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, ( y e x y + cos x ) d x + x e x y d y = 0 Here, M = ( y e x y + cos x ) and N = x e x y . Substitute ( y e x y + cos x ) for M and x e x y for N in the equation ( 1 ) . ∂ M ∂ y = ∂ ( y e x y + cos x ) ∂ y = ∂ ( y e x y ) ∂ y + ∂ ( cos x ) ∂ y = ( 1 + x y ) e x y Substitute x e x y for N in the equation ( 2 ) . ∂ N ∂ x = ∂ ( x e x y ) ∂ x = ( 1 + x y ) e x y Hence, the given differential equation is exact. Substitute M = ( y e x y + cos x ) for M in the equation ( 4 ) . ∂ ϕ ∂ x = ( y e x y + cos x ) Integrate the above equation. ∫ ∂ ϕ = ∫ ( y e x y + cos x ) ∂ x ϕ ( x , y ) = ∫ y e x y ∂ x + ∫ cos x ∂ x ϕ ( x , y ) = e x y + sin x + h ( y ) …… ( 6 ) Here, h is an arbitrary function of y

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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Chapter 1.9, Problem 18P

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The solution of the differential equation (yexy+cosx)dx+xexydy=0 with initial boundary condition y(π/2)=0.

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Chapter 1.9, Problem 17P

Chapter 1.9, Problem 19P

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Chapter 1 Solutions

Differential Equations and Linear Algebra (4th Edition)

Ch. 1.1 - True-False review For items a-n, decide if the...Ch. 1.1 - For Problems 14 determine the order of the...Ch. 1.1 - For Problems 14 determine the order of the...Ch. 1.1 - For Problems 1-4 determine the order of the...Ch. 1.1 - For Problems 14 determine the order of the...Ch. 1.1 - Verify that, for t0, y(t)=lnt is a solution to the...Ch. 1.1 - Verify that y(x)=x/(x+1) is a solution to the...Ch. 1.1 - Verify that y(x)=exsinx is a solution to the...Ch. 1.1 - By writing Equation 1.1.7 in the form 1TTmdTdt=k...Ch. 1.1 - A glass of water whose temperature is 50F is taken...

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The solution of the differential equation ( y e x y + cos x ) d x + x e x y d y = 0 with initial boundary condition y ( π / 2 ) = 0 .

The solution of the differential equation (yexy+cosx)dx+xexydy=0 with initial boundary condition y(π/2)=0.

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Chapter 1 Solutions