Explanation Formula used: Integrating factor: The integrating factor of the First-Order linear differential equation y ′ + f ( x ) y = g ( x ) is I ( x ) = e ∫ f ( x ) d x . Theorem used: The solution of the differential equation, f ( y ) y ′ = g ( x ) is given implicitly by the equation, ∫ f ( y ) d y = ∫ g ( x ) d x . Calculation: Obtain the solution of the differential equation by using integrating factor. Rewrite the differential equation y ′ = 2 x + 2 x y 1 + x 2 as, y ′ = 2 x + 2 x y 1 + x 2 y ′ = 2 x 1 + x 2 + 2 x y 1 + x 2 y ′ − 2 x y 1 + x 2 = 2 x 1 + x 2 . Comparing the equations, y ′ − 2 x y 1 + x 2 = 2 x 1 + x 2 and y ′ + f ( x ) y = g ( x ) , it is observed that f ( x ) = − 2 x 1 + x 2 and g ( x ) = 2 x 1 + x 2 . I ( x ) = e − ∫ 2 x 1 + x 2 d x = e − ln ( 1 + x 2 ) = e ln 1 ( 1 + x 2 ) = 1 1 + x 2 Thus, the integrating factor of the differential equation y ′ − 2 x y 1 + x 2 = 2 x 1 + x 2 is 1 1 + x 2 . Multiply both sides of the equation y ′ − 2 x y 1 + x 2 = 2 x 1 + x 2 by the integrating factor 1 1 + x 2 . 1 1 + x 2 y ′ − 1 1 + x 2 2 x y 1 + x 2 = 1 1 + x 2 2 x 1 + x 2 ( 1 1 + x 2 y ) ′ = 2 x ( 1 + x 2 ) 2 [ ∵ ( x y ) ′ = x y ′ + x ′ y ] Integrate both sides of the equation

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Author: Pearson

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Chapter 9.3, Problem 51E

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To find: The general solution of the equation y′=2x+2y1+x2 using integrating factor and separation of variables.

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Chapter 9.3, Problem 50E

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

Solution Summary: The author explains how to obtain the solution of the differential equation by using integrating factor and separation of variables.

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To find: The general solution of the equation y′=2x+2y1+x2 using integrating factor and separation of variables.

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