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I) Let us for simplicity put the physical constants $\hbar=1=m$ to one. OP is considering the usual transcription $u(r)\equiv rR(r)$ of the 3D radial TISE into a 1D TISE, $$\tag{A} - \frac{1}{2} u^{\prime\prime}(r)+U_{\ell}(r)u(r) ~=~E u(r),$$ where the total potential energy $$\tag{49.8b} U_{\ell}(r)~:=~ U(r) + \frac{C_{\ell}}{2r^2} $$ is a sum of a ...

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