Explanation Formula used: 1. If g ( t ) and f ( t ) are the function of t and y is defined as, y = { g ( t ) , 0 ≤ t < a f ( t ) , a ≤ t < b 0 , t ≥ b Then the unit step function for the above function is express as follows, y = g ( t ) − g ( t ) U ( t − a ) + f ( t ) U ( t − a ) − f ( t ) U ( t − b ) Here U is a unit step function. 2. If F ( s ) = L { f ( t ) } and a > 0 , then L { f ( t − a ) U ( t − a ) } = e − a s F ( s ) L { U ( t − a ) } = e − a s s 3. If L { f ( t ) } = F ( s ) and a is any real number, then L { e a t f ( t ) } = F ( s − a ) 4. The Laplace transformation of t n and 1 are as follows, L { t n } = n s n + 1 L { 1 } = 1 s Here n is a positive integer. Calculation: The step function for the given graph is, f ( t ) = { t , 0 ≤ t < 1 1 , 1 ≤ t < 4 0 , t ≥ 4 Here, f ( t ) is t when t is in between 0 to 1 , f ( t ) is 1 when t is in between 1 to 4 and f ( t ) = 0 when t ≥ 4 . Use the above definition to write the unit step function for the above step function. f ( t ) = t − t U ( t − 1 ) + U ( t − 1 ) − U ( t − 4 ) = t − ( t − 1 ) U ( t − 1 ) − U ( t − 4 ) Consider the above expression as equation (1)

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 4, Problem 29CR

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The equation for the graph in terms of the unit step function and then find L{f(t)} and L{etf(t)} using the unit step function.

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Chapter 4, Problem 28CR

Chapter 4, Problem 30CR

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Chapter 4 Solutions

Advanced Engineering Mathematics

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The equation for the graph in terms of the unit step function and then find L { f ( t ) } and L { e t f ( t ) } using the unit step function.

Solution Summary: The author explains the unit step function for the graph and the formula used.

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The equation for the graph in terms of the unit step function and then find L{f(t)} and L{etf(t)} using the unit step function.

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Chapter 4 Solutions