Explanation Given: The differential equation x 2 y ″ − 2 x y ′ + 2 y = x 4 e x . It can be written as x 2 d 2 y d x 2 − 2 x d y d x + 2 y = x 4 e x . Since the equation is nonhomogeneous, we first solve the associated homogeneous equation. Calculations: Assume y = x m be a solution of the homogeneous differential equation. Differentiating y = x m twice, to get d y d x = m x m − 1 , d 2 y d x 2 = m ( m − 1 ) x m − 2 Substitute y and d 2 y d x 2 into the differential equation yields: x 2 . m ( m − 1 ) x m − 2 − 2 x m x m − 1 + 2 x m = 0 [ m ( m − 1 ) − 2 m + 2 ] x m = 0 Solve the auxiliary equation m ( m − 1 ) − 2 m + 2 = 0 m 2 − 3 m + 2 = 0 ( m − 1 ) ( m − 2 ) = 0 m 1 = 1 and m 2 = 2 Thus, the complimentary solution is y c = c 1 x + c 2 x 2 . Hence, y 1 = x , y 2 = x 2 . Before we start solving by variation of parameters, the coefficient of y ' ' should be 1. Divide the equation by x 2 . y ' ' − 2 y ' x + 2 y x 2 = x 2 e x f ( x ) = x 2 e x Recall that the formulas u 1 ' = W 1 W and u 2 ' = W 2 W where W 1 , W 2 and W are determinants given by

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 4.7, Problem 22E

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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 general solution of the given differential equation using variation of parameters

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