This paper studies the response cumulants for a kind of special nonholonomic systems under non-Gaussian, delta-correlated excitations. We present a new methodology for formulating the equations goveing the evolution of theresponse cumulants of the stochastic dynamic systems. The response cumulant differential equations (CDEs) derivedcan be used to calculate the response cumulants for both linear and nonlinear systems. One example is given to illustratehow to use the CDEs for calculating response cumulants.