Explanation Procedure used: General solution of a differential equation by undermined coefficients: 1. Consider the given differential equation a 1 y ( n ) + ... + a n − 1 y ′ + a n y = g ( x ) and determine the homogeneous differential equation associated with the DE. 2. Calculate the roots of auxiliary equation of homogeneous equation. 3. Determine g ( x ) . 4. Assume the particular solution to the differential equation in form of an equation. 5. Substitute, the particular solution into the differential equation. 6. Compare the coefficients and determine the particular solution. Consider differential equation, y ″ + y = sin x cos 2 x . The associated homogenous equation of differential equation is y ″ + y = 0 . The auxiliary equation corresponding to the homogenous equation, y ″ + y = 0 is m 2 + 1 = 0 . Find the roots of auxiliary equation. Therefore, m 2 + 1 = 0 m 2 = − 1 m = ± i Hence the complimentary function is y c = ( c 1 cos x + c 3 sin x ) . Compare the differential equation y ″ + y = sin x cos 2 x to y " + y = sin x cos 2 x . Therefore, g ( x ) = sin x cos 2 x g ( x ) = sin x ( 1 − 2 sin 2 x ) g ( x ) = sin x − 2 sin 3 x Assume that the particular solution of the given differential equation is also in in the form of a quadratic function, y p = A x sin x + B x cos x + C sin 3 x . Therefore, y p = A x sin x + B x cos x + C sin 3 x y ′ p = A sin x + A x cos x + B cos x − B x sin x + 3 C sin 2 x cos x y ″ p = A cos x − A x sin x + A cos x − B sin x + B x cos x − B x cos x + 3 C sin 3 x + 6 C sin x cos 2 x Therefore, y ″ p = 2 ( A x cos x − B x sin x ) − ( A x sin x + B x cos x ) − 3 C sin 3 x + 6 C sin x cos 2 x Substitute these values into the differential equation

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 4.4, Problem 44E

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To explain: How the method explained in the chapter can be used to determine the particular solution of the differential equation y″+y=sinxcos2x.

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Chapter 4.4, Problem 43E

Chapter 4.4, Problem 45E

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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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How the method explained in the chapter can be used to determine the particular solution of the differential equation y ″ + y = sin x cos 2 x .

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To explain: How the method explained in the chapter can be used to determine the particular solution of the differential equation y″+y=sinxcos2x.

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