Based on the Helmholtz free energy function of the dielectric elastomer (DE) with consideration of the dissipation processes and thermal effects, the dynamic equations derived from the work done by the inertia forces and Euler-Lagrange equation along with the quasi-static equation can be established. Under typical loading patterns, including linear and sinusoidal voltages and a sinusoidal force coupled with a constant voltage, the distinctions between the performances of DE induced by the two different governing equations can be fully discussed. Therefore, the effects of the inertia forces in both plane and thickness directions can also be revealed in detail. Furthermore, the mechanisms of failures are briefly studied. The numerical results have indicated that the deformations calculated from the quasi-static and dynamic models are identical under small amplitudes and frequencies of actuation. However, when the external loading is large enough to cause failures, obvious distinction appears and the DE suffers from rupture sooner using the quasi-static model, especially under linear actuation. In addition, no resonance can be detected via the quasi-static equation and the mean stretch calculated from the dynamic equation coincides with the stretch obtained from the equivalent voltage (static) of the applied sinusoidal voltage in some extents. Under a sinusoidal force with a static voltage, the stretch calculated from the dynamic equation oscillates around the curve induced by the quasi-static model at first and finally coincides with it. When power source is cut off, the current leakage lowers the mean stretch of the oscillation.