Boundary conditions for Green's function?

Question

I have been told that when using green's function we need boundary conditions of that are homogenous and of the form: $$\alpha y(a)+\beta y'(a)+\gamma y(b)+\epsilon y'(b)=0$$ But I cannot see the reason for the need of boundary conditions like this, why won't any boundary conditions work?

Answer 1

It seems likely, as @TrialAndError notes in the comments, that you are considering Green's functions for second order ODEs on $[a,b]$. In this case, boundary conditions involving derivatives higher than first order (I assume that this is what you mean by "any boundary conditions") can be rewritten in the form stated by using the differential equation (a relation giving $y''$ in terms of $y$ and $y'$) to eliminate the higher order derivatives.