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