The Schrodinger equation (time-independent) is for sure the most important equation in quantum mechanics;
$$-\frac{ℏ^2}{2m}∇^2ψ(x)+V(x)ψ(x)=Eψ(x)$$
let consider the one-dimensional equation;
$$\frac{d^2ψ(x)}{dx^2}-\frac{2m}{ℏ^2}(V(x)-E)ψ(x)=0$$
Is there any exact general eignvalue-eignfucntion solution for such equation? And if it's not possible to get the exact eignvalue-eignfucntion solution, can the equation be solved in a pure mathematical sense? Where we can rewrite the equation as;
$$\frac{d^2ψ(x)}{dx^2}+ S(x)ψ(x)=0$$
wherein $S(x)=-\frac{2m}{ℏ^2}(V(x)-E)$. I'm not talking here about the WKB approximation method, I'm talking about an exact and general solution, so does this equation has analytical solutions?