We develop a novel transformation that maps the linear, nonhomogeneous, multidimensional population balance equation (PBE) into an advection equation that is readily solved using the method of characteristics. The PBEs targeted by this transformation exclude aggregation, breakage, and time dependent birth and death rates. In addition, internal coordinates are assumed to grow independently of each other. The ensuing general formulation is then used to recover closed-form analytical solutions for problems with monosurface and bulk-diffusion growth-rates as well as Gaussian-type nucleation. For completeness, we derive the multidimensional Green's functions for our approach. This is followed by a brief discussion on how the proposed framework may be used for code verification of moment methods such as the quadrature method of moments and the direct quadrature method of moments. Finally, a sample Mathematica code is provided to derive analytical solutions for the single-internal-coordinate case given user-specified growth, birth, and death rates. (c) 2015 American Institute of Chemical Engineers AIChE J, 61: 1691-1698, 2015