Explanation Given that, the differential equation is y ‴ + y ″ = 8 x 2 . The corresponding homogeneous differential is y ‴ + y ″ = 0 . The auxiliary equation of the homogeneous equation is m 3 + m 2 = 0 . Solve the auxiliary equation m 3 + m 2 = 0 as follows. m 3 + m 2 = 0 m 2 ( m + 1 ) = 0 Here, the values of m 1 = m 2 = 0 , and m 3 = − 1 . Therefore, the complementary function is y c = c 1 + c 2 x + c 3 e − x . Since 8 x 2 is annihilate by the differential operator D 3 , apply the differential operator D 3 to both sides of the equation y ‴ + y ″ = 8 x 2 . D 3 ( D 3 + D 2 ) y = D 2 ( 8 x 2 ) D 3 ( D 3 + D 2 ) y = 0 [ where D 2 ( 8 x 2 ) = 0 ] The corresponding auxiliary equation is m 2 ( m 2 + 4 m + 4 ) = 0 . m 3 ( m 3 + m 2 ) = 0 m 5 ( m + 1 ) = 0 m 1 = m 2 = m 3 = m 4 = m 5 = 0 , m 6 = − 1 Therefore, the general solution is y = c 1 + c 2 x + c 3 x 2 + c 4 x 3 + c 5 x 4 + c 6 e − x . In order to find the particular solutions y 1 = c 1 , y 2 = c 3 x and y 6 = c 6 e − x are duplicates within the set of solutions to the homogeneous equation above. So neglect them, which leaves only y 3 = c 3 x 2 , c 4 x 3 and y 5 = c 5 x 4 . Now replace c 3 , c 4 and c 5 by A , B , and C , to get y p = A x 2 + B x 3 + C x 4

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 41E

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To solve: The differential equation y‴+y″=8x2 by undetermined coefficients.

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

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The differential equation y ‴ + y ″ = 8 x 2 by undetermined coefficients.

Solution Summary: The author explains how to solve the differential equation y′′′′+Y′′=8x2 by undetermined coefficients.

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To solve: The differential equation y‴+y″=8x2 by undetermined coefficients.

Chapter 4 Solutions