Explanation The given Bernoulli equation is y ' = r y − k y 2 with the boundary t > 0 and k > 0 . Divide the equation y ' = r y − k y 2 by y 2 . y ' y 2 = r y y 2 − k y 2 y 2 y ' y − 2 = r y − − k Let u = y − , and differentiate it as follows: d u d t = d d t ( y − ) = − y − 2 d y d t The above equation can be written as follows: d u d t = − y − 2 y ' y − 2 y ' = − d u d t Substitute the value of y − 2 y ' = − d u d t in y ' y − 2 = r y − − k . − d u d t = r u − k d u d t + r u = k Compare the equation d u d t + r u = k with the linear equation where p ( t ) = r , g ( t ) = k . Solve normally by selecting appropriate μ ( t ) = e ∫ p ( t ) d t . Substitute the value of p ( t ) = r in μ ( t ) = e ∫ p ( t ) d t

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Question

Chapter 2.4, Problem 29P

To determine

To solve: The Bernoulli equation by using the substitution.

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Chapter 2.4, Problem 28P

Chapter 2.4, Problem 30P

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Chapter 2 Solutions

Elementary Differential Equations

Ch. 2.1 - In each of Problems 1 through 12: Draw a direction...Ch. 2.1 - In each of Problems 1 through 12: Draw a direction...Ch. 2.1 - Prob. 3P Ch. 2.1 - Prob. 4P Ch. 2.1 - Prob. 5P Ch. 2.1 - Prob. 6P Ch. 2.1 - In each of Problems 1 through 12: Draw a direction...Ch. 2.1 - Prob. 8P Ch. 2.1 - Prob. 9P Ch. 2.1 - Prob. 10P

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