Answer
$y=-e^{4x}+3e^{2x}$
Work Step by Step
$y''-6y'+8y=0$
Use auxiliary equation
$r^{2}-6r+8=0$
$(r-4)(r-2)=0$
$r_{1}=4$
$r_{2}=2$
Formula 8
$y=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}$
$y=c_{1}e^{4x}+c_{2}e^{2x}$
Given: $y(0)=2$
$e^{0}=1$
$y(0)=2=c_{1}e^{4(0)}+c_{2}e^{2(0)}$
$c_{1}+c_{2}=2$
$y'(0)=2=4c_{1}e^{4(0)}+2c_{2}e^{2(0)}$
$4c_{1}+2c_{2}=2$
$c_{1}+c_{2}=2$
$4c_{1}+2c_{2}=2$
Use substitution to solve for $c_{1}$ and $c_{2}$
$c_{1}=2-c_{2}$
$4(2-c_{2})+2c_{2}=2$
$8-4c_{2}+2c_{2}=2$
$-2c_{2}=-6$
$c_{2}=3$
$c_{1}=2-c_{2}$
$c_{1}=3-2$
$c_{1}=-1$
$y=-e^{4x}+3e^{2x}$
