Davidson potentials of the form $β^2 +β_0^4/β^2$, when used in the original Bohr Hamiltonian for $γ$-independent potentials bridge the U(5) and O(6) symmetries. Using a variational procedure, we determine for each value of angular momentum $L$ the value of $β_0$ at which the derivative of the energy ratio $R_L=E(L)/E(2)$ with respect to $β_0$ has a sharp maximum, the collection of $R_L$ values at these points forming a band which practically coincides with the ground state band of the E(5) model, corresponding to the critical point in the shape phase transition from U(5) to O(6). The same potentials, when used in the Bohr Hamiltonian after separating variables as in the X(5) model, bridge the U(5) and SU(3) symmetries, the same variational procedure leading to a band which practically coincides with the ground state band of the X(5) model, corresponding to the critical point of the U(5) to SU(3) shape phase transition. A new derivation of the Holmberg-Lipas formula for nuclear energy spectra is obtained as a by-product. Comment: LaTeX, 12 pages, 4 postscript figures