Explanation Given information: The given differential equation is F ( x , y ) = ( 1 + x 2 ) i − 2 x y j . Graph: At the point ( x , y ) in the plane, the vector F( x , y ) has a slope of d y d x = − ( 2 x y ) 1 + x 2 d y d x = − ( 2 x y 1 + x 2 ) The above equation can be written in differential form as 2 x y d x + ( 1 + x 2 ) d y = 0 . We will compare differential equation 2 x y d x + ( 1 + 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 = 1 + x 2 ∂ M ∂ y = ∂ ( 2 x y ) ∂ y ∂ M ∂ y = 2 x ∂ N ∂ x = ∂ ( 1 + x 2 ) ∂ x ∂ N ∂ x = 2 x Since ∂ M ∂ y = ∂ N ∂ x , hence this differential equation is exact

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ISBN: 9781337604796

Author: Larson

Publisher: CENGAGE L

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Chapter 16.1, Problem 42E

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To graph: The family of curvestangent to the force field F(x,y)=(1+x2)i−2xyj.

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Chapter 16.1, Problem 41E

Chapter 16.1, Problem 43E

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To graph: The family of curvestangent to the force field F ( x , y ) = ( 1 + x 2 ) i − 2 x y j .

Solution Summary: The author explains the family of curvestangent to the force field F(x,y)=(1+x2)i-2xyj.

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To graph: The family of curvestangent to the force field F(x,y)=(1+x2)i−2xyj.

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Chapter 16 Solutions