Explanation Given information: The given differential equation is e − ( x 2 + y 2 ) ( x d x + y 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) 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 e − ( x 2 + y 2 ) ( x d x + y d y ) = 0 with the M ( x , y ) d x + N ( x , y ) d y = 0 . We have M = x e − ( x 2 + y 2 ) N = y e − ( x 2 + y 2 ) ∂ M ∂ y = ∂ ( x e − ( x 2 + y 2 ) ) ∂ y ∂ M ∂ y = − 2 x y ⋅ e − ( x 2 + y 2 ) ∂ N ∂ x = ∂ ( y e − ( x 2 + y 2 ) ) ∂ x ∂ N ∂ x = − 2 x y ⋅ e − ( x 2 + y 2 ) Since ∂ M ∂ y = ∂ N ∂ x , hence this differential equation is exact

11th Edition

ISBN: 9781337275378

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 16.1, Problem 12E

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Whether the differential equation e−(x2+y2)(xdx+ydy)=0 is exact or not and also finds the general solution.

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Chapter 16.1, Problem 11E

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Whether the differential equation e − ( x 2 + y 2 ) ( x d x + y d y ) = 0 is exact or not and also finds the general solution.

Solution Summary: The author compares the differential equation e-(x2+y 2)(xdx+ydy)=0 with the general solution.

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Whether the differential equation e−(x2+y2)(xdx+ydy)=0 is exact or not and also finds the general solution.

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Chapter 16 Solutions