For any real numbers \( \alpha \) and \( \beta \), let \( y_{\alpha,\beta}(x), \, x \in \mathbb{R} \), be the solution of the differential equation \( \frac{dy}{dx} + \alpha y = x e^{\beta x}, \; y(1) = 1 \).
Let \( S = \{ y_{\alpha,\beta}(x) : \alpha, \beta \in \mathbb{R} \} \). Then which of the following functions belong(s) to the set \( S \)?
