Explanation The first order differential equation is written in the differential form M ( x , y ) d y + N ( x , y ) d y = 0 . The solution of the differential equation is defined by ϕ ( x , y ) = c where c is arbitrary constant ϕ is the potential function. Approach Used: Partial differentiate M with respect to y . ∂ M ∂ y ….. ( 1 ) Here, M is the coefficient of d x . Partial differentiate N with respect to x . ∂ N ∂ x …… ( 2 ) Here, N is the coefficient of d y . The following is the expression for the differential equation to be exact. ∂ M ∂ y = ∂ N ∂ x ……. ( 3 ) If the given differential equation is exact then there exist a potential function ϕ such that ∂ ϕ ∂ x = M …… ( 4 ) ∂ ϕ ∂ y = N …… ( 5 ) Calculation: Consider the given differential equation, ( 1 x − y x 2 + y 2 ) d x + x x 2 + y 2 d y = 0 Substitute ( 1 x − y x 2 + y 2 ) for M in the equation ( 1 ) . ∂ M ∂ y = ∂ ( 1 x − y x 2 + y 2 ) ∂ y = ∂ ( 1 x ) ∂ y − ∂ ( y x 2 + y 2 ) ∂ y = − x 2 + y 2 ( x 2 + y 2 ) 2 Substitute x x 2 + y 2 for N in the equation ( 2 ) . ∂ N ∂ x = ∂ ( x x 2 + y 2 ) ∂ x = ( x 2 + y 2 ) ∂ ( x ) ∂ x − x ∂ ( x 2 + y 2 ) ∂ x ( x 2 + y 2 ) 2 = − x 2 + y 2 ( x 2 + y 2 ) 2 Hence the given differential equation is exact. Substitute ( 1 x − y x 2 + y 2 ) for M in the equation ( 4 ) . ∂ ϕ ∂ x = ( 1 x − y x 2 + y 2 ) Integrate the above equation

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ISBN: 9780321990167

Author: ANNIN

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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Chapter 1.9, Problem 8P

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The solution of the differential equation $(1x−yx2+y2)dx+xx2+y2dy=0$ .

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Chapter 1.9, Problem 7P

Chapter 1.9, Problem 9P

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Chapter 1 Solutions

EBK DIFFERENTIAL EQUATIONS AND LINEAR A

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The solution of the differential equation ( 1 x − y x 2 + y 2 ) d x + x x 2 + y 2 d y = 0 .

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Chapter 1 Solutions