Explanation Given: The function f a is, f a ( t ) = { f ( t − a ) , if t ≥ a , 0 , i f 0 ≤ t < a . Here, f ∈ E ( 0 , ∞ ) and a is a positive real number. Approach: Consider figure ( 1 ) , f ( t ) = u a ( t ) And u a ( t ) = { 0 , 0 ≤ t < a , 1 , t ≥ a . Here, u a ( t ) is a unit step function or heaviside step function, a is any positive integer. Figure ( 1 ) In figure ( 2 ) , f ( t ) = u a ( t ) − u b ( t ) ... ( 1 ) Here, b > a Figure ( 2 ) Unit step function f ( t ) = u a ( t ) − u b ( t ) Use properties of the Laplace transform and the table ( 1 ) of Laplace transforms to determine L { f a ( t ) } . The Laplace transform satisfies the linearity properties as follows. L [ f + g ] = L [ f ] + L [ g ] L [ c f ] = c L [ f ] The properties of Laplace transformation for all transformable functions f and g and constants c are given below. Function f ( t ) Laplace Transform F ( s ) f ( t ) = t n , n is a nonnegative integer F ( s ) = n ! s n + 1 , s > 0 . f ( t ) = e a t , a constant F ( s ) = 1 s − a , s > a . f ( t ) = sin b t , b constant F ( s ) = b s 2 + b 2 , s > 0 . f ( t ) = cos b t , b constant F ( s ) = s s 2 + b 2 , s > 0 . f ( t ) = t − 1 / 2 F ( s ) = ( π / s ) 1 / 2 , s > 0 . f ( t ) = u a ( t ) (see Section 10

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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Chapter 10.10, Problem 29AP

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The Laplace transformation of the given function fa(t) is L{fa(t)}=f(s+a).

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Chapter 10.10, Problem 28AP

Chapter 10.10, Problem 30AP

Chapter 10 Solutions

Differential Equations and Linear Algebra (4th Edition)

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The Laplace transformation of the given function f a ( t ) is L { f a ( t ) } = f ( s + a ) .

Solution Summary: The author explains the Laplace transformation of the given function f_a(t).

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The Laplace transformation of the given function fa(t) is L{fa(t)}=f(s+a).

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Chapter 10 Solutions

The Laplace transformation of the given function f a ( t ) is L { f a ( t ) } = f ( s + a ) .

Solution Summary: The author explains the Laplace transformation of the given function f_a(t).

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To show: The Laplace transformation of the given function fa(t) is L{fa(t)}=f(s+a).

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Chapter 10 Solutions

To show:

The Laplace transformation of the given function fa(t) is L{fa(t)}=f(s+a).