Decomposing and solving quasilinear second-order differential equations

Decompositions of linear ordinary differential equations (ode's) into components of lower order have successfully been employed for determining its solutions. Here this method is generalized to certain classes of quasilinear equations of second order, i.e. equations that are linear w.r.t. the second derivative, and rational otherwise. Often it leads to simple expressions for the general solution that hardly can be obtained otherwise, i.e. it is a genuine extension of Lie's symmetry analysis. Due to its efficiency it is suggested that it is applied always as a first step in an ode solver.