A non-homogeneous membrane is a membrane that has variable thickness or material density. Several recent publications deal with the axisymmetric and antisymmetric modes of transverse vibration for a composite doubly connected and solid membrane with constant or variable density. In this paper, exact solutions for both the axisymmetric and antisymmetric modes of circular and annular membranes with any piecewise polynomial variation of the density are given using a power series solution. The dynamic problem is solved exactly using a recurrence relationship up to any accuracy desired, by deriving the dynamic stiffness matrix for circular and annular membrane elements. Many results for linear, parabolic, and cubic variation of complete and annular membranes are solved using the dynamic stiffness method and presented in the Tables.