Elementary Differential Equations and Boundary Value Problems 9th Edition

Published by Wiley
ISBN 10: 0-47038-334-8
ISBN 13: 978-0-47038-334-6

Chapter 4 - Higher Order Linear Equations - 4.2 Homogenous Equations with Constant Coefficients - Problems - Page 232: 16


$y=C_{1}e^{2t}+C_{2} e^{-2t}+C_{3}e^{t} + C_{4}e^{-t}$

Work Step by Step

Let $\;\;\;\;y=e^{rt}\\\\$ $y^{(4)}-5{y}''+4y=0 \;\;\;\; \Rightarrow \;\;\;\;r^4e^{rt}-5r^2e^{rt}+4e^{rt}=0\\\\$ $r^4-5r^2+4=(r^2-4)(r^2-1)=0 \;\;\;\;\rightarrow \;\;\;\;r_{1}=2,r_{2}=-2\;\;\;\;or\;\;\;\; r_{3}=1,r_{4}=-1\\\\$ $y=C_{1}e^{2t}+C_{2} e^{-2t}+C_{3}e^{t} + C_{4}e^{-t}$
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