Axisymmetric displacements and stresses in functionally-graded hollow cylinders, disks and spheres subjected to uniform internal pressure are determined using plane elasticity theory and Complementary Functions method. The material is assumed to be functionally graded in the radial direction. Variations in the material properties such as Young's modulus and Poisson's ratio may be arbitrary functions of the radial coordinate. This assumption yields a two-point boundary value problem with a governing differential equation of variable coefficients. General analytical solutions of such equations are not available. Infusion of Complementary Functions method into the analysis of stresses in pressure vessels is a novel approach. Complementary Functions method reduces the boundary value problem to an initial-value problem which can be solved accurately by one of many efficient methods such as Runge-Kutta method. Various material models from the literature are used and corresponding radial displacement and stresses are computed. Verification of the proposed method is done using benchmark solutions available in the literature for some special cases and virtually exact results are obtained using the fifth-order Runge-Kutta method (RK5).