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 2 y ‴ − 3 y ″ − 3 y ′ + 2 y = ( e x + e − x ) 2 . Find the roots of auxiliary equation of homogeneous DE 2 y ‴ − 3 y ″ − 3 y ′ + 2 y = 0 associate to 2 y ‴ − 3 y ″ − 3 y ′ + 2 y = ( e x + e − x ) 2 . Therefore, 2 y ‴ − 3 y ″ − 3 y ′ + 2 y = 0 2 m 3 − 3 m 2 − 3 m + 2 = 0 The solutions of the auxiliary equation are m 1 = − 1 , m 2 = 1 2 and m 3 = 2 . Thus, the complimentary function is y c ( x ) = c 1 e − x + c 2 e 1 2 x + c 3 e 2 x . Consider the right-hand side of the differential equation ( e x + e − x ) 2 . Thus, ( e x + e − x ) 2 = e 2 x + e − 2 x + 2 . The differential operator ( D − α ) n annihilates the exponential functions. The differential operator ( D − 2 ) annihilates the term e 2 x . The differential operator ( D + 2 ) annihilates the term e − 2 x . The constant term 2 by D. Therefore, D ( D − 2 ) ( D + 2 ) ( e x + e − x ) 2 = 0 Apply D ( D − 2 ) ( D + 2 ) to the differential equation. D ( D − 2 ) ( D + 2 ) ( 2 y ‴ − 3 y ″ − 3 y ′ + 2 y ) = 0 D ( D − 2 ) ( D + 2 ) ( 2 D 3 − 3 D 2 − 3 D + 2 ) ( y ) = 0 The auxiliary equation is m ( m − 2 ) ( m + 2 ) ( 2 m 3 − 3 m 2 − 3 m + 2 = 0 ) = 0 . Thus, the roots of the auxiliary equation are m 1 = − 1 , m 2 = 1 2 , m 3 = 2 , m 4 = 2 , m 5 = − 2 and m 6 = 0 . Thus, the solution of the DE is y ( x ) = c 1 e − x + c 2 e 1 2 x + c 3 e 2 x + c 4 x e 2 x + c 5 e − 2 x + c 6 . Since the terms c 1 e − x , c 2 e 1 2 x and c 3 e 2 x are repeated. Therefore, the particular solution is y p ( x ) = A x e 2 x + B e − 2 x + C . Differentiate y p ( x ) = A x e 2 x + B e − 2 x + C with respect to x

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 4.5, Problem 62E

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To solve: The differential equation 2y‴−3y″−3y′+2y=(ex+e−x)2 by undetermined coefficients.

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Chapter 4.5, Problem 61E

Chapter 4.5, Problem 63E

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A First Course in Differential Equations with Modeling Applications (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 - Given that x(t) = c1 cos t + c2 sin t is the...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 2 y ‴ − 3 y ″ − 3 y ′ + 2 y = ( e x + e − x ) 2 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 2y‴−3y″−3y′+2y=(ex+e−x)2 by undetermined coefficients.

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Chapter 4 Solutions