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 ) = e a t cosh b t . Compute the Laplace transformation of f ( t ) = e a t cosh b t as follows. Substitute f ( t ) = e a t cosh b t in F ( s ) = ∫ 0 ∞ e − s t f ( t ) d t and evaluate further. F ( s ) = ∫ 0 ∞ e − s t e a t cosh b t d t . Let F ( s ) = lim A → ∞ ∫ 0 A e − s t e a t cosh b t d t (1) It is given that, cosh b t = e b t + e − b t 2 . So, the above equation will become as, F ( s ) = lim A → ∞ ∫ 0 A e − s t e a t ( e b t + e − b t 2 ) d t = lim A → ∞ ∫ 0 A e − s t ( e ( a + b ) t + e ( a − b ) t 2 ) d t = lim A → ∞ [ 1 2 ∫ 0 A e ( a + b ) t ⋅ e − s t d t + 1 2 ∫ 0 A e ( a − b ) t ⋅ e − s t d t ] = lim A → ∞ [ 1 2 ∫ 0 A e ( a − s + b ) t d t + 1 2 ∫ 0 A e − ( b + s − a ) t d t ] Let F ( s ) = lim A → ∞ [ 1 2 ∫ 0 A e ( a − s + b ) t d t + 1 2 ∫ 0 A e − ( b + s − a ) t d t ] be equation (2). First integrate ∫ 0 A e ( a − s + b ) t d t with respect to t as follows. Let u = ( a − s + b ) t , then d u = ( a − s + b ) d t . Note that, for t = 0, u = 0 and for t = A , u = ( a − s + b ) A . That implies, ∫ 0 A e ( a − s + b ) t d t = ∫ 0 ( a − s + b ) A e u a − s + b d u = 1 a − s + b ∫ 0 ( a − s + b ) A e u d u = 1 a − s + b [ e u ] 0 ( a − s + b ) A = 1 a − s + b [ e ( a − s + b ) A − e 0 ] = 1 a − s + b [ e ( a − s + b ) A − 1 ] That is, ∫ 0 A e ( a − s + b ) t d t = 1 a − s + b [ e ( a − s + b ) A − 1 ]

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Chapter 6.1, Problem 9P

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The Laplace transformation of the function f(t)=eatcoshbt.

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can I see the steps for how you got the same answers already provided for μ1->μ4. this is a homework that provide you answers for question after attempting it three tries

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Chapter 6 Solutions

Elementary Differential Equations

Ch. 6.1 - In each of Problems 1 through 4, sketch the graph...Ch. 6.1 - In each of Problems 1 through 4, sketch the graph...Ch. 6.1 - In each of Problems 1 through 4, sketch the graph...Ch. 6.1 - Prob. 4P Ch. 6.1 - Find the Laplace transform of each of the...Ch. 6.1 - Find the Laplace transform of f (t) = cos at,...Ch. 6.1 - Recall that cosh bt = (ebt + e−bt)/2 and sinh bt =...Ch. 6.1 - Prob. 8P Ch. 6.1 - Prob. 9P Ch. 6.1 - Recall that cosh bt = (ebt + e−bt)/2 and sinh bt =...

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The Laplace transformation of the function f ( t ) = e a t cosh b t .

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