Explanation Given: The functions are, F ( s ) = ∫ 0 T e − s t f ( t ) d t 1 − e − s T f ( t ) = sin t The period is 2π and 4 π . Approach: 1) The Laplace transform satisfies the property: L { sin a t } ( s ) = a a 2 + s 2 The integral ∫ e a x sin b x d x is given by e a x ( a sin b x − b cos b x ) a 2 + b 2 + C . Here, C is an arbitrary constant. Calculation: Consider, L { f ( t ) } ( s ) = ∫ 0 T e − s t f ( t ) d t 1 − e − s T Take the period 2π for f ( t ) = sin t . L { sin t } ( s ) = ∫ 0 2 π e − s t sin t d t 1 − e − 2 π s = 1 1 − e − 2 π s [ e − s t s 2 + 1 ( − s sin t − cos t ) ] 0 2 π = ( 1 − e − 2 π s ) ( 1 − e − 2 π s ) ( s 2 + 1 ) = 1 s 2 + 1 Take the period 4π for f ( t ) = sin t

Fundamentals of Differential Equations [With CDROM] - 7th Edition

7th Edition

ISBN: 9780321410481

Author: Saff, Edward B., Snider, Arthur David, Nagle, R. Kent

Publisher: Addison Wesley

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Chapter 7.7, Problem 21E

To determine

To prove:

The equality F(s)=∫0Te−stf(t)dt1−e−sT for the function f(t)=sint by taking the period 2π and 4π.

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Chapter 7.7, Problem 20E

Chapter 7.7, Problem 22E

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Fundamentals of Differential Equations [With CDROM] - 7th Edition

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To prove: The equality F ( s ) = ∫ 0 T e − s t f ( t ) d t 1 − e − s T for the function f ( t ) = sin t by taking the period 2π and 4 π .

Solution Summary: The author explains how the Laplace transform satisfies the property: Lleftmathrmsintright

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To prove:

The equality F(s)=∫0Te−stf(t)dt1−e−sT for the function f(t)=sint by taking the period 2π and 4π.

Want to see the full answer?

Chapter 7 Solutions

To prove: The equality F ( s ) = ∫ 0 T e − s t f ( t ) d t 1 − e − s T for the function f ( t ) = sin t by taking the period 2π and 4 π .

Solution Summary: The author explains how the Laplace transform satisfies the property: Lleftmathrmsintright

Videos

To prove: The equality F(s)=∫0Te−stf(t)dt1−e−sT for the function f(t)=sint by taking the period 2π and 4π.

Want to see the full answer?

Chapter 7 Solutions

To prove:

The equality F(s)=∫0Te−stf(t)dt1−e−sT for the function f(t)=sint by taking the period 2π and 4π.