Equivalence Classes and Symmetries of Abel’s Equation

Fritz Schwarz GMD, Institut SCAI 53754 Sankt Augustin, Germany Email: fritz.schwarz@gmd.de

A scheme for solving Abel’s equation is described that is based on a certain infinite group of point transformations, the so-called structure invariance group, and a normal form wrt. this group, the so-called rational normal form (RNF). The key objects in this scheme are the equivalence classes of equations wrt. this group. They are identified by means of an absolute invariant. The equations of an equivalence class may allow one or two point symmetries that are applied for the solution procedure. If an equation does not have a symmetry, membership in an equivalence class may be decided by functional decomposition in the coefficient field of the given equation. Based on this scheme a solution algorithm is developed that generates most of the closed form solutions described in the literature as a consequence of non-trivial symmetries. It is suggested that similar schemes may be developed for other classes of equations.