The dynamic behavior of the spin-1 Ising Blume-Emery-Griffiths model Hamiltonian with bilinear and biquadratic nearest-neighbor exchange interactions and a single-ion potential is studied by using the path probability method of Kikuchi. First the equilibrium behavior of the model is given briefly in order to understand the dynamic behavior. Then, the path probability method is applied to the model and the set of nonlinear differential equations, which is also called the dynamic or rate equations, is obtained. The dynamic equations are solved by using the Runge-Kutta method in order to study the relaxation of order parameters. The relaxation of the order parameters are investigated for the system which undergoes the first- and second-order phase transitions, especially near and far from the transition temperatures. From this investigation, the "flatness" property of metastable states and the "overshooting" phenomenon are seen explicitly. On the other hand, the solutions of the dynamic equations are also expressed by means of a flow diagram for temperatures near and far from the transition temperatures. The stable, metastable and unstable solutions are shown in the flow diagrams, explicitly and the role of the unstable points, as separators between the stable and metastable points, is described. The dynamic behavior of the model is also studied by using the kinetic equations based on the Zwanzig-Nakajima projection operator formalism and the variational principle. Finally, it is found that one can investigate the dynamic behavior of the system by the path probability method more comprehensively than via the kinetic equations based on the Zwanzig-Nakajima projection operator formalism and the variational principle. (C) 2000 American Institute of Physics. [S0021-9606(00)50714-4].