Explanation Using integration by parts, ∫ − ∞ ∞ f ( t ) δ ′ ( t ) d t = ( f ( t ) δ ( t ) ) | − ∞ ∞ − ∫ − ∞ ∞ f ′ ( t ) δ ′ ( t ) d t = 0 − 0 − − f ′ ( 0 ) = − f ′ ( 0 ) Therefore, ∫ − ∞ ∞ f ( t ) δ ′ ( t ) d t = − f ′ ( 0 ) . Also, ∫ − ∞ ∞ f ( t ) δ ( n ) ( t ) d t = ( f ( t ) δ ( n − 1 ) ( t ) ) | − ∞ ∞ − ∫ − ∞ ∞ f ′ ( t ) δ ( n − 1 ) ( t ) d t = 0 − 0 − ∫ − ∞ ∞ f ′ ( t ) δ (

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 7.9, Problem 34E

To determine

To Prove:

The following relations.

∫−∞∞f(t)δ′(t) dt=−f′(0) and

∫−∞∞f(t)δ(n)(t) dt=(−1)nf(n)(0).

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Chapter 7.9, Problem 33E

Chapter 7.9, Problem 35E

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Chapter 7 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

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To Prove: The following relations. ∫ − ∞ ∞ f ( t ) δ ′ ( t ) d t = − f ′ ( 0 ) and ∫ − ∞ ∞ f ( t ) δ ( n ) ( t ) d t = ( − 1 ) n f ( n ) ( 0 ) .

Solution Summary: The author explains how integration by parts proves the following relations: rll

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To Prove:

The following relations.

∫−∞∞f(t)δ′(t) dt=−f′(0) and

∫−∞∞f(t)δ(n)(t) dt=(−1)nf(n)(0).

Want to see the full answer?

Chapter 7 Solutions

To Prove: The following relations. ∫ − ∞ ∞ f ( t ) δ ′ ( t ) d t = − f ′ ( 0 ) and ∫ − ∞ ∞ f ( t ) δ ( n ) ( t ) d t = ( − 1 ) n f ( n ) ( 0 ) .

Solution Summary: The author explains how integration by parts proves the following relations: rll

Concept explainers

Videos

To Prove: The following relations. ∫−∞∞f(t)δ′(t) dt=−f′(0) and ∫−∞∞f(t)δ(n)(t) dt=(−1)nf(n)(0).

Want to see the full answer?

Chapter 7 Solutions

To Prove:

The following relations.

∫−∞∞f(t)δ′(t) dt=−f′(0) and

∫−∞∞f(t)δ(n)(t) dt=(−1)nf(n)(0).