Program Explanation Given information: The first-order differential equation is ( x + e y ) y ′ = x e − y − 1 . Explanation: The given differential equation can be written as, x y ′ + e y y ′ = x e − y − 1 Consider e y as t and differentiate with respect to x . e y d y d x = d t d x d y d x = 1 e y d t d x = 1 t d t d x The given differential equation can be written as, x t d t d x + d t d x = x t − 1 d t d x ( x t + 1 ) = x t − 1 d t d x = x − t x + t = 1 − t x 1 + t x Substitute z = t x and differentiate with respect to x as, x z = t d d x ( x z ) = d t d x z d x d x + x d z d x = d t d x z + x d z d x = d t d x Now, the given differential equation can be written as, z + x d z d x = 1 − z 1 + z x d z d x = 1 − z 1 + z − z = 1 − z − z ( 1 + z ) 1 + z = 1 − 2 z − z 2 1 + z Simplify the above equation as, 1 + z 1 − 2 z − z 2 d z = d x x Integrate on both sides as, ∫ 1 + z 1 − 2 z − z 2 d z = ∫ d x x + c ......

5th Edition

ISBN: 8220100802454

Author: Calvis

Publisher: PEARSON

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Problems on Growing Functions

For a given set of functions f and g, the growth of functions states that the function g grows as fast as the function f. The growth rate of algorithms helps determine the algorithm's efficiency and compare its performance to other algorithms.

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Chapter 1.6, Problem 30P

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Program Description: Purpose ofproblem is to calculate the general solution of the differential equation (x+ey)y′=xe−y−1 .

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Chapter 1.6, Problem 29P

Chapter 1.6, Problem 31P

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Chapter 1 Solutions

EBK DIFFERENTIAL EQUATIONS

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Program Description: Purpose ofproblem is to calculate the general solution of the differential equation ( x + e y ) y ′ = x e − y − 1 .

Solution Summary: The author explains the purpose of the program, which is to calculate the general solution to the first-order differential equation.

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Program Description: Purpose ofproblem is to calculate the general solution of the differential equation (x+ey)y′=xe−y−1 .

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Chapter 1 Solutions