For any real numbers $\alpha$ and $\beta$, let $y_{\alpha, \beta}(x), x \in R$, be the solution of the differential equation
$\frac{d y}{d x}+\alpha y=x e^{\beta x}, y(1)=1$
Let $S=\left\{y_{\alpha \beta}(x): \alpha, \beta \in R \right\}$. Then which of the following functions belong(s) to the set $S$ ?
$(A)$ $f( x )=\frac{ x ^2}{2} e ^{- x }+\left( e -\frac{1}{2}\right) e ^{- x }$
$(B)$ $f( x )=-\frac{ x ^2}{2} e ^{- x }+\left( e +\frac{1}{2}\right) e ^{- x }$
$(C)$ $f( x )=\frac{ e ^{ x }}{2}\left( x -\frac{1}{2}\right)+\left( e -\frac{ e ^2}{4}\right) e ^{- x }$
$(D)$ $f( x )=\frac{ e ^{ x }}{2}\left(\frac{1}{2}- x \right)+\left( e +\frac{ e ^2}{4}\right) e ^{- x }$