Explanation Formula used: Integrating factor: The integrating factor of a function f ( x ) is given by I ( x ) = e ∫ f ( x ) d x . Calculation: First find the solution by using an integrating factor. The given equation can be written as follows. y ′ = 2 x ( y + 1 ) y ′ = 2 x y + 2 x y ′ − 2 x y = 2 x Find the integrating factor of this first order linear equation where f ( x ) = − 2 x and g ( x ) = 2 x . I ( x ) = e ∫ − 2 x d x = e − 2 ∫ x d x = e − 2 ( x 2 2 ) = e − x 2 Multiply both sides of the equation y ′ − 2 x y = 2 x by the integrating factor. e − x 2 y ′ − 2 e − x 2 x y = 2 e − x 2 x ( e − x 2 y ) ′ = 2 e − x 2 x [ ∵ ( x y ) ′ = x y ′ + x ′ y ] Integrating on both sides, ∫ ( e − x 2 y ) ′ d x =

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Differential Equations

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Chapter 9.3, Problem 53E

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To find: The general solution of the equation y′=2x(y+1) using integrating factor and separation of variables.

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Chapter 9.3, Problem 52E

Chapter 9.3, Problem 54E

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Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

Ch. 9.1 - Show that y=Cx+1 is the general solution of the...Ch. 9.1 - Prob. 2MP Ch. 9.1 - Prob. 3MP Ch. 9.1 - Prob. 4MP Ch. 9.1 - Prob. 5MP Ch. 9.1 - Prob. 1ED Ch. 9.1 - Prob. 2ED Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E

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The general solution of the equation y ′ = 2 x ( y + 1 ) using integrating factor and separation of variables.

Solution Summary: The author explains how to find the general solution of the equation yprime, using integrating factor and separation of variables.

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To find: The general solution of the equation y′=2x(y+1) using integrating factor and separation of variables.

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Chapter 9 Solutions