  We obtain exact solutions of the 2D Schr\"odinger equation with the central
potentials $V(r)=ar^2+br^{-2}+cr^{-4}$ and $V(r)=ar^{-1}+br^{-2}$ in a
non-commutative space up to the first order of non-commutativity parameter
using the power-series expansion method similar to the 2D Schr\"odinger
equation with the singular even-power and inverse-power potentials respectively
in commutative space. We derive the exact non-commutative energy levels and
show that the energy is shifted to $m$ levels, as in the Zeeman effect.
