Explanation Given: Let f ( t ) be piecewise continuous on ( 0 , ∞ ) and of exponential order α and assume lim t → 0 1 [ f ( t ) t ] exist and also F ( s ) = L { f ( s ) } . [Hint: First show that d d s L { f ( t ) t } ( s ) = − F ( s ) and then use the result of Problem 2 6 .] Approach: a) Consider F ( u ) is the Laplace transform of f ( s ) . b) Use definition to write F ( u ) = ∫ 0 ∞ e − st f ( t ) d t . c) Integrate above equation on both side with respect to s and take limits s and ∞ . Calculation: Consider F ( u ) is the Laplace transform of f ( s ) . Use definition to write F ( u ) = ∫ 0 ∞ e − st f ( t ) d t . Integrate above equation on both side with respect to s and take limits s and ∞ . Then, ∫ s ∞ F ( u ) d u = ∫ s ∞ [ ∫ 0 ∞ e − s t f ( t ) d t ] d s Change the order of the integration on right side, ∫ s ∞ F ( s ) d s = ∫ s ∞ [ ∫ 0 ∞ e − st d s ] f ( t ) d t Solve the above integration,

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 7.3, Problem 27E

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To prove:

L{f(t)t}(s)=∫s∞F(u)du.

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Chapter 7.3, Problem 26E

Chapter 7.3, Problem 28E

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Chapter 7 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

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To prove: L { f ( t ) t } ( s ) = ∫ s ∞ F ( u ) d u .

Solution Summary: The author explains how to prove that f(t)=displaystyle int

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To prove:

L{f(t)t}(s)=∫s∞F(u)du.

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Chapter 7 Solutions

To prove: L { f ( t ) t } ( s ) = ∫ s ∞ F ( u ) d u .

Solution Summary: The author explains how to prove that f(t)=displaystyle int

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To prove: L{f(t)t}(s)=∫s∞F(u)du.

Want to see the full answer?

Chapter 7 Solutions

To prove:

L{f(t)t}(s)=∫s∞F(u)du.