Explanation Given: Bernoulli differential equation is of the form d y d x + P ( x ) y = Q ( x ) y n (1) u = y 1 − n (2) Product rule d ( f ⋅ g ) d x = f ⋅ d ( g ) d x + g ⋅ d ( f ) d x Calculation: Obtain the solution of the differential equation x y ′ + y = − x y 2 . Rewrite the given differential equation as follows, x y ′ + y = − x y 2 Divide the differential equation by x , ( x y ′ + y ) x = − x y 2 x x y ′ x + y x = − x y 2 x y ′ + 1 x y = − y 2 Thus, the redefined differential equation is y ′ + 1 x y = − y 2 . This is a Bernoulli differential equation of the form d y d x + P ( x ) y = Q ( x ) y n . Comparing the coefficients of y and constant terms, P ( x ) = 1 x , Q ( x ) = − y 2 and n = 2 Substitute n = 2 in equation (2) u = y 1 − n = y 1 − 2 = y − 1 Differentiate the above equation with respect to x d u d x = d ( y − 1 ) d x d y d x = − 1 y 2 d y d x d y d x = − y 2 d u d x Substitute d y d x = − y 2 d u d x the redefined differential equation is y ′ + 1 x y = − y 2 − y 2 d u d x + 1 x y = "

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ISBN: 9780357019788

Author: Stewart

Publisher: CENGAGE L

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Chapter 9.5, Problem 24E

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To solve: The differential equation xy′+y=−xy2 using Bernoulli differential equation.

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Chapter 9.5, Problem 23E

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The differential equation x y ′ + y = − x y 2 using Bernoulli differential equation.

Solution Summary: The author explains how to solve the differential equation xyprime +y=-x

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To solve: The differential equation xy′+y=−xy2 using Bernoulli differential equation.

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