Degenerate optical parametric oscillators (OPOs) pumped by continuous-wave (cw) lasers in bulk or integrated format represent a promising platform for the study of classical as well as quantum phenomena, enabling interesting applications such as squeezing and random number generation. When subjected to phase modulation, such devices can generate ultrashort output pulses. In this report, we study pulse generation and spectral broadening in degenerate OPOs with intracavity electro-optic modulator (EOM). We numerically investigate ultrashort pulse generation from a cw-driven degenerate OPO based on MgO:PPLN as the nonlinear gain medium. The phase sensitive gain provided by the quadratic nonlinear medium in a detuned cavity and the inherent self-phase-locking of the degenerate field with that of the pump govern the signal amplification. Using high-performance computing tools, we study the full intracavity complex field evolution over 10000 round trips, as the OPO reaches the steady state. Taking advantage of the high nonlinear gain and low threshold of degenerate OPO, and using an intracavity EOM synchronized to the free-spectral-range of the cavity in combination with spectral filtering, we show that a stable, uniform, and periodic train of picosecond pulses of 5 ps duration with FWHM spectral bandwidth of 0.4 THz can be generated. The results of our theoretical model are valid for OPOs in bulk and integrated format.