A set of one-group space-dependent neutron kinetics equations for reactor cores with spatially variable moderator density is developed. The solution to this set of differential equations is obtained numerically using an IBM-7094 digital computer. Employing the variational technique of von Neumann, a numerical stability criterion for space-dependent neutron kinetics equations is established. The present analysis is useful in the determination of the core open-loop response as well as the reactor system transient behavior. The open-loop response of a typical boiling-water reactor core for several values of step change in reactivity was determined using the present analysis. These are shown to be in agreement with the results of the classical space-independent neutron kinetics equations. The open-loop characteristic of the reactor core due to a step change in density distribution is also presented. The main distinguishing feature of the present study is the ability to determine the open-loop response due to disturbances (such as a series of successive step changes in density distribution) for which the classical space-independent approach provides no solution. Characteristics of this type are necessary in the dynamic analysis of boiling-water reactors where the system density distribution varies in time and space. A simple approximate method for the solution of the space-dependent neutron kinetics equations is also presented.