CAPS 2019 Problem 4

Given a real number $\alpha$ , find all pairs $(f,g)$ of functions $f,g :\mathbb{R} \to \mathbb{R}$ such that $$xf(x+y)+\alpha \cdot yf(x-y)=g(x)+g(y) \;\;\;\;\;\;\;\;\;\; ,\forall x,y \in \mathbb{R}.$$