Variation of Parameters

1. The problem statement, all variables and given/known data

$$y'' - 2y = x + 1$$





2. Relevant equations

$$ y_{o} = Ae^{√(2)x} + Be^{-√(2)x} $$

$$ v_{1}'e^{√(2)x} + v_{2}'e^{-√(2)x}\equiv 0 $$

$$ √(2)v_{1}'e^{√(2)x}-√(2) - v_{2}'e^{-√(2)x} = x + 1 $$





3. The attempt at a solution



$$ v_{2}' = \frac{x+1}{-2√(2)e^{-√(2)x}} $$

$$ v_{2} = (\frac{-x}{4}+\frac{√2}{8}-\frac{1}{4})*e^{√(2)x} $$



$$v_{1}' = (\frac{x+1}{-2√(2)e^{-√(2)x}})(e^{-2√(2)x})$$

$$v_{1} = (\frac{-x}{4}-\frac{√2 + 2}{8})*e^{-√(2)x}$$



$$ y = Ae^{√(2)x} + Be^{-√(2)x} + (\frac{-x}{4}-\frac{√2 + 2}{8})*e^{-√(2)x} + (\frac{-x}{4}+\frac{√2}{8}-\frac{1}{4})*e^{√(2)x} $$



Where's my error? The correct solution is $$ y = Ae^{√(2)x} + Be^{-√(2)x} - \frac{x}{2} - \frac{1}{2}$$