This study primarily deals with introducing an efficient numerical technique called the Complementary Functions Method (CFM) for the solutions of the initial value problem for the linear elastic analysis of anisotropic rotating uniform discs. To bring the performance of the method to light, first, closed form formulas are derived for such discs. The governing equation of the problem at stake is solved analytically with the help of the Euler-Cauchy technique under three types of boundary conditions namely free-free, fixed-free, and fixed-guided constraints. Secondly, the CFM is applied to the same problem. It was found that both numerical and analytical results coincide with each other up to a desired numerical accuracy. Third, after verifying the results with the literature, a parametric study with CFM on the elastic behavior of discs made up of five different materials which physically exist is performed. And finally, by using hypothetically chosen anisotropy degrees from 0.3 through 5, the effects of the anisotropy on the elastic response of such structures are investigated analytically. Useful graphs are provided for readers.