Explanation Formula used: The Laplace transformation of f : “The Laplace transform of f which is dented by L { f ( t ) } = F ( s ) = ∫ 0 ∞ e − s t f ( t ) d t ”. Calculation: The given function is f ( t ) = t n e a t . Compute the Laplace transformation of f ( t ) = t n e a t as follows. Substitute f ( t ) = t n e a t in F ( s ) = ∫ 0 ∞ e − s t f ( t ) d t and evaluate further. F ( s ) = ∫ 0 ∞ e − s t t n e a t d t = ∫ 0 ∞ t n e ( − s + a ) t d t . Here, u = t n , then d u = n t n − 1 d t and d v = e ( − s + a ) t d t . That implies, ∫ d v = ∫ e ( − s + a ) t d t v = ∫ e w ( − s + a ) d w [ Let w = ( − s + a ) t , then d w = ( − s + a ) d t ] v = 1 ( − s + a ) ∫ e w d w v = 1 ( − s + a ) [ e w ] v = 1 ( − s + a ) e ( − s + a ) t [ ∵ w = ( − s + a ) t ] That is, v = 1 ( − s + a ) e ( − s + a ) t . That implies, F ( s ) = lim A → ∞ ( [ t n ⋅ 1 ( − s + a ) e ( − s + a ) t ] 0 A − ∫ 0 A n ( − s + a ) t n − 1 e ( − s + a ) t d t ) = [ − t n ⋅ 1 ( s − a ) e ( − s + a ) t ] 0 ∞ + n ( s − a ) ∫ 0 ∞ t n − 1 e ( − s + a ) t d t = ( 0 − 0 ) + n ( s − a ) ∫ 0 ∞ t n − 1 e ( − s + a ) t d t = n ( s − a ) ∫ 0 ∞ t n − 1 e ( − s + a ) t d t Note that, the limit and the integral exist only if e ( a − s ) t approaches to 0 as faster than lim t → ∞ t n approaches to ∞ . That implies, F ( s ) = n ( s − a ) ⋅ n − 1 ( s − a ) ⋅ ... ⋅ 2 ( s − a ) F ( s ) [ ∵ F ( s ) = ∫ 0 ∞ t e a t e − s t d t ] (1) Compute the Laplace transformation of F ( s ) = ∫ 0 ∞ t e a t e − s t d t as follows. F ( s ) = ∫ 0 ∞ t e a t e − s t d t = ∫ 0 ∞ t e ( a − s ) t d t Here, u = t , then d u = d t and d v = e ( a − s ) t d t

Elementary Differential Equations and Boundary Value Problems, Enhanced

11th Edition

ISBN: 9781119381648

Author: Boyce

Publisher: WILEY

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Chapter 6.1, Problem 14P

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The Laplace transformation of the function f(t)=tneat by using the Laplace transform.

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Chapter 6.1, Problem 13P

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Chapter 6 Solutions

Elementary Differential Equations and Boundary Value Problems, Enhanced

Ch. 6.1 - Prob. 1P Ch. 6.1 - Prob. 2P Ch. 6.1 - Prob. 3P Ch. 6.1 - Prob. 4P Ch. 6.1 - Prob. 5P Ch. 6.1 - Prob. 6P Ch. 6.1 - Prob. 7P Ch. 6.1 - Prob. 8P Ch. 6.1 - Prob. 9P Ch. 6.1 - Prob. 10P

Ch. 6.1 - Prob. 11P Ch. 6.1 - Prob. 12P Ch. 6.1 - Prob. 13P Ch. 6.1 - Prob. 14P Ch. 6.1 - Prob. 15P Ch. 6.1 - Prob. 16P Ch. 6.1 - Prob. 17P Ch. 6.1 - Prob. 18P Ch. 6.1 - Prob. 19P Ch. 6.1 - Prob. 20P Ch. 6.1 - Prob. 21P Ch. 6.1 - Prob. 22P Ch. 6.1 - Prob. 23P Ch. 6.1 - Prob. 24P Ch. 6.2 - Prob. 1P Ch. 6.2 - Prob. 2P Ch. 6.2 - Prob. 3P Ch. 6.2 - Prob. 4P Ch. 6.2 - Prob. 5P Ch. 6.2 - Prob. 6P Ch. 6.2 - Prob. 7P Ch. 6.2 - Prob. 8P Ch. 6.2 - Prob. 9P Ch. 6.2 - Prob. 10P Ch. 6.2 - Prob. 11P Ch. 6.2 - Prob. 12P Ch. 6.2 - Prob. 13P Ch. 6.2 - Prob. 14P Ch. 6.2 - Prob. 15P Ch. 6.2 - Prob. 16P Ch. 6.2 - Prob. 17P Ch. 6.2 - Prob. 18P Ch. 6.2 - Prob. 19P Ch. 6.2 - Prob. 20P Ch. 6.2 - Prob. 21P Ch. 6.2 - Prob. 22P Ch. 6.2 - Prob. 23P Ch. 6.2 - Prob. 24P Ch. 6.2 - Prob. 25P Ch. 6.2 - Prob. 26P Ch. 6.2 - Prob. 27P Ch. 6.2 - Prob. 28P Ch. 6.2 - Prob. 29P Ch. 6.3 - Prob. 1P Ch. 6.3 - Prob. 2P Ch. 6.3 - Prob. 3P Ch. 6.3 - Prob. 4P Ch. 6.3 - Prob. 5P Ch. 6.3 - Prob. 6P Ch. 6.3 - Prob. 7P Ch. 6.3 - Prob. 8P Ch. 6.3 - Prob. 9P Ch. 6.3 - Prob. 10P Ch. 6.3 - Prob. 11P Ch. 6.3 - Prob. 12P Ch. 6.3 - Prob. 13P Ch. 6.3 - Prob. 14P Ch. 6.3 - Prob. 15P Ch. 6.3 - Prob. 16P Ch. 6.3 - Prob. 17P Ch. 6.3 - Prob. 18P Ch. 6.3 - Prob. 19P Ch. 6.3 - Prob. 20P Ch. 6.3 - Prob. 21P Ch. 6.3 - Prob. 22P Ch. 6.3 - Prob. 23P Ch. 6.3 - Prob. 24P Ch. 6.3 - Prob. 25P Ch. 6.3 - Prob. 26P Ch. 6.3 - Prob. 27P Ch. 6.3 - Prob. 28P Ch. 6.3 - Prob. 29P Ch. 6.3 - Prob. 30P Ch. 6.4 - Prob. 1P Ch. 6.4 - Prob. 2P Ch. 6.4 - Prob. 3P Ch. 6.4 - Prob. 4P Ch. 6.4 - Prob. 5P Ch. 6.4 - Prob. 6P Ch. 6.4 - Prob. 7P Ch. 6.4 - In each of Problems 1 through 8: a. Sketch the...Ch. 6.4 - Prob. 9P Ch. 6.4 - Prob. 10P Ch. 6.4 - Prob. 11P Ch. 6.4 - Prob. 12P Ch. 6.4 - Prob. 13P Ch. 6.4 - Prob. 14P Ch. 6.4 - Prob. 15P Ch. 6.4 - Prob. 16P Ch. 6.4 - Prob. 17P Ch. 6.5 - Prob. 1P Ch. 6.5 - Prob. 2P Ch. 6.5 - Prob. 3P Ch. 6.5 - Prob. 4P Ch. 6.5 - Prob. 5P Ch. 6.5 - Prob. 6P Ch. 6.5 - Prob. 7P Ch. 6.5 - Prob. 8P Ch. 6.5 - Prob. 9P Ch. 6.5 - Prob. 10P Ch. 6.5 - Prob. 11P Ch. 6.5 - Prob. 12P Ch. 6.5 - Prob. 13P Ch. 6.5 - Prob. 14P Ch. 6.5 - Prob. 15P Ch. 6.5 - Prob. 16P Ch. 6.5 - Prob. 17P Ch. 6.5 - Prob. 18P Ch. 6.5 - Prob. 19P Ch. 6.6 - Prob. 1P Ch. 6.6 - Prob. 2P Ch. 6.6 - Prob. 3P Ch. 6.6 - Prob. 4P Ch. 6.6 - Prob. 5P Ch. 6.6 - In each of Problems 4 through 7, find the Laplace...Ch. 6.6 - Prob. 7P Ch. 6.6 - Prob. 8P Ch. 6.6 - Prob. 9P Ch. 6.6 - Prob. 10P Ch. 6.6 - Prob. 11P Ch. 6.6 - Prob. 12P Ch. 6.6 - Prob. 13P Ch. 6.6 - Prob. 14P Ch. 6.6 - Prob. 15P Ch. 6.6 - Prob. 16P Ch. 6.6 - Prob. 17P Ch. 6.6 - Prob. 18P Ch. 6.6 - Prob. 19P Ch. 6.6 - Prob. 20P Ch. 6.6 - Prob. 21P

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The Laplace transformation of the function f ( t ) = t n e a t by using the Laplace transform.

The Laplace transformation of the function f(t)=tneat by using the Laplace transform.

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Chapter 6 Solutions