Answer
$y = C_1e^{3x} + C_2e^{-2x}$
Work Step by Step
Question: $y"-y'-6y=0$
We know:
$y = e^{rx}$
$y' = re^{rx}$
$y" = r^{2}e^{rx}$
This results in $(r^{2}-r-6)e^{rx}=0$
The corresponding characteristic equation is:
$r^{2}-r-6=0$
Factoring gives:
$(r-3)(r+2) = 0$
$r=3$ and $r=-2$
Adding the constants gives the following general solution:
$y = C_1e^{3x}$ + $C_2e^{-2x}$
