I was wondering, if it is possible to obtain somehow a solution expressed with elementary functions from an equation, which is in non closed-form at first sight. For istance let us consider an implicit function F(x,y)=0 with Fx(0,0)≠0, which can't be explicitly solved for y. Nevertheless, according to the implicit function theorem, a function y(x) exists in the neighbourhood of (0,0). We also know from the same theorem that yx(x)=-Fx(x,y)/Fy(x,y). One could now claim that the differential equation dy/dx=g(x,y) (whereby g(x,y)=-Fx(x,y)/Fy(x,y)) could have a closed-form solution for y! The odds are that this method would never yield any result. If that is the case, can sb, more familiar with the Galois theory or the theory of functions, provide us with a justification/proof?