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Solved: In 1 and 2, use the definition of the Laplace

Fundamentals of Differential Equations | 8th Edition | ISBN: 9780321747730 | Authors: R. Kent Nagle, Edward B. Saff, Arthur David Snider ISBN: 9780321747730 43

Solution for problem 1RP Chapter 7.9

Fundamentals of Differential Equations | 8th Edition

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Fundamentals of Differential Equations | 8th Edition | ISBN: 9780321747730 | Authors: R. Kent Nagle, Edward B. Saff, Arthur David Snider

Fundamentals of Differential Equations | 8th Edition

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Problem 1RP

Problem 1RP

In Problems 1 and 2, use the definition of the Laplace transform to determine  

Step-by-Step Solution:
Step 1 of 3

Auto-correlation (Wiener-Khinchin’s theorem) The auto-correlation, R(x), of a function, f(x), is de▯ned as ∫ ∞ R(x) ▯ f(x + y)f(y)dy: (1) −∞ Recalling that the convolution between f(x) and g(x) is de▯ned as ∫ ∞ f(x) ▯ g(x) ▯−∞ f(x ▯ y)g(y)dy (2) it can be shown, by setting g(x) ▯ f(▯x), that R(x) = f(x) ▯ f(▯x): (3) According to the Fourier convolution theorem, F(f(x) ▯ g(x)) = F(!)G(!): (4) ∫ ∞ G(!) = g(x)

Step 2 of 3

Chapter 7.9, Problem 1RP is Solved
Step 3 of 3

Textbook: Fundamentals of Differential Equations
Edition: 8
Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider
ISBN: 9780321747730

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Solved: In 1 and 2, use the definition of the Laplace