Explanation Given: The given differential equation is y 1 2 d y d x + y 3 2 = 1 and the initial boundary condition is y ( 1 ) = 1 2 . Procedure used: Bernoulli’s differential equation: d y d x + P ( x ) y = f ( x ) y n Consider the following substitution. y 1 − n = u General differential equation: d y d x + P y = Q General solution equation: I u = ∫ I Q d x Here, the integrating factor is I . Calculation: Write the expression for Bernoulli’s differential equation. d y d x + P ( x ) y = f ( x ) y n (1) Consider the following substitution. y 1 − n = u Consider the given differential equation y 1 2 d y d x + y 3 2 = 1 and write the equation in the form of Bernoulli’s differential equation. y 1 2 d y d x + y 3 2 = 1 d y d x + y 3 2 y 1 2 = 1 y 1 2 d y d x + y = y − 1 2 (2) Compare the equation (2) with the Bernoulli’s differential equation to obtain n as − 1 2 . Substitute − 1 2 for n in equation (1). y 1 − n = u y 1 − ( − 1 2 ) = u y 3 2 = u y = u 2 3 (3) Differentiate equation (3) with respect to t. y = u 2 3 d y d t = 2 3 u 2 3 − 1 d u d t d y d t = 2 3 u − 1 3 d u d t Substitute 2 3 u − 1 3 d u d t for d y d t and u 2 3 for y in equation (2). 2 3 u − 1 3 d u d t + u 2 3 = u − 1 2 × 2 3 2 3 u − 1 3 d u d t + u 2 3 = u − 1 3 2 3 d u d t + u = 1 d u d t + 3 2 u = 3 2 (4) Write the expression to calculate the integrating factor

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 2.5, Problem 22E

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Chapter 2.5, Problem 21E

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Advanced Engineering Mathematics

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