I have found the general solution for the equation of $\displaystyle y'' - 9y = (x+1) e^{-x}$ to be

$\displaystyle y(x) = A e^{3x} + B e^{-3x} - \frac{1}{8} e^{-x} x - \frac{3 e^{-x}}{32}$

by the method of undetermined coefficients.

However I need to use themethod of variation of parameters, but I am totally confused.

I have the homogeneous equation as: $\displaystyle y_{h} = A e^{3x} + B e^{-3x}$ and so $\displaystyle y_{p} = A(x) e^{3x} + B(x) e^{-3x}$

and

$\displaystyle A'(x) e^{3x} + B'(x) e^{-3x} = 0$

$\displaystyle 3A'(x) e^{3x} - 3B'(x) e^{-3x} = (x+1) e^{-x}$

But where do I go now? Can anyone help me with the steps to solve this?

Thank you all in advance.