a. Explanation Given information: The wave equation in physics is given by the equation ∂ 2 z ∂ t 2 = c 2 ∂ 2 z ∂ x 2 and heat equation in the physics is given by the equation ∂ z ∂ t = c 2 ∂ 2 z ∂ x 2 . Formula used: The derivative with respect to a variable for the function of two or more than two variables is partially differentiating with respect to that variable and keeping all other variables as constants. The second order partial derivatives for a function of two variables z = g ( x , y ) are as follows: With respect to x , ∂ 2 f ∂ x 2 g ( x , y ) = ∂ ∂ x ( ∂ g ∂ x ) = ( g x ) x = g x x With respect to y , ∂ 2 f ∂ y 2 g ( x , y ) = ∂ ∂ y ( ∂ g ∂ y ) = ( g y ) y = g y y The derivative of the function e a x is, d d x e a x = a e a x . Trigonometric derivatives are as follows: d d x sin a x = a cos a x and d d x cos a x = − a sin a x Calculation: Consider the function, z = e − t ( sin x c + cos x c ) . First we will check whether it satisfies the wave equation or not. The wave equation is, ∂ 2 z ∂ t 2 = c 2 ∂ 2 z ∂ x 2 . Since, we know he derivative with respect to a variable for the function of two or more than two variables is partially differentiating with respect to that variable and keeping all other variables as constants. Left hand side of the equation is, ∂ 2 ∂ t 2 ( e − t ( sin x c + cos x c ) ) = ∂ ∂ t ( ∂ ∂ t ( e − t ( sin x c + cos x c ) ) ) = ∂ ∂ t ( − e − t ( sin x c + cos x c ) ) = e − t ( sin x c + cos x c ) Right hand side of the equation is, c 2 ∂ 2 z ∂ x 2 = c 2 ∂ ∂ x ( ∂ ∂ x ( e − t ( sin x c + cos x c ) b. To determine To check: Whetherthe function sin 3 c t sin 3 x satisfies the heat and wave equation in physics. c. To determine To check: Whetherthe function sin 5 c t cos 5 x satisfies the heat and wave equation in physics.

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

6th Edition

ISBN: 9781524908102

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Question

Chapter 11.3, Problem 50PS

To determine

To chck: Whetherthe function z=e−t(sinxc+cosxc) satisfies the heat and wave equation in physics.

To determine

To check: Whetherthe function sin3ctsin3x satisfies the heat and wave equation in physics.

To determine

To check: Whetherthe function sin5ctcos5x satisfies the heat and wave equation in physics.

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Chapter 11.3, Problem 49PS

Chapter 11.3, Problem 51PS

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Chapter 11 Solutions

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

Ch. 11.1 - Prob. 1PS Ch. 11.1 - Prob. 2PS Ch. 11.1 - Prob. 3PS Ch. 11.1 - Prob. 4PS Ch. 11.1 - Prob. 5PS Ch. 11.1 - Prob. 6PS Ch. 11.1 - Prob. 7PS Ch. 11.1 - Prob. 8PS Ch. 11.1 - Prob. 9PS Ch. 11.1 - Prob. 10PS

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To chck: Whetherthe function z = e − t ( sin x c + cos x c ) satisfies the heat and wave equation in physics.

Solution Summary: The author explains that the function z=e-tleft satisfies the heat equation and not the wave equation in physics.

To chck: Whetherthe function z=e−t(sinxc+cosxc) satisfies the heat and wave equation in physics.

To check: Whetherthe function sin3ctsin3x satisfies the heat and wave equation in physics.

To check: Whetherthe function sin5ctcos5x satisfies the heat and wave equation in physics.

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Chapter 11 Solutions