Explanation Given information: The given differential equation is F ( x , y ) = 4 x 2 y i − ( 2 x y 2 + x y 2 ) 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 2 + x y 2 ) 4 x 2 y d y d x = − ( 2 y 4 + 1 4 x y 3 ) The above equation can be written in differential form as ( 2 y 4 + 1 ) d x + 4 x y 3 d y = 0 . We will compare differential equation ( 2 y 4 + 1 ) d x + 4 x y 3 d y = 0 with the M ( x , y ) d x + N ( x , y ) d y = 0 . We have M = 2 y 4 + 1 N = 4 x y 3 ∂ M ∂ y = ∂ ( 2 y 4 + 1 ) ∂ y ∂ M ∂ y = 8 y 3 ∂ N ∂ x = ∂ ( 4 x y 3 ) ∂ x ∂ N ∂ x = 4 y 3 Since ∂ M ∂ y ≠ ∂ N ∂ x , hence this differential equation is not exact. Now, we can calculate M y ( x , y ) − N x ( x , y ) N ( x , y ) h ( x ) = 8 y 3 − 4 y 3 4 x y 3 h ( x ) = 1 x The integrating factor is given as: e ∫ h ( x ) d x = e ∫ ( 1 x ) d x e ∫ h ( x ) d x = e log x e ∫ h ( x ) d x = x If we multiply the given differential equation by x , we will produce the exact differential equation

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ISBN: 9781337604796

Author: Larson

Publisher: CENGAGE L

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Chapter 16.1, Problem 41E

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To graph: The family of curvestangent to the force field F(x,y)=4x2yi−(2xy2+xy2)j.

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Chapter 16.1, Problem 40E

Chapter 16.1, Problem 42E

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To graph: The family of curvestangent to the force field F ( x , y ) = 4 x 2 y i − ( 2 x y 2 + x y 2 ) j .

Solution Summary: The author explains the family of curvestangent to the force field F(x,y)=4x2yi-(2x

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To graph: The family of curvestangent to the force field F(x,y)=4x2yi−(2xy2+xy2)j.

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Chapter 16 Solutions