Explanation The given differential equation is y ″ + 3 y ′ = 4 x − 5 . The corresponding homogeneous differential is y ″ + 3 y ′ = 0 . The auxiliary equation of the homogeneous equation is m 2 + 3 m = 0 . Solve the auxiliary equation as follows. m 2 + 3 m = 0 m ( m + 3 ) = 0 m 1 = 0 , m 2 = − 3 Therefore, the complementary function is y c = c 1 + c 2 e − 3 x . Since, 4 x − 5 is annihilate by the differential operator D 2 , apply the differential operator D 2 to both sides of the equation y ″ + 3 y ′ = 4 x − 5 . D 2 ( D 2 + 3 D ) y = D 2 ( 4 x − 5 ) D 2 ( D 2 + 3 D ) y = 0 [ where D 2 ( 4 x − 5 ) = 0 ] The corresponding auxiliary equation is m 2 + 3 m and factors are given by m 2 + 3 m = 0 m 3 ( m + 3 ) = 0 m 1 = m 2 = m 3 = 0 , m 4 = − 3 Therefore, the general solution is y = c 1 + c 2 x + c 3 x 2 + c 4 e − 3 x . In order to find the particular solution y p of the given differential equation, observe that solutions y 1 = c 1 and y 4 = c 4 e − 3 x are duplicates within the set of solutions to the homogeneous equation above

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 4.5, Problem 40E

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To solve: The differential equation by undetermined coefficients.

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Chapter 4.5, Problem 39E

Chapter 4.5, Problem 41E

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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 differential equation by undetermined coefficients.

Solution Summary: The author explains how to solve the differential equation by undetermined coefficients.

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Chapter 4 Solutions