Explanation Procedure used: The Wronskian, W is calculated as W = | y 1 y 2 y 1 ′ y 2 ′ | . The equations to obtain the particular solution is as follows. u 1 = ∫ u 1 ′ d t u 2 = ∫ u 2 ′ d t Where u 1 ′ = − y 2 f ( x ) W u 2 ′ = y 1 f ( x ) W Calculation: The given nonhomogeneous equation is y ″ − 8 y ′ + 17 y = e 4 t sec t . Note that the characteristic equation of the corresponding homogeneous equation is r 2 − 8 r + 17 = 0 . Now, solve the characteristic equation as follows. r 2 − 8 r + 17 = 0 r = 8 ± 64 − 4 ( 17 ) 2 r = 8 ± 2 i 2 r = 4 ± i Therefore, the above equation has the roots r = 4 ± i . Thus, the general solution will be y h ( t ) = c 1 e 4 t cos t + c 2 e 4 t sin t . Note that the particular solution is of the form y p = u 1 y 1 + u 2 y 2 . IT is to be noted that here y 1 = e 4 t cos t , y 2 = e 4 t sin t . Calculate the Wronskian as follows. W = | y 1 y 2 y 1 ′ y 2 ′ | W = | e 4 t cos t e 4 t sin t − e 4 t sin t + 4 e 4 t cos t e 4 t cos t + 4 e 4 t sin t | W = 4 e 8 t sin t cos t + e 8 t cos 2 t − 4 e 8 t sin t cos t + e 8 t sin 2 t W = e 8 t Now, obtain the value of u 1 ′ as follows

5th Edition

ISBN: 9780128149485

Author: Abell, Martha L. L.

Publisher: Elsevier Science

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Differential Equations

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Chapter 4 Solutions

Introductory Differential Equations

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Chapter 4 Solutions