Explanation Procedure used: General solution to differential equation subjected to boundary conditions: 1. Consider the differential equation L and find the roots of auxiliary equation f ( m ) . 2. Find the general solution. 3. Find the particular solution and use y = y c + y p to find the general solution. 4. Substitute, the boundary values to determine the unknown coefficients. 5. Substitute, the coefficients into the general solution. Calculation: Consider the differential equation y ″ − y = x + sin x and the homogeneous equation to y ″ − y = x + sin x is y ″ − y = 0 . The auxiliary equation associated to y ″ − y = 0 is m 2 − 1 = 0 and determine the roots of m 2 − 1 = 0 , m 2 = 1 m = ± 1 Hence, the roots of the auxiliary equation are m 1 = 1 and m 2 = − 1 . Since both the roots are distinct, the complementary function is y c = c 1 e x + c 2 e − x . Consider the particular solution of the form y p = A x + B + C sin x . Hence, y ′ p = A + C cos x and y ″ = − C sin x and substitute these values into y ″ − y = x + sin x . − C sin x − A x − B − C sin x = x + sin x − 2 C sin x − A x − B = x + sin x Compare the coefficients of similar terms on both sides, − A = 1 A = − 1 , B = 0 and − 2 C = 1 C = − 1 2 Hence, the particular solution is y p = − x − 1 2 sin x . The general solution is in the form y = y c + y p , y = c 1 e x + c 2 e − x − x − 1 2 sin x . The given boundary conditions y ( 0 ) = 2 and y ′ ( 0 ) = 3

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

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Chapter 4, Problem 41RE

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To solve: The differential equation y″−y=x+sinx subjected to the boundary conditions y(0)=2 and y′(0)=3.

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Chapter 4, Problem 40RE

Chapter 4, Problem 42RE

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Differential Equations with Boundary-Value Problems

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 differential equation y ″ − y = x + sin x subjected to the boundary conditions y ( 0 ) = 2 and y ′ ( 0 ) = 3 .

Solution Summary: The author explains the procedure used to solve the differential equation subjected to boundary conditions.

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To solve: The differential equation y″−y=x+sinx subjected to the boundary conditions y(0)=2 and y′(0)=3.

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