Solving Second Order Ordinary Differential Equations with Maximal Symmetry Group

Abstract

Second order ordinary differential equations of the form $y''+A{y'}^3+B{y'}^2+Cy'+D$ with $y\equiv y(x)$ 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.