The comments people have been posting look excellent, and if you are already on your way to solving and understanding n(x) feel free to ignore this. I just thought I would make the mistake of posting a possible picture and coordinate system (setting 0 coincident with Ev on the far right), etc. Using deltaE separates the problem into two distinct parts and subsumes everything you have done with rho, Electric field and potential into the parameter, deltaE, which is a coefficient in the quadratic dependence of the band edge energies ...
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