The etching of semiconductor wafers is modeled for etching by radicals (isotropic etching) and etching by both radicals and vertically incident cold ions (anisotropic etching) in a glow-discharge plasma. Explicit analytical expressions for evolving two-dimensional etched surfaces are found by the method of characteristics. These parametric relations are expressed in terms of the position along the initial exposed wafer surface. Exact surface equations are given for the radical etching with two different ansatz for the evolution equation. The superposition of radical etching and etching due to vertically incident, bombarding ions is solved by approximate analytical expressions for the etched surface. Two-dimensional etched surfaces are displayed graphically for various times.