+ct})}{\partial t}=-15c \sin (3x+3ct) +ce^{x+ct}$ and $\dfrac{\partial w}{\partial x}= -15 \sin (3x+3ct) +e^{x+ct}$ Now, $\dfrac{\partial^2 w}{\partial t^2}= \dfrac{\partial}{\partial t}( -15c \sin (3x+3ct) +ce^{x+ct})= -45c^2 \cos (3x+3ct) +c^2e^{x+ct}...(1)$ and $c^2 \dfrac{\partial^2 w}{\partial x^2}=c^2 \dfrac{\partial}{\partial x}( -15 \sin (3x+3ct) +e^{x+ct})=-45c^2 \cos (3x+3ct) +c^2e^{x+ct} ......(2)$ Thus, equations (1) and (2) are equal, so the wave equation is satisfied.

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