Solving Second Order Ordinary Differential Equations with Maximal Symmetry Group

Second order ordinary differential equations of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$y''+A{y'}^3+B{y'}^2+Cy'+D$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$y\equiv y(x)$\end{document} and A, B, C and D functions of x and y are of special interest because they may allow the largest possible group of point symmetries if its coefficients satisfy certain constraints. For large classes of these equations a solution algorithm is described that determines its general solution in closed form by reducing it to a linear third-order equation. If the results obtained by Sophus Lie in the last century are supplemented by more recent concepts like Janet bases and Loewy decompositions, a systematic solution procedure is obtained that is easily implemented in a computer algebra system.