To determine To show: The equality ∂ 2 y ∂ t 2 = a 2 ∂ 2 y ∂ x 2 with a = 2 holds. Explanation It is given that the displacement y of a string is y ( x , t ) = 2 sin 2 x cos 4 t , x is the distance from one end of the string and t is the time. The wave equation demands second order derivatives with respect to x and t . Differentiate y ( x , t ) = 2 sin 2 x cos 4 t partially with respect to x as follows. ∂ y ∂ x = ∂ ∂ x ( 2 sin 2 x cos 4 t ) = 2 cos 4 t ∂ ∂ x ( sin 2 x ) = 2 cos 4 t ( cos 2 x ) ∂ ∂ x ( 2 x ) use chain rule = 2 cos 4 t ( cos 2 x ) ( 2 ) That is, ∂ y ∂ x = 4 cos 2 x cos 4 t . Differentiate ∂ y ∂ x = 4 cos 2 x cos 4 t partially with respect to x again as follows. ∂ 2 y ∂ x 2 = ∂ ∂ x ( 4 cos 2 x cos 4 t ) = 4 cos 4 t ∂ ∂ x ( cos 2 x ) = 4 cos 4 t ( − sin 2 x ) ∂ ∂ x ( 2 x ) use chain rule = − 4 cos 4 t ( sin 2 x ) ( 2 ) That is, ∂ 2 y ∂ x 2 = − 8 sin 2 x cos 4 t . Differentiate y ( x , t ) = 2 sin 2 x cos 4 t partially with respect to t as follows. ∂ y ∂ t = ∂ ∂ t ( 2 sin 2 x cos 4 t ) = 2 sin 2 x ∂ ∂ t ( cos 4 t ) = 2 sin 2 x ( − sin 4 t ) ∂ ∂ t ( 4 t ) = − 2 sin 2 x sin 4 t ( 4 ) That is, ∂ y ∂ t = − 8 sin 2 x sin 4 t

Basic Technical Mathematics with Calculus (11th Edition)

11th Edition

ISBN: 9780134437736

Author: Allyn J. Washington, Richard Evans

Publisher: PEARSON

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Chapter 29.3, Problem 50E

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To show: The equality ∂2y∂t2=a2∂2y∂x2 with a=2 holds.

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Chapter 29.3, Problem 49E

Chapter 29.4, Problem 1PE

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Chapter 29 Solutions

Basic Technical Mathematics with Calculus (11th Edition)

Ch. 29.1 - Practice Exercise If f(x, y) = 4xy2 – 3x2y, find...Ch. 29.1 - Prob. 1E Ch. 29.1 - Prob. 2E Ch. 29.1 - Prob. 3E Ch. 29.1 - Prob. 4E Ch. 29.1 - Prob. 5E Ch. 29.1 - Prob. 6E Ch. 29.1 - Prob. 7E Ch. 29.1 - Prob. 8E Ch. 29.1 - Prob. 9E

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