Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 17 - Section 17.1 - Second-Order Linear Equations - 17.1 Exercise - Page 1160: 11


$y=c_{1}e^{\frac{-1-\sqrt 3}{2}x}+c_{2}e^{\frac{-1+\sqrt 3}{2}x}$

Work Step by Step

$2\frac{d^{2}y}{dt^{2}}+2\frac{dy}{dt}-y=0$ $2y''+2y'-y=0$ Use auxiliary equation, $2r^{2}+2r-1=0$ $2r^{2}+2r-1=0$ $r=\frac{-2±\sqrt ((2)^{2}-4(2)(-1))}{2(2)}$ $r=\frac{-2±\sqrt (4+8)}{4}$ $r=\frac{-2±2\sqrt 3}{4}=\frac{-1±\sqrt 3}{2}$ $r_{1}=\frac{-1+\sqrt 3}{2}$ $r_{2}=\frac{-1-\sqrt 3}{2}$ Formula 8 $y=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}$ $y=c_{1}e^{\frac{-1-\sqrt 3}{2}x}+c_{2}e^{\frac{-1+\sqrt 3}{2}x}$
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