(a) Explanation Given: L { f } = F ( s ) . lim k → ∞ e − s k f ( k ) = 0 if s > 0 . The Laplace transform: F ( s ) = L { f ( t ) } = ∫ 0 ∞ e − s t f ( t ) d t Proof: Show that: L { f ' } = s F ( s ) − f ( 0 ) By definition of Laplace transform, L { f ' ( t ) } = ∫ 0 ∞ e − s t f ' ( t ) d t = lim N → ∞ ∫ 0 N e − s t f ' ( t ) d t Integrate by parts, L { f ' ( t ) } = lim N → ∞ ∫ 0 N e − s t ︸ u f ' ( t ) ︸ v d t = lim N → ∞ [ { e − s t (b) To determine To find: the value of L { y } by applying the result in part a (c) To determine To solve: the initial value problem in part b by using inverse Laplace transforms to obtain y .

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 7.7, Problem 60PS

(a)

To determine

To prove: L{f'}=sF(s)−f(0) .

(b)

To determine

To find:the value of L{y} by applying the result in part a

(c)

To determine

To solve: the initial value problem in part b by using inverse Laplace transforms to obtain y .

Expert Solution & Answer

Chapter 7 Solutions

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

Ch. 7.1 - Prob. 1PS Ch. 7.1 - Prob. 2PS Ch. 7.1 - Prob. 3PS Ch. 7.1 - Prob. 4PS Ch. 7.1 - Prob. 5PS Ch. 7.1 - Prob. 6PS Ch. 7.1 - Prob. 7PS Ch. 7.1 - Prob. 8PS Ch. 7.1 - Prob. 9PS Ch. 7.1 - Prob. 10PS

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To prove: L { f ' } = s F ( s ) − f ( 0 ) .

Solution Summary: The author explains the Laplace transform and how it works.

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To prove: L{f'}=sF(s)−f(0) .

To find:the value of L{y} by applying the result in part a

To solve: the initial value problem in part b by using inverse Laplace transforms to obtain y .

Chapter 7 Solutions