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

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.

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Chapter 3.4, Problem 44E

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Chapter 3.4, Problem 43E

Chapter 3.4, Problem 45E

Chapter 3 Solutions

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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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