Program Explanation Given information: The differential equation is ( x + tan − 1 y ) d x + x + y 1 + y 2 d y = 0 . Explanation: Compare the given differential equation with the general differential equation to obtain the value of M ( x , y ) and N ( x , y ) as, M ( x , y ) = x + tan − 1 y N ( x , y ) = x + y 1 + y 2 Substitute x + tan − 1 y for M and x + y 1 + y 2 for N in equation (1) as, ∂ ∂ y ( x + tan − 1 y ) = ? ∂ ∂ x ( x + y 1 + y 2 ) 0 + ∂ ∂ y ( tan − 1 y ) = ? ∂ ∂ x ( x 1 + y 2 + y 1 + y 2 ) 1 1 + y 2 = ? 1 1 + y 2 + 0 1 1 + y 2 = 1 1 + y 2 Thus, the given differential equation is exact. The value of M ( x , y ) can be written as, M ( x , y ) = ∂ F ∂ x The value of F ( x , y ) can be obtained if both the sides of above equation are integrated with respect to x as, ∫ M ( x , y ) d x = F ( x , y ) Substitute x + tan − 1 y in for M ( x , y ) in above equation as, F ( x , y ) = ∫ ( x + tan − 1 y ) d x = x 2 2 + x tan − 1 y + g ( y )

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

5th Edition

ISBN: 9780321816252

Author: C. Henry Edwards, David E. Penney, David Calvis

Publisher: PEARSON

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A naïve solution for the above-stated problem is to evaluate all of the possible combinations of different pieces of varying size to find the maximum revenue-generating combination. Although this method seems the easiest and quick to practice, it has exp…

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Chapter 1.6, Problem 38P

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Program Description: Purpose of problem is to showthat the differential equation (x+tan−1y)dx+x+y1+y2dy=0 is exact and obtain the solution for it.

Summary introduction: If the differential equation is M(x,y)dx+N(x,y)dy=0 then the below condition must satisfies,

∂Mdy=∂N∂x ....... (1)

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Chapter 1.6, Problem 37P

Chapter 1.6, Problem 39P

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Chapter 1 Solutions

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

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Solution Summary: The author explains that the differential equation (x+mathrmtan-1y) is exact and obtains the solution for it.

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Program Description: Purpose of problem is to showthat the differential equation (x+tan−1y)dx+x+y1+y2dy=0 is exact and obtain the solution for it. Summary introduction: If the differential equation is M(x,y)dx+N(x,y)dy=0 then the below condition must satisfies, ∂Mdy=∂N∂x ....... (1)

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Chapter 1 Solutions

Program Description: Purpose of problem is to showthat the differential equation (x+tan−1y)dx+x+y1+y2dy=0 is exact and obtain the solution for it.

Summary introduction: If the differential equation is M(x,y)dx+N(x,y)dy=0 then the below condition must satisfies,

∂Mdy=∂N∂x ....... (1)