New exact general solutions of Abel equation of the second kind

摘要

The Abel equation of the second kind[g0(x)+g1(x)u]u′=f0(x)+f1(x)u+f2(x)u2is reduced to the equivalent equation of the Abel normal form uux′-u=Φ(x) by means of various admissible functional transformations. These equivalent equations do not admit exact solutions in terms of known functions, since only very special cases of the above type of Abel equation can be solved in parametric form, see [5], [6], [8]. In this paper, we introduce a new functional relation between the variable coefficients that can lead to the general solutions of the above Abel equation.