To analyze pulse switching in two-nonidentical-core couplers, we derive a pair of new generalized coupled-mode equations that describe the switching dynamics of optical pulses in two nonidentical parallel waveguides by considering dispersive properties of the coupling coefficient. We study the influence of the first-order coupling coefficient dispersion (i.e. intermodal dispersion) on pulse propagation in a directional coupler which consists of two nonidentical cores. It is found that the intermodal dispersion can cause severe pulse distortion and affect the coupler functioning as an optical switch. To show how the pulse switching dynamics are affected by the nonlinearity of the coupler, we study the pulse propagation under different input conditions. It is studied that an increase in the threshold amplitude is obtained for the case of nonidentical cores.