Instability is observed at certain speeds for rotating disks when the radial displacement in the centrifugal force is taken into consideration. The critical angular velocity values which cause unbounded displacement and stresses will be determined. Analysis of functionally-graded variable-thickness disks requires solving variable-coefficient governing differential equations. Analytical solutions of such equations can be obtained only for certain material grading functions and uniform thickness profiles. In the present study, disks with non-uniform thickness profiles and heterogeneous material properties graded in the radial direction are considered. Solution of the resulting variable-coefficient governing equation of displacement requires the implementation of numerical methods. To this end, a novel approach, Complementary Functions Method is employed as the solution scheme. Complementary Functions Method provides an accurate and efficient solution procedure by reducing the two-point boundary value problem to a system of initial value problems. It is applicable for any continuous material grading functions and thickness profiles. The presented method is validated using a benchmark disk problem with uniform-thickness and material properties graded by a simple power function. Also, the efficiency of the solution scheme is demonstrated by comparing the results with those of finite element method (ANSYS). (C) 2018 Elsevier Ltd. All rights reserved.