Explanation Procedure used: Solution of the differential equation by method of reduction of order: 1. Consider the differential equation L ( x ) . 2. Make a proper substitution. 3. Reduce the order and find the solution. Calculation: The given differential equation is, 2 y ′ y ″ = 1 , y ( 0 ) = 2 , y ′ ( 0 ) = 1 . Now, substitute u = y ′ then u ′ = y ″ to transform the differential equation an equation in which the independent variable is the basic variable y and dependent variable is substitution variable u . Thus, 2 y ′ y ″ = 1 , y ( 0 ) = 2 , y ′ ( 0 ) = 1 2 u u ′ = 1 2 u d u d x = 1 2 u d u = d x Take integral on both sides. Therefore, 2 ∫ u d u = ∫ d x u 2 = x + c 1 [ c 1 is the integral constant . ] ( y ′ ) 2 = x + c 1 Introduce the initial value y ′ ( 0 ) = 1 . Thus, ( y ′ ( x ) ) 2 = x + c 1 1 = 0 + c 1 1 = c 1 Hence, reduced DE is ( y ′ ) 2 = x + 1

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

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 4.10, Problem 11E

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To solve: The initial value problem 2y′y″=1,y(0)=2,y′(0)=1.

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Chapter 4.10, Problem 10E

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Chapter 4 Solutions

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

Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - Given that y=c1+c2x2 is a two-parameter family of...Ch. 4.1 - Find two members of the family of solutions in...Ch. 4.1 - Prob. 7E Ch. 4.1 - Use the general solution of x + 2x = 0 given in...Ch. 4.1 - In Problems 9 and 10 find an interval centered...Ch. 4.1 - In Problems 9 and 10 find an interval centered...

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The initial value problem 2 y ′ y ″ = 1 , y ( 0 ) = 2 , y ′ ( 0 ) = 1 .

Solution Summary: The author explains the solution of the differential equation by method of reduction of order.

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To solve: The initial value problem 2y′y″=1,y(0)=2,y′(0)=1.

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