Explanation Integrating factor: I ( x ) = e ∫ P ( x ) d x (1) 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 ′ − 2 y = x 2 , x > 0 . Express the given differential equation x y ′ − 2 y = x 2 . Divide the differential equation by x , x y ′ − 2 y x = x 2 x x y ′ x − 2 y x = x 2 x y ′ − 2 y x = x Thus, the redefined differential equation is y ′ − 2 x y = x . This is a first order linear differential equation of the form y ′ + P ( x ) y = Q ( x ) . Comparing the coefficients of y and constant terms, P ( x ) = − 2 x and Q ( x ) = x The integrating factor is computed as follows, Substitute P ( x ) = − 2 x in equation (1), I ( x ) = e ∫ ( − 2 x ) d x = e − 2 ln x =

Student Solutions Manual for Stewart's Single Variable Calculus: Early Transcendentals, 8th (James Stewart Calculus)

8th Edition

ISBN: 9781305272422

Author: James Stewart, Jeffrey A. Cole, Daniel Drucker, Daniel Anderson

Publisher: Cengage Learning

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Chapter 9.5, Problem 11E

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To solve: The differential equation xy′−2y=x2,x>0 .

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Student Solutions Manual for Stewart's Single Variable Calculus: Early Transcendentals, 8th (James Stewart Calculus)

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