One equation, three variables tangled together: kinetic energy split into a radial part and an angular part, the proton's pull, and the total energy E we want to find.
Show the algebra
r21∂r∂(r2∂r∂ψ)=∂r2∂2ψ+r2∂r∂ψ
Product rule: the radial Laplacian is the second derivative plus a 2/r term from the spreading of spherical shells.
V(r)=−4πε0re2
The potential depends on r alone. This is what makes separation possible.