Find the general solution of $ y'''- y'' - 9y' +9y = 0 $ Find the general solution of  $ y'''- y'' - 9y' +9y = 0 $
The answer is
$y=c_{1}e^{-3x}+c_{2}e^{3x}+c_{3}e^{x}$
how do i approach this problem?

You have the following differential equation:
$$ y^{'''}-y^{''}-9y^{'}+9y=0 $$
Applying the Laplace transform:
$$ Y(s)(s^{3}-s^{2}-9s+9)=K_{1}+sK_{2}+s^{2}K_{3} $$
Rearranging:
$$ Y(s)=\frac{K_{1}+sK_{2}+s^{2}K_{3}}{(s+3)(s-3)(s-1)} $$
I can decompose the division into simple fractions:
$$ Y(s)=\frac{C_{1}}{s+3}+\frac{C_{2}}{s-3}+\frac{C_{1}}{s-1}$$
Doing the inverse laplace transform:
$y=C_{1}e^{-3x}+C_{2}e^{3x}+C_{3}e^{x}$
