Stability of Polynomial Planar Systems (differential equations)

Question

Consider the general system of differential equations $\dot{x}=f(x,y)$ and $\dot{y}=g(x,y)$ where $f$ and $y$ are polynomial functions and $f(0,0)=g(0,0)=0$. What are the conditions on $f$ and $g$ such that the Lyapunov function $V(x,y)= \alpha x^2+ \beta y^2$ where $\alpha, \beta \ge 0$ can certify the stability of the the origin? Thank you, in advance, for your response!