Explanation Procedure used: Solution of the differential equation with undetermined coefficients-Annihilator Approach: The Annihilator Approach represents the differential equation L ( y ) = g ( x ) , where, the function g ( x ) may consist any or the entire finite sum, product of multiple polynomials, sine and cosine functions and exponential functions as well. Steps: 1. Look for the complementary solution y c for the homogeneous equation L ( y ) = 0 . 2. Operate both sides of the equation L ( y ) = g ( x ) by the differential operator which annihilates the function g ( x ) . 3. Solve for a homogeneous differential equation by looking for the general solution of the differential equation obtained L 1 L ( y ) = 0 . 4. Solve for the particular solution of the non-homogeneous differential equation L ( y ) = g ( x ) by eliminating the duplicated term in the complementary solution of L 1 L ( y ) = 0 . 5. Determine the undetermined coefficients by substituting the particular solution into non-homogeneous differential equation L ( y ) = g ( x ) . 6. Write the general solution from complementary and particular solution in the form y = y c + y p . Calculation: Consider the differential equation y ‴ − y ″ + y ′ − y = x e x − e − x + 7 . Find the roots of auxiliary equation of homogeneous DE y ‴ − y ″ + y ′ − y = 0 associate to y ‴ − y ″ + y ′ − y = x e x − e − x + 7 . Therefore, y ‴ − y ″ + y ′ − y = 0 m 3 − m 2 + m − 1 = 0 m 2 ( m − 1 ) + 1 ( m − 1 ) = 0 ( m − 1 ) ( m 2 + 1 ) = 0 The solutions of the auxiliary equation are m 1 = 1 , m 2 = i and m 3 = − i . Thus, the complimentary function is y c ( x ) = c 1 e x + c 2 cos x + c 3 sin x . The constant term 7 can be annihilated by the differential operator D . The operator ( D − 1 ) 2 annihilates x e x and ( D + 1 ) annihilates e − x . Therefore, D ( D + 1 ) ( D − 1 ) 2 ( x e x − e − x + 7 ) = 0 Thus, D ( D + 1 ) ( D − 1 ) 2 ( D 3 − D 2 + D − 1 ) ( y ) = 0 The auxiliary equation is m ( m + 1 ) ( m − 1 ) 2 ( m 3 − m 2 + m − 1 ) = 0 . Thus, the roots of the auxiliary equation are m 1 = 1 , m 2 = i , m 3 = − i , m 4 = 1 , m 5 = 1 , m 6 = − 1 and m 7 = 0 . Thus, the solution of the DE is y ( x ) = c 1 e x + c 2 cos x + c 3 sin x + c 4 x e x + c 5 x 2 e x + c 6 e − x + c 7 . Here, the terms c 1 e x , c 2 cos x and c 3 sin x are repeated

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ISBN: 9781337652469

Author: ZILL

Publisher: CENGAGE L

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Chapter 4.5, Problem 60E

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To solve: The differential equation y‴−y″+y′−y=xex−e−x+7 by undetermined coefficients.

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Chapter 4.5, Problem 59E

Chapter 4.5, Problem 61E

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The differential equation y ‴ − y ″ + y ′ − y = x e x − e − x + 7 by undetermined coefficients.

Solution Summary: The author explains the Annihilator Approach, which solves the differential equation with undetermined coefficients.

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To solve: The differential equation y‴−y″+y′−y=xex−e−x+7 by undetermined coefficients.

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