Explanation Calculation: For the differential equation y ' ' − 2 y ' + 5 y = e x sin ( x ) , the corresponding homogeneous differential equation is y ' ' − 2 y ' + 5 y = 0 The corresponding auxiliary equation is: m 2 − 2 m + 5 = 0 m 2 − 2 m + 5 = 0 m = 2 ± 4 − 20 2 = 1 ± − 16 = 1 ± 4 i m 1 = 1 + 2 i and m 2 = 1 − 2 i Therefore, the complementary function is: y c = e x ( c 1 sin 2 x + c 2 cos 2 x ) Since, e x sin x is annihilated by the differential operator ( D 2 − 2 D + 2 ) . Therefore, apply the differential operator ( D 2 − 2 D + 2 ) to both sides of the equation y ' ' − 2 y ' + 5 y = e x sin x ( D 2 − 2 D + 2 ) ( D 2 − 2 D + 5 ) y = ( D 2 − 2 D + 2 ) ( e x sin x ) ( D 2 − 2 D + 2 ) ( D 2 − 2 D + 5 ) y = 0 The corresponding auxiliary equation is: ( m 2 − 2 m + 2 ) ( m 2 − 2 m + 5 ) = 0 ( ( m − 1 ) 2 + 1 ) ( ( m − 1 ) 2 + 4 ) = 0 The roots are: m 1 = 1 + 2 i , m 2 = 1 − 2 i , m 3 = 1 − i and m 4 = 1 + i . Therefore, its general solution is y = e x ( c 1 sin 2 x + c 2 cos 2 x ) + e x ( c 3 e x sin x + c 4 cos 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

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 53E

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The general solution of the differential equation y''−2y'+5y=exsin(x) by undetermined coefficients.

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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 differential equation y ' ' − 2 y ' + 5 y = e x sin ( x ) by undetermined coefficients.

Solution Summary: The author explains the general solution of the differential equation y' ''s +5y=exmathrmsin(x) by undetermined coefficients

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The general solution of the differential equation y''−2y'+5y=exsin(x) by undetermined coefficients.

Chapter 4 Solutions