Solving Third Order Differential Equations with Maximal Symmetry Group

Abstract

The largest group of Lie symmetries that a third-order ordinary differential equation (ode) may allow has seven parameters. Equations sharing this property belong to a single equivalence class with a canonical representative v′′′(u)=0. Due to this simple canonical form, any equation belonging to this equivalence class may be identified in terms of certain constraints for its coefficients. Furthermore a set of equations for the transformation functions to canonical form may be set up for which large classes of solutions may be determined algorithmically. Based on these steps a solution algorithm is described for any equation with this symmetry type which resembles a similar scheme for second order equations with projective symmetry group.