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. Express the given differential equation as follows, Divide the differential equation by x 2 , x 2 y ′ − y x 2 = 2 x 3 e − 1 x x 2 x 2 y ′ x 2 − 1 x 2 y = 2 x 3 e − 1 x x 2 y ′ − 1 x 2 y = 2 x e − 1 x Thus, the redefined differential equation is y ′ − 1 x 2 y = 2 x e − 1 x . This is a first order linear differential equation of the form d y d x + P ( x ) y = Q ( x ) . Comparing the coefficients of y and constant terms, P ( x ) = − 1 x 2 and Q ( x ) = 2 x e − 1 x The integrating factor is computed as follows, Substitute P ( x ) = − 1 x 2 in equation (1), I ( x ) = e ∫ − 1 x 2 d x = e − ∫ x − 2 d x = e − ( x − 2 + 1 − 2 + 1 ) </

8th Edition

ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

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Chapter 9, Problem 8RE

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To solve: The differential equation x2y′−y=2x3e−1x .

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The differential equation x 2 y ′ − y = 2 x 3 e − 1 x .

Solution Summary: The author explains how to solve the differential equation by dividing it by x2 and obtaining the solution.

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To solve: The differential equation x2y′−y=2x3e−1x .

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