Explanation Given that, the differential equation is y ″ − 2 y ′ + y = x 3 + 4 x . The corresponding homogeneous differential is y ″ − 2 y ′ + y = 0 . The auxiliary equation of the homogeneous equation is m 2 − 2 m + 1 = 0 . Solve the auxiliary equation m 2 − 2 m + 1 = 0 as follows. m 2 − 2 m + 1 = 0 ( m − 1 ) ( m − 1 ) = 0 Here, the values of m 1 = 1 and m 2 = 1 . Therefore, the complementary function is y c = c 1 e x + c 2 x e x . Since x 3 + 4 x is annihilate by the differential operator D 4 , apply the differential operator D 4 to both sides of the equation y ″ − 2 y ′ + y = x 3 + 4 x . D 4 ( D 2 − 2 D + 1 ) y = D 4 ( x 3 + 4 x ) D 4 ( D 2 − 2 D + 1 ) y = 0 [ where D 4 ( x 3 + 4 x ) = 0 ] The corresponding auxiliary equation is m 4 ( m 2 − 2 m + 1 ) = 0 . m 4 ( m 2 − 2 m + 1 ) = 0 m 4 ( m − 1 ) ( m − 1 ) = 0 m 1 = m 2 = m 3 = m 4 = 0 , m 5 = m 5 = 1 Therefore, the general solution is y = c 1 + c 2 x + c 3 x 2 + c 4 x 3 + c 5 e x + c 6 x e x . In order to find the particular solutions y = c 5 e x and y = c 6 x e x are duplicates within the set of solutions to the homogeneous equation above. So neglect them, which leaves only c 1 , c 2 x , c 3 x 2 and c 4 x 3 . Now replace c 1 , c 2 , c 3 and c 4 by A , B , C and D , to get y p = A + B x + C x 2 + D x 3

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 4.5, Problem 42E

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To solve: The differential equation y″−2y′+y=x3+4x by undetermined coefficients.

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Chapter 4.5, Problem 41E

Chapter 4.5, Problem 43E

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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 y ″ − 2 y ′ + y = x 3 + 4 x by undetermined coefficients.

Solution Summary: The author explains how to solve the differential equation y′′-2xprime by undetermined coefficients, and the auxiliary equation of the homogeneous equation.

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To solve: The differential equation y″−2y′+y=x3+4x by undetermined coefficients.

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