Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 17 - Second-Order Differential Equations - 17.3 Applications of Second-Order Differential Equations - 17.3 Exercises: 3

Answer

$x(t)=-\frac{1}{5}e^{-6t}+1\frac{1}{5}e^{-t}$

Work Step by Step

$k(0.5)=6$ $k=12$ $2\frac{d^2x}{dt^2}+14\frac{dx}{dt}+12x=0$ $r^2+7r+6=0$ $r=-6$ and $r=-1$ $y=c_{1}e^{-6t}+c_{2}e^{-t}$ $x(0)=1$ $y=c_{1}+c_{2}=1$ $x'(0)=0$ $y'=-6c_{1}e^{-6t}-c_{2}e^{-t}$ $-6c_{1}-c_{2}=0$ Use the two equations to find $c_{1}=-\frac{1}{5}$ $c_{2}=1\frac{1}{5}$ $x(t)=-\frac{1}{5}e^{-6t}+1\frac{1}{5}e^{-t}$
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