Explanation Given: The given Bernoulli’s equation is, d y d t − 1 2 y = t y − 1 Procedure used: The standard form of Bernoulli’s equation is, d y d t + p ( t ) y = q ( t ) y n Substitute w for y 1 − n in the above equation to transform into the linear equation. d w d t + ( 1 − n ) p ( t ) w = ( 1 − n ) q ( t ) The integrating factor of the above equation is, μ ( t ) = e ∫ ( 1 − n ) p ( t ) d t Multiply the above equation with integrating factor and solve for w . w μ ( t ) = ∫ ( 1 − n ) q ( t ) μ ( t ) d t + C Calculation: Consider the given Bernoulli’s equation. d y d t − 1 2 y = t y − 1 (1) Compare the above equation with standard form of Bernoulli’s equation. Then, p ( t ) = − 1 2 , q ( t ) = t and n = − 1 . Thus, w = y 1 − n = y 1 − ( − 1 ) = y 2 Differentiate w = y 2 on both sides with respect to t as, d w d t = 2 y d y d t d y d t = 1 2 y − 1 d w d t Substitute w for y 2 and 1 2 y − 1 d w d t for d y d t in equation (1)

5th Edition

ISBN: 9780128149485

Author: Abell, Martha L. L.

Publisher: Elsevier Science

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Chapter 2.6, Problem 1E

To determine

The solution for the given Bernoulli’s equation.

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Chapter 2.5, Problem 60E

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Chapter 2 Solutions

Introductory Differential Equations

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