Symmetry Analysis of Abel's Equation

A solution algorithm for Abel's equation and some generalizations based on a nontrivial Lie symmetry of a particular kind, i.e., so-called structure-preserving symmetry, is described. For the existence of such a symmetry a criterion in terms of the coefficients of the so-called rational normal form of the given equation is derived. If it is affirmative, solving Abel's equation is reduced to a well-defined integration problem. It is shown that almost all known ad hoc methods for obtaining closed form solutions are consequences of this type of symmetry. Possible extensions of this scheme to more general classes of first-order ordinary differential equations are pointed out.