An analytical solution to the nonlinear space- and time-dependent coupled wave equations describing a pulsed doubly-resonant optical parametric oscillator (DROPO) with arbitrary cavity mirror reflectivities and arbitrary nonlinear interaction strength in an absorptive medium has been obtained, assuming degenerate operation and neglecting dispersive effects. This solution makes it possible to investigate the spatio-temporal behavior of the optical fields in the cavity when the DROPO is pumped by a source having any desired spatio-temporal intensity profile for any values of cavity mirror reflectivity. The influence of pumping source parameters such as pulse duration and peak pulse intensity on the behavior of the DROPO is studied for different mirror reflectivities, as are the effects of absorption and cavity length. Steady-state solutions exhibit multivaluedness when the cavity Q is high; temporally, bistability and hysteresis are predicted, leading to asymmetric temporal compression of the transmitted pump pulse. The steady-state solutions provide closed-form expressions for the oscillation threshold, and it is shown that a DROPO can be designed which retains the advantage over a singly resonant OPO (SROPO) of a lower threshold while still providing the high efficiency of the SROPO, but without the extreme sensitivity to cavity length variations and pump bandwidth usually associated with DROPOs. Some popular misconceptions with regard to the temporal modeling of optical parametric oscillators are examined and clarified, and the predictions of the theory are compared to some existing experimental data with excellent agreement.