Answer
y = $C_1e^{3x}$ + $C_2xe^{3x}$
Work Step by Step
Question: y"-6y'+9y=0
We know:
y = $e^{rx}$
y' = $re^{rx}$
y" = $r^{2}e^{rx}$
This results in $(r^{2}-6r+9)e^{rx}$=0
The corresponding characteristic equation is:
$r^{2}-6r+9=0$
Factoring gives:
$(r-3)(r-3) = 0$
r=3 and r=3
Because we have the same solution twice we need to add an extra x to one of the answers.
Adding the constants gives the following general solution:
y = $C_1e^{3x}$ + $C_2xe^{3x}$
