In this chapter we consider the solution of the generalized population-balance equation (GPBE) with a generic set of internal coordinates ξ = (ξ1, ξ2, …, ξM) under the hypothesis of spatial homogeneity. Under this hypothesis, all spatial gradients are null and the GPBE depends only on time t and on ξ. As discussed earlier, we refer here to the GPBE as a general equation describing the evolution of a number-density function (NDF) in space (neglected in this chapter), in time, and in the phase space generated by the internal coordinates. The extension to inhomogeneous systems is discussed in Chapter 8. It is also worth mentioning that the GPBE is given different names in different fields. It is called the population-balance equation (PBE) in crystallization, precipitation, and, in general, in the literature on particulate processes. In the simulation of aerosols, it is called the general aerosol dynamic equation (Friedlander, 2000), whereas in the simulation of sprays it is generally known as the Williams–Boltzmann equation (Williams, 1985) and, especially in this case, the droplet velocity is included as one of the internal coordinates. When dealing with solid particles, if the particle velocity is included as the only internal coordinate, the GPBE is called the Boltzmann equation. In general, when the particle velocity is the only internal coordinate, the GPBE is also called the kinetic equation (KE). Many of the challenges associated with the KE occur with inhomogeneous systems, which are discussed in Chapter 8.