Explanation Procedure used: Find a homogenous equation with the constant coefficients whose general solution is provided. 1 Determine the roots of the auxiliary equation from the solution given and obtain the auxiliary equation. 2 Determine the homogenous linear equation from the auxiliary equation. Calculation: Consider the given general solutions y = c 1 cos x + c 2 sin x + c 3 cos 2 x + c 4 sin 2 x . The roots of the auxiliary equation are m 1 = i , m 2 = − i , m 3 = 2 i and m 4 = − 2 i . Denote both the roots variables as m . Therefore, the factors of the auxiliary equation are ( m − i ) , ( m + i ) , ( m − 2 i ) and ( m + 2 i ) . Multiply the factors and equate to zero. ( m − i ) ( m + i ) ( m − 2 i ) ( m + 2 i ) = 0 ( m 2 + 1 ) ( m 2 + 4 ) = 0 m 4 + m 2 + 4 m 2 + 4 = 0 m 4 + 5 m 2 + 4 = 0 For an auxiliary equation y = e m 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 4.3, Problem 58E

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To find: The homogeneous linear differential equation with constant coefficient whole general solution is y=c1cosx+c2sinx+c3cos2x+c4sin2x.

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Chapter 4.3, Problem 57E

Chapter 4.3, Problem 59E

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Chapter 4 Solutions

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 homogeneous linear differential equation with constant coefficient whole general solution is y = c 1 cos x + c 2 sin x + c 3 cos 2 x + c 4 sin 2 x .

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To find: The homogeneous linear differential equation with constant coefficient whole general solution is y=c1cosx+c2sinx+c3cos2x+c4sin2x.

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