Explanation Given information: The given differential equation is 2 x y d x + ( y 2 − x 2 ) d y = 0 . Concept Used: 1) Let M and N have continuous partial derivatives on an open disk R . The differential equation M ( x , y ) d x + N ( x , y ) d y = 0 is exact if and only if ∂ M ∂ y = ∂ N ∂ x . 2) If M y ( x , y ) − N x ( x , y ) N ( x , y ) = h ( x ) is a function of x alone, then e ∫ h ( x ) d x is an integrating factor. 3) If N x ( x , y ) − M y ( x , y ) M ( x , y ) = k ( y ) is a function of y alone, then e ∫ k ( y ) d y is an integrating factor. 4) The equation M ( x , y ) d x + N ( x , y ) d y = 0 is an exact differential equation when there exists a function f of two variables x and y having continuous partial derivatives such that f x ( x , y ) = M ( x , y ) and f y ( x , y ) = N ( x , y ) The general solution of the equation is f y ( x , y ) = C Calculation: First we will compare differential equation 2 x y d x + ( y 2 − x 2 ) d y = 0 with the M ( x , y ) d x + N ( x , y ) d y = 0 . We have M = 2 x y N = y 2 − x 2 ∂ M ∂ y = ∂ ( 2 x y ) ∂ y ∂ M ∂ y = 2 x ∂ N ∂ x = ∂ ( y 2 − x 2 ) ∂ x ∂ N ∂ x = − 2 x Since ∂ M ∂ y ≠ ∂ N ∂ x , hence this differential equation is not exact

11th Edition

ISBN: 9781337604796

Author: Larson

Publisher: CENGAGE L

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Chapter 16, Problem 14RE

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The integrating factor that is function of x or y alone and use it to find the general solution of the differential equation 2xydx+(y2−x2)dy=0.

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Chapter 16, Problem 13RE

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The integrating factor that is function of x or y alone and use it to find the general solution of the differential equation 2 x y d x + ( y 2 − x 2 ) d y = 0 .

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The integrating factor that is function of x or y alone and use it to find the general solution of the differential equation 2xydx+(y2−x2)dy=0.

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