Abstract <p>Slender elastic elements with non-uniform cross-sections are extensively utilized in engineering applications owing to their structural efficiency. Mathematically, these elements occupy a non-prismatic, inhomogeneous cylindrical region within the three-dimensional Euclidean space, significantly complicating the analytical determination of their mechanical response compared to prismatic homogeneous counterparts. Their state of stress and strain is described by a system of partial differential equations (PDEs) and boundary conditions derived via a variational principle. Analytical, closed-form solutions to these equations are only available in a few cases. Analytical solutions are presented in this paper for tapered cylinders with square and circular cross-sections, and axially varying material properties, subjected to external actions applied at the ends. Beyond their theoretical significance, analytical solutions provide practical benefits in engineering analysis, facilitating the rapid preliminary design of aerospace structures, mechanical components, biomedical implants, and smart elastic elements for energy harvesting, among other applications. By examining paradigmatic cases solved in closed form, this study also highlights the limitations of commonly used engineering approaches that approximate the geometry and material properties as piecewise constant along the cylinder’s axis, an inadequate practice often employed in technical applications.</p>