A simplified method to study the response of embedded piles in a non-homogeneous Pasternak elastic soil is developed. The governing differential equation (DE) derived from equilibrium is solved using the Differential Transformation Method (DTM), providing a compact solution for static and stability analyses that is easy to implement. The solution derived with the DTM overcomes the limitation of conventional approaches used to solve the proposed structural model, where the mathematical derivation and solution become cumbersome and tedious to implement. The proposed formulation is a continuation of the work recently presented by the authors for frictionless beam-column elements, and now incorporates the magnitude and distribution of friction along the element. The applied axial load is carried by both end-bearing resistance and skin friction, so that purely end-bearing, partially frictional, and purely frictional piles can be studied. In practice, the influence of side friction on pile stability is typically neglected, but can have a substantial influence on the buckling capacity of embedded elements---a practical benefit that has received little attention. This work provides an analytical solution for the governing DE with the flexibility to account for different distributions and magnitudes of skin friction, different combinations of geotechnical end- and side-resistance, and generalized end-boundary conditions. The model is applied to elucidate the effects of intermediate end-boundary conditions, as well as the distribution and magnitude of soil-pile interface friction on pile stability.