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