Explanation Given: The given functionis y ( t ) = e a t sin ( ω t ) . Formula used: From the definition, the Laplace transform function Y of the function y is Y ( s ) = ∫ 0 ∞ y ( t ) e − s t d t Use integration byparts formula. ∫ f ( x ) g ' ( x ) d x = f ( x ) g ( x ) − ∫ g ( x ) f ' ( x ) d x Calculation: L [ y ( t ) ] = L [ e a t sin ( ω t ) ] = ∫ 0 ∞ ( e a t sin ( ω t ) ) e − s t d t = ∫ 0 ∞ e ( a − s ) t sin ( ω t ) d t Apply the property of improper integrals. lim b → ∞ ∫ 0 ∞ e ( a − s ) t sin ( ω t ) d t Find the solution of ∫ e ( a − s ) t sin ( ω t ) d t Apply integration by parts formula. f ( x ) = e ( a − s ) t , f ' ( x ) = e ( a − s ) t a − s g ' ( x ) = sin ( ω t ) , g ( x ) = − 1 ω cos ( ω t ) ∫ e ( a − s ) t sin ( ω t ) d t = − 1 ω e ( a − s ) t cos ( ω t ) − ∫ 1 a − s ⋅ e ( a − s ) t ( − 1 ω cos ( ω t ) ) d t = − 1 ω e ( a − s ) t cos ( ω t ) + 1 ω ( a − s ) ∫ e ( a − s ) t ( cos ( ω t ) ) d t Apply integration by parts formula. f ( x ) = e ( a − s ) t , f ' ( x ) = e ( a − s ) t a − s g ' ( x ) = cos ( ω t ) , g ( x ) = 1 ω sin ( ω t ) = − 1 ω e ( a − s ) t cos ( ω t ) + 1 ω ( a − s ) { 1 ω e ( a − s ) t sin ( ω t ) − ∫ 1 ( a − s ) e ( a − s ) t ⋅ 1 ω sin ( ω t ) d t } = − 1 ω e ( a − s ) t cos ( ω t ) + 1 ω 2 ( a − s ) e ( a − s ) t sin ( ω t ) − 1 ω 2 ( a − s ) 2 ∫ e ( a − s ) t sin ( ω t ) d t = − 1 ω e ( a − s ) t cos ( ω t ) + 1 ω 2 ( a − s ) e ( a − s ) t sin ( ω t ) − 1 ω 2 ( a − s ) 2 ∫ e ( a − s ) t sin ( ω t ) d t ⇒ ∫ e ( a − s ) t sin ( ω t ) d t + 1 ω 2 ( a − s ) 2 ∫ e ( a − s ) t sin ( ω t ) d t = − 1 ω e ( a − s ) t cos ( ω t ) + 1 ω 2 ( a − s ) e ( a − s ) t sin ( ω t ) − 1 ω 2 ( a − s

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Author: Paul Blanchard, Robert L. Devaney, Glen R. Hall

Publisher: Cengage Learning

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Chapter 6.3, Problem 3E

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Find the Laplace transform of the given function using definition.

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Ch. 6.1 - In Exercises 1-4, compute the Laplace transform of...Ch. 6.1 - In Exercises 1-4, compute the Laplace transform of...Ch. 6.1 - In Exercises 1-4, compute the Laplace transform of...Ch. 6.1 - Verify that L[tn]=n!sn+1(s0) [Hint: A rigorous...Ch. 6.1 - Using L[tn]=n!sn+1(s0) give a formula for the...Ch. 6.1 - In Exercises 7-14, find the inverse Laplace...Ch. 6.1 - Prob. 8E Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 11E Ch. 6.1 - Prob. 13E

Ch. 6.1 - In Exercises 15-24 (a) compute the Laplace...Ch. 6.1 - In Exercises 15-24 (a) compute the Laplace...Ch. 6.1 - Prob. 25E Ch. 6.2 - In Exercises 8-13, solve the given initial-value...Ch. 6.3 - Prob. 3E Ch. 6.3 - Prob. 5E Ch. 6.3 - Prob. 7E Ch. 6.3 - In Exercises 11-14, write the given quadratic in...Ch. 6.3 - In Exercises 1518 , compute the inverse Laplace...Ch. 6.3 - In Exercises 2734 (a) compute the Laplace...Ch. 6.3 - Prob. 29E Ch. 6.3 - In Exercises 27-34 (a) compute the Laplace...Ch. 6.5 - Prob. 1E Ch. 6.5 - Prob. 2E Ch. 6.5 - In Exercises 14, compute the convolution f*g for...Ch. 6.5 - In Exercises 14, compute the convolution f*g for...Ch. 6.5 - Prob. 5E

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Intro to the Laplace Transform & Three Examples; Author: Dr. Trefor Bazett;https://www.youtube.com/watch?v=KqokoYr_h1A;License: Standard YouTube License, CC-BY

Find the Laplace transform of the given function using definition.

Solution Summary: The author explains how to find the Laplace transform of the given function using definition.

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Find the Laplace transform of the given function using definition.

Chapter 6 Solutions