Answer
$y(x) = C_1•e^{6x} + C_2•e^{2x}$
Work Step by Step
We assume a solution of the form $y=e^{mx}$,where $m$ Is a constant.
$y = e^{mx}$ $\implies$ $y' = m(e^{mx})$ and $y'' = m^2(e^{mx})$
$y'' -8y' + 12y = 0$
$\implies$ $m^2(e^{mx}) - 8m(e^{mx}) + 12(e^{mx}) = 0 $
$\implies$ $e^{mx}(m^2-8m+12) = 0$
since $e^{mx}$ cant be 0
$\implies$ $m^2 -8m + 12= 0$
$\implies$ $m^2 + 6m + 2m + 12 = 0$
$\implies$ $(m+6)(m+2) = 0$
$\implies$ $ m = 6$ or $ m = 2$
since we have distinct roots,we have that :
$y(x) = C_1e^{6x} + C_2e^{2x}$
