The paper describes the process of applying mathematical methods for obtaining analytic solutions in dynamic problems of free vibrations of composite laminate materials which are used in automotive industry. For selected high-order deformation theories the paper presents mathematical method for analytic solving of partial differential equations of motion. The procedure of obtaining results by combining symbolic and numerical values of relevant variables was implemented. Through a kind of comparative analysis, the emphasis was put on the use of HSDTs based on shape functions. Comparative values of results obtained using different shape functions are presented in tabular form. The paper considers simply supported thin, moderately thick and thick laminate plates. Moreover, it shows the influence of the laminate class of symmetry on the method of obtaining results. Simply supported laminate plates were primarily considered, specifically symmetric and anti-symmetric cross-ply laminate plates.