Solving second order ordinary differential equations with maximal symmetry group (Q1293452)

The author investigates the second-order ordinary differential equation \(y''+{A y'}^3+ {B y'}^2+C y' + D =0\) with \(A,B,C,D\in Q(x,y)\). By a nonlinear change of coordinates this is equivalent to a simpler equation. The functions involved in the transformation satisfy a system of linear partial differential equations. The proof is based on Lie symmetry and Janet bases. This algorithmic approach is illustrated by several examples. In the appendix a short introduction into Loewy decomposition for the solution of partial differential equations is presented.