We numerically and experimentally demonstrate the generation of broad bandwidth mode-locked dual-wavelength pulses with diverse-pattern from a dispersion managed erbium-doped (Er-doped) fiber laser. The two-peak gain profile of the Er-doped fiber is shown to have advantages in achieving broadband dual-wavelength pulses compared to a comb filter in our cavity. Our obtained bandwidths of 24 nm and 11.5 nm represent the broadest achieved in an Er-doped dual-wavelength fiber laser to date. In addition, the weak third-order dispersion (TOD) of the fibers facilitates two dispersion-pattern pulses (one stretched pulse and one dissipative soliton) generated in the near zero dispersion regime. Our results provide a convenient, effective way to obtain such sources for potential applications, such as in dual-comb metrology and multicolor pulses in nonlinear microscopy.