The purpose of this manuscript is to introduce a new class of functional equations and to provide their general solutions. Specifically, we consider \( f(x^2-\alpha y^2)= xf(x)-\alpha y f(y), \quad x,y\in \mathbb {R}, \) where f is an unknown mapping from the field of real numbers \(\mathbb {R}\) into a real vector space \(\mathcal {X}\) , and \(\alpha > 0\) is a fixed real number. We also study the stability problem and the Hyers-Ulam stability of this equation on a restricted domain. Furthermore, we obtain several results concerning the asymptotic behavior of the equation, as well as its hyperstability in the case \(\alpha =1\) .
