Free vibration analysis of homogeneous and isotropic circular thin plates with variable distribution of parameters by using Green's functions (solution to homogeneous ordinary differential equations with variable coefficients) is considered. The formula of Green's function (called the influence function) depends on the Poisson ratio and the coefficient of distribution of plate flexural rigidity, and the thickness is obtained in a closed-form. The limited independent solutions to differential Euler equations are expanded in the Neumann power series using the Volterra integral equations of the second kind. This approach allows one to obtain the analytical frequency equations as the power series rapidly convergens to exact eigenvalues for different values of the power index and different values of the Poisson ratio. The six lower natural dimensionless frequencies of axisymmetric vibration of circular plates of constant and variable thickness are calculated for different boundary conditions. The obtained results are compared with selected results presented in the literature.