Explanation Given: The integral equation is f ( t ) + ∫ 0 t f ( τ ) d τ = 1 . Theorems used: 1. If the functions f and g are piecewise continuous on the interval [ 0 , ∞ ) , then the convolution of f and g, denoted by the symbol f ∗ g , is a function defined by the integral f ∗ g = ∫ 0 t f ( τ ) g ( t − τ ) d τ , where f & g are function of t . 2. If f ( t ) and g ( t ) are piecewise continuous on [ 0 , ∞ ) , and of exponential order, then L { f ∗ g } = L { f ( t ) } L { g ( t ) } . Calculation: Consider f ( t ) = 1 − ∫ 0 t f ( τ ) d τ . Take Laplace transform of integral of each term and use the above mentioned theorems as, L { f ( t ) } = L { 1 } − L { ∫ 0 t 1 ∗ f ( τ ) d τ } L { f ( t ) } = L { 1

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 4.4, Problem 45E

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To solve: The integral equation with the use of Laplace transforms.

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Chapter 4.4, Problem 44E

Chapter 4.4, Problem 46E

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The integral equation with the use of Laplace transforms.

Solution Summary: The author explains the Laplace transformation of the integral equation, which is f(t)+displaystyle int

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Chapter 4 Solutions