Starting from a generalized diffusion equation and the Collins-Kimball boundary condition, we investigated the inertia and the non-Markovian effects on the time-dependent rate constant of a diffusion-controlled reaction at short times. In the short-time limit, we obtained the rate constant analytically, and found that the rate constant was independent of the friction coefficient, and was always smaller than the result of the classical Smoluchowski-Collins-Kimball (SCK) theory in which both of the inertia and the non-Markovian effects were neglected. At finite times, we obtained the rate constant numerically, and found that the decay of the rate constant was slower than that of the SCK result. When the non-Markovian effect became larger, the decay became much slower. Our results were consistent with a relevant theory based on a generalized Fokker-Planck equation. The results were compared with computer simulations, and a good agreement was obtained for the case of the maximum reactivity.