Method of Variation of Parameters

Consider the 2nd order linear ODE in standard form:

y^{″} (x) + p (x) y^{'} (x) + q (x) y (x) = g (x)

where $p (x)$ , $q (x)$ and $g (x)$ , are continuous on the interval of interest

Put ODE in standard form
Look for a particular solution of the form $y_{p} (x) = u_{1} (x) y_{1} (x) + u_{2} (x) y_{2} (x)$
with $y_{1}, y_{2}$ as linearly independent solutions to the homogenous problem
$u_{1}^{'} = \frac{- y_{2} g (x)}{W (y_{1}, y_{2})}, u_{2}^{'} = \frac{y_{1} g (x)}{W (y_{1}, y_{2})}$
$u_{1} (x) = \int u_{1}^{'} (x) d x and u_{2} (x) = \int u_{2}^{'} (x) d x$
$y (x) = y_{g e n} (x) = \underset{y_{c} (x)}{\underset{⏟}{c_{1} y_{1} (x) + c_{2} y_{2} (x)}} + \underset{y_{p} (x)}{\underset{⏟}{u_{1} (x) y_{1} (x) + u_{2} (x) y_{2} (x)}}$