## Calculus 8th Edition

$x(t)=-\frac{1}{5}e^{-6t}+1\frac{1}{5}e^{-t}$
$k(0.5)=6$ $k=12$ $2\frac{d^2x}{dt^2}+14\frac{dx}{dt}+12x=0$ $r^2+7r+6=0$ $r=-6$ 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 $c_{1}=-\frac{1}{5}$ $c_{2}=1\frac{1}{5}$ $x(t)=-\frac{1}{5}e^{-6t}+1\frac{1}{5}e^{-t}$