To determine The Laplace transformation of the function f ( t ) = t e a t by using the Laplace transform. 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 e a t . Compute the Laplace transformation of f ( t ) = t e a t as follows. Substitute f ( t ) = t e a t in F ( s ) = ∫ 0 ∞ e − s t f ( t ) d t and evaluate further. F ( s ) = ∫ 0 ∞ e − s t t e a t d t (1) Use integration by parts and simplify (1) as follows. In (1), u = t , then du = dt and d v = e − s t ⋅ e a t d t , that is, d v = e ( a − s ) t d t . That implies, ∫ d v = ∫ e ( a − s ) t d t v = ∫ e w ( a − s ) d w [ Let w = ( a − s ) t , then d w = ( a − s ) d t ] v = 1 a − s ∫ e w d w v = 1 a − s [ e w ] v = 1 a − s e ( a − s ) t [ ∵ w = ( a − s ) t ] That is, v = 1 a − s e ( a − s ) t

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ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 6.1, Problem 15P

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

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Elementary Differential Equations

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