Explanation Calculation: For the differential equation y ' ' + 2 y ' + y = x 2 e − x , the corresponding homogeneous differential equation is y ' ' + 2 y ' + y = 0 The corresponding auxiliary equation is: m 2 + 2 m + 1 = 0 m 2 + 2 m + 1 = 0 ( m + 1 ) 2 = 0 m 1 = m 2 = − 1 Therefore, the complementary function is: y c = c 1 e − x + c 2 x e − x Since, x 2 e − x is annihilated by the differential operator ( D + 1 ) 3 . Therefore, apply the differential operator ( D + 1 ) 3 to both sides of the equation y ' ' + 2 y ' + y = x 2 e − x ( D + 1 ) 3 ( D 2 + 2 D + 1 ) y = ( D + 1 ) 3 ( x 2 e − x ) ( D + 1 ) 3 ( D 2 + 2 D + 1 ) y = 0 The corresponding auxiliary equation is: ( m + 1 ) 3 ( m 2 + 2 m + 1 ) = 0 ( m + 1 ) 5 = 0 The roots are: m 1 = m 2 = m 3 = m 4 = m 5 = − 1 Therefore, its general solution is y = c 1 e − x + c 2 x e − x + c 3 x 3 e − x + c 4 x 4 e − x The particular solution y p of the given differential equation consists of the terms remaining in the above general equation which are not present in the complementary function of the original equation. Let y p = A x 3 e − x + B x 4 e − x

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 52E

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The solution of the differential equation y''+2y'+y=x2e−x by undetermined coefficients.

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Chapter 4.5, Problem 51E

Chapter 4.5, Problem 53E

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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 solution of the differential equation y ' ' + 2 y ' + y = x 2 e − x by undetermined coefficients.

Solution Summary: The author explains the solution of the differential equation y''+2y's +y=x2'e'-x by undetermined coefficients.

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The solution of the differential equation y''+2y'+y=x2e−x by undetermined coefficients.

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