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: 22


$y= [\;C_{1} +tC_{2}\;] cos(t)+ \;[C_{3} +tC_{4}\; ]sin(t)\;$

Work Step by Step

Let $\;\;\;\;y=e^{rt}\\\\$ $y^{(4)}+2{y}''+y=0 \;\;\;\;\Rightarrow \;\;\;\; r^4e^{rt}+2r^2e^{rt}+e^{rt}=0\\\\$ $r^4+2r^2+1=(r^2+1)^2=(r-i)^2(r+i)^2=0 $$ \rightarrow\;\;\;\; r_{1},r_{2}= i\;\;\;or\;\;\;r_{3},r_{4}=- i \;\;\;\;\;\;\\\\$ So the 8 roots are: $\;\;\;r_{1},r_{2}=\pm i \;\;\;,\;\;\;r_{3},r_{4}=\pm i $ The general solution for complex roots is: $y= C_{1}e^{\alpha t}cos(\beta t)+C_{2}e^{\alpha t}sin(\beta t)\\$ $y= [\;C_{1}cos(t)+C_{2} sin(t)\;]+ t[\;C_{3}cos(t)+C_{4} sin(t)\;]$ $y= [\;C_{1} +tC_{2}\;] cos(t)+ \;[C_{3} +tC_{4}\; ]sin(t)\;$
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