Hybrid-Trefftz elements are a non-conforming breed of finite elements that use domain approximations that satisfy locally the homogeneous form of the differential equations governing the problem. This option builds relevant physical information into the kernel of the elements and is responsible for the super-convergence that typifies hybrid-Trefftz finite elements applied to wave propagation problems (e.g. insensitivity to small wavelengths and large solution gradients). Moreover, the same option reduces the formulation of the hybrid-Trefftz elements to the boundaries of the mesh. On the other hand, Trefftz-compliant approximation functions are unable to recover the particular solution of a non-homogeneous problem, as they are complementary solutions themselves. In such cases, the particular solution needs to be approximated using some specific procedure, most frequently the Dual Reciprocity Method. A typical non-homogeneous problem occurs when the governing equations describing the propagation of an elastic wave are discretized in time using finite difference methods, as the non-trivial initial conditions force the emergence of a source (non-homogeneous) term at each time step. In the recent years, the authors have developed a novel, Trefftz-based Dual Reciprocity Method, where the functions used to approximate the particular solution are of the same type as the functions used to construct the complementary solution (with different wave numbers), which led to important algorithmic simplifications and superior computational efficiency. This technique is now extended to plane elasticity problems and implemented in FreeHyTE, currently the largest public computational platform featuring hybrid-Trefftz finite elements.