Explanation Given data: The differential equation is, y ″ + 2 y ′ + 10 y = x 2 e − x cos 3 x (1) Consider the auxiliary equation is, r 2 + 2 r + 10 = 0 (2) Roots of equation (2) are, r = − ( 2 ) ± ( 2 ) 2 − 4 ( 1 ) ( 10 ) 2 ( 1 ) { ∵ r = − b ± b 2 − 4 a c 2 a for the equation of a r 2 + b r + c = 0 } = − 2 ± i 6 2 = − 1 ± i 3 Write the expression for the complementary solution for the complex roots, y c ( x ) = e α x ( c 1 cos β x + c 2 sin β x ) Substitute − 1 for α and 3 for β , y c ( x ) = e − 1 x ( c 1 cos 3 x + c 2 sin 3 x ) = e − x ( c 1 cos 3 x + c 2 sin 3 x ) Since G ( x ) = x 2 e − x cos 3 x is a polynomial degree of 2. If Right hand side (RHS) of a differential equation contains a product of an exponential function, a cosine function, and n th order polynomial, the trail solution of the differential equation also should be a product of these corresponding choices. So, the trail solution y p ( x ) for this case can be expressed as follows. y ( x ) = ( A x n + B x n − 1 ⋯ + Z ) e − x cos 3 x + ( A 1 x n + B 1 x n − 1 ⋯ + Z 1 ) e − x sin 3 x (3) Here, n is the order of a polynomial, and A , B , Z , A 1 , B 1 , Z 1 , are the constants

Calculus, Early Transcendentals, International Metric Edition

8th Edition

ISBN: 9781305272378

Author: James Stewart

Publisher: Cengage Learning

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Differential Equations

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Chapter 17.2, Problem 17E

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To write: The trail solution for the method of undetermined coefficients.

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The trail solution for the method of undetermined coefficients.

Solution Summary: The author explains the trail solution for the method of undetermined coefficients.

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To write: The trail solution for the method of undetermined coefficients.

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