# Thread: Solving a DE by method of variation of parameters

1. ## Solving a DE by method of variation of parameters

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

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

by the method of undetermined coefficients.

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

I have the homogeneous equation as: $y_{h} = A e^{3x} + B e^{-3x}$ and so $y_{p} = A(x) e^{3x} + B(x) e^{-3x}$
and
$A'(x) e^{3x} + B'(x) e^{-3x} = 0$
$3A'(x) e^{3x} - 3B'(x) e^{-3x} = (x+1) e^{-x}$

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

Thank you all in advance.

2. ## Re: Solving a DE by method of variation of parameters

Try multiplying the first equation by 3, then adding the two equations, as this will eliminate $B'(x)e^{-3x}$.