Energy of Hydrogen-like atoms using Shcordinger's formula

[Lucas]

Hello,


Really need help for an homework, here is what I need to do:

For Hydrogen-like atoms, using Schrodinger's formula: Eψnlm=-(h²/2m)d²Eψnlm/dr²+VEψnlm (where V=-ke²/r)
a) Derive a formula for E=1,0,0 , E=2,0,0 and E=3,0,0 (where E=n,l,m) and show the second derivative for each wave function.
b) Calculate the value for each in Joules.

Please advise on how to do this. I have spent the last day looking at various resources on the topic (Schrödinger equation for the hydrogen and hydrogen-like atoms, Schrodinger equation in spherical coordinates, etc), at these places:
hyperphysics.phy-astr.gsu.edu/hbase/quantum/sch3d.html#c2
www.nyu.edu/classes/tuckerman/adv.chem/lectures/lecture_8/node5.html
en.wikipedia.org/wiki/Hydrogen-like_atom
onlinelibrary.wiley.com/doi/10.1002/9780470747414.app9/pdf
en.citizendium.org/wiki/Hydrogen-like_atom

I understand the wave functions, and atomic orbitals and probability. But this above looks very unclear to me. Where do I start? Do I need to use spherical coordinates? In the given equation, ψnlm cancels out on each side of the equation, this makes no sense. Why? What do I need to derive exactly? Thank you very much