Explanation Given: The given differential equation is t 2 d y d t + y 2 = t y . Procedure used: Bernoulli’s differential equation: d y d x + P ( x ) y = f ( x ) y n Substitute the following. y 1 − n = u General differential equation: d y d x + P y = Q General solution equation: I u = ∫ I Q d t 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) Substitute the following. y 1 − n = u Consider the given differential equation t 2 d y d t + y 2 = t y and write the equation in the form of Bernoulli’s differential equation. t 2 d y d t + y 2 = t y t 2 d y d t − t y = − y 2 d y d t − y t = − y 2 t 2 (2) Compare the equation (2) with the Bernoulli’s differential equation to obtain n as 2 . Substitute 2 for n in equation (1). y 1 − n = u y 1 − 2 = u y − 1 = u (3) Differentiate equation (3) with respect to t. y − 1 = u d y d t = d ( u − 1 ) d t d y d t = − u − 2 d u d t Substitute − u − 2 d u d t for d y d t and u -1 for y in equation (2)

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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The solution of the given differential equation by using appropriate substitution.

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Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - dydx=1xy (a) y(0) = 0 (b) y(1) = 0 (c) y(2) = 2...Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E

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The solution of the given differential equation by using appropriate substitution.

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