Explanation Given: The given convolution is ( 3 sin t ) ∗ ( cos 2 t ) . Formula Used: The convolution for the functions f ( t ) and g ( t ) (both are causal functions) are ( f ∗ g ) ( t ) = ∫ 0 t f ( t − x ) g ( x ) d x The commutative property of convolution is also true. ( g ∗ f ) ( t ) = ∫ 0 t g ( t − x ) f ( x ) d x The trigonometric identity 2 cos ( p ) sin ( q ) = sin ( p + q ) − sin ( p − q ) Calculation: f ( t ) = 3 sin t f ( t − x ) = 3 sin ( t − x ) g ( t ) = cos 2 t g ( x ) = cos 2 x Now, ( 3 sin t ) ∗ ( cos 2 t ) = ∫ 0 t 3 sin ( t − x ) ⋅ cos ( 2 x ) d x = 3 ∫ 0 t [ cos ( 2 x ) ⋅ sin ( t − x ) ] d x = 3 ∫ 0 t 1 2 [ sin ( 2 x + t − x ) − sin { 2 x − ( t − x ) } ] d x = 3 2 ∫ 0 t [ sin ( x + t ) − sin ( 3 x − t ) ] d x = 3 2 [ ∫ 0 t sin ( x + t ) d x − ∫ 0 t sin ( 3 x − t ) d x ] Apply u-substitution in both parts

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ISBN: 9780495561989

Author: Paul Blanchard, Robert L. Devaney, Glen R. Hall

Publisher: Cengage Learning

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Chapter 6.5, Problem 5E

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Find the convolution f∗g for the given functions.

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Differential Equations

Ch. 6.1 - In Exercises 1-4, compute the Laplace transform of...Ch. 6.1 - In Exercises 1-4, compute the Laplace transform of...Ch. 6.1 - In Exercises 1-4, compute the Laplace transform of...Ch. 6.1 - Verify that L[tn]=n!sn+1(s0) [Hint: A rigorous...Ch. 6.1 - Using L[tn]=n!sn+1(s0) give a formula for the...Ch. 6.1 - In Exercises 7-14, find the inverse Laplace...Ch. 6.1 - Prob. 8E Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 11E Ch. 6.1 - Prob. 13E

Ch. 6.1 - In Exercises 15-24 (a) compute the Laplace...Ch. 6.1 - In Exercises 15-24 (a) compute the Laplace...Ch. 6.1 - Prob. 25E Ch. 6.2 - In Exercises 8-13, solve the given initial-value...Ch. 6.3 - Prob. 3E Ch. 6.3 - Prob. 5E Ch. 6.3 - Prob. 7E Ch. 6.3 - In Exercises 11-14, write the given quadratic in...Ch. 6.3 - In Exercises 1518 , compute the inverse Laplace...Ch. 6.3 - In Exercises 2734 (a) compute the Laplace...Ch. 6.3 - Prob. 29E Ch. 6.3 - In Exercises 27-34 (a) compute the Laplace...Ch. 6.5 - Prob. 1E Ch. 6.5 - Prob. 2E Ch. 6.5 - In Exercises 14, compute the convolution f*g for...Ch. 6.5 - In Exercises 14, compute the convolution f*g for...Ch. 6.5 - Prob. 5E

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Find the convolution f ∗ g for the given functions.

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Find the convolution f∗g for the given functions.

Chapter 6 Solutions