For any real numbers $$\alpha$$ and $$\beta$$, let $$y_{\alpha, \beta} \left(x\right), x \epsilon R$$, be the solution of the differential equation
$$\frac{dy}{dx} + \alpha y = x e^{\beta x}, y \left(1\right) = 1$$.
Let $$S = \left\{y_{\alpha, \beta} \left(x \right) : \alpha, \beta \epsilon R \right\}$$. Then which of the following function 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

$$f(x) = \frac{e^{x}}{2} \left( \frac{1}{2} -x\right) + \left(e + \frac{e^{2}}{4}\right) e^{-x}$$