How to derive Schrödinger wave equation with radial probabprobablity function and what are its solutions

Its derived with the help of total energy E = ½ mv^2 + V

Then you assume a wave function psi as e^i(kx-wt) and double differentiate it to get the schrodinger wave eqn. You will find many videos available on youtube.

And quantum numbers are the solutions of schrodinger wave eqn.