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 general solution. Calculation: The given differential equation is, y ″ + 2 y ′ + y = sin x + 3 cos 2 x . Determine the associate homogenous equation of differential equation y ″ + 2 y ′ + y = sin x + 3 cos 2 x . That is, y ″ + 2 y ′ + y = 0 . The Auxiliary equation corresponding to the homogenous equation, y ″ + 2 y ′ + y = 0 is m 2 + 2 m + 1 = 0 . Obtain the roots of auxiliary equation as, y ″ + 2 y ′ + y = 0 m 2 + 2 m + 1 = 0 m 2 + m + m + 1 = 0 m ( m + 1 ) + 1 ( m + 1 ) = 0 Simplify further as, ( m + 1 ) ( m + 1 ) = 0 ( m + 1 ) 2 = 0 m = − 1 That is, the roots of the differential equation are m 1 = − 1 and m 2 = − 1 which are repetitive roots. Therefore, the complementary function is y c = ( c 1 e − x + c 2 x e − x ) . Comparing the differential equation y ″ + 2 y ′ + y = sin x + 3 cos 2 x to y ″ + 2 y ′ + y = g ( x ) . Therefore, g ( x ) = sin x + 3 cos 2 x . Assume that a particular solution of the given differential equation is also in the form of a quadratic function y p = A cos x + B sin x + C cos 2 x + D sin 2 x . Compute the first and the second derivative of the above equation. y ′ p = [ A ( − sin x ) + B ( cos x ) − 2 C sin 2 x + 2 D cos 2 x ] y ″ p = − A sin x + B cos x − 2 C sin 2 x + 2 D cos 2 x y ″ p = − ( A cos x + B sin x ) − 4 ( C cos 2 x + D sin 2 x ) Substitute these values in the differential equation, y ″ + 2 y ′ + y = sin x + 3 cos 2 x . Thus, [ − 3 C cos 2 x + D sin x + 2 ( − A sin x + B cos x ) − 4 C sin 2 x + 4 D cos 2 x ] = sin x + 3 cos 2 x ( − 2 A ) sin x + ( 2 B ) cos x + ( 4 D − 3 C ) cos 2 x + ( − 4 C − 3 D ) sin 2 x = sin x + 3 cos 2 x Equate the coefficient of sin x

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Differential Equations

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Chapter 3.4, Problem 19E

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To solve: The differential equation y″+2y′+y=sinx+3cos2x by undetermined coefficient.

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Chapter 3.4, Problem 18E

Chapter 3.4, Problem 20E

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Chapter 3 Solutions

Advanced Engineering Mathematics

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The differential equation y ″ + 2 y ′ + y = sin x + 3 cos 2 x by undetermined coefficient.

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Chapter 3 Solutions