Longitudinal mode selection in diode lasers by injection locking has been studied using the standard multimode rate equations with an optical injection term, and with the effects of spontaneous emission included. The phenomena of spectral narrowing and power switching between the dominant free-running mode and the injected target mode are illustrated and discussed. In general, injection locking into the long-wavelength side of the spectrum is more effective than into the short-wavelength side because of the gain peak shift. The relaxation oscillation frequency is estimated using a small-signal analysis of the single-mode rate equations, an approach that is valid provided the external injection is strong enough. Our calculations indicate that around the gain peak the relaxation oscillation frequency strongly depends on the wavelength of the injected target mode, because of the changing differential gain. Modes with shorter wavelengths have faster relaxation oscillations due to larger differential gain. This can be helpful in increasing the transmission speed in optical communication applications.