## Calculus 8th Edition

$y=2e^{-2x}-2xe^{-2x}$
$y''+4y'+4y=0$ Use auxiliary equation $r^{2}+4r+4=0$ $r=\frac{-4±\sqrt ((4)^{2}-4(1)(4))}{2(1)}$ $r=\frac{-4±\sqrt 0}{2}$ $r=-2$ $y=c_{1}e^{r_{1}x}+c_{2}xe^{r_{2}x}$ $y=c_{1}e^{-2x}+c_{2}xe^{-2x}$ $y(0)=2$ $2=c_{1}e^{-2(0)}+c_{2}(0)e^{-2(0)}$ $c_{1}=2$ $y(1)=0$ $0=2e^{-2}+c_{2}e^{-2}$ $2e^{-2}=-c_{2}e^{-2}$ $c_{2}=-2$ $y=2e^{-2x}-2xe^{-2x}$