Explanation Given: The initial value problem is, d y d x = x tan − 1 x y y ( 0 ) = 3 Calculation: Separate the variables of the given differential equation. d y d x = x tan − 1 x y y d y = x tan − 1 x d x (1) Integrate equation (1). ∫ y d y = ∫ x tan − 1 x d x (2) Apply integration by parts to solve right hand side integral of equation (2). ∫ x tan − 1 x d x = tan − 1 x ∫ x d x − ∫ { d ( tan − 1 x ) d x ∫ x d x } d x = tan − 1 x ( x 2 2 ) − ∫ ( 1 1 + x 2 ) ( x 2 2 ) d x = tan − 1 x ( x 2 2 ) − 1 2 ∫ ( x 2 1 + x 2 ) d x = tan − 1 x ( x 2 2 ) − 1 2 ∫ ( x 2 + 1 − 1 1 + x 2 ) d x Further, simplify the above equation, ∫ x tan − 1 x d x = tan − 1 x ( x 2 2 ) − 1 2 ∫ ( 1 − 1 1 + x 2 ) d x = tan − 1 x ( x 2 2 ) − 1 2 [ x − tan − 1 x ] + C Here, constant of integration is C

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

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Differential Equations

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Chapter 2.2, Problem 50E

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The explicit solution of the given differential equation.

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Chapter 2.2, Problem 49E

Chapter 2.2, Problem 51E

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Chapter 2 Solutions

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 explicit solution of the given differential equation.

Solution Summary: The author explains the explicit solution of the given differential equation: underset_y2=mathrmtan

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Chapter 2 Solutions