Explanation Formula used: The differential equation d y d x + P ( x ) y = f ( x ) y n where n is any real number, is called Bernoulli’s equation . Calculation: The given differential equation is x y ″ = y ′ + ( y ′ ) 3 . Let u = y ′ so that u ′ = y ″ . The given equation becomes x u ′ = u + u 3 u ′ = ( 1 x ) u + ( 1 x ) u 3 Rearranging the terms, to get x u ′ = u + u 3 u ′ − ( 1 x ) u = ( 1 x ) u 3 Which is Bernoulli. Put w = u − 2 in the above equation, to get d w d x + ( 2 x ) w = − 2 x

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 3.7, Problem 13E

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To solve: The equation xy″=y′+(y′)3 and show that the substitution u=y′ leads to a Bernoulli equation.

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Chapter 3.7, Problem 12E

Chapter 3.7, Problem 14E

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Chapter 3 Solutions

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The equation x y ″ = y ′ + ( y ′ ) 3 and show that the substitution u = y ′ leads to a Bernoulli equation.

Solution Summary: The author explains that the substitution u=yprime leads to a Bernoulli equation.

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To solve: The equation xy″=y′+(y′)3 and show that the substitution u=y′ leads to a Bernoulli equation.

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Chapter 3 Solutions