Nonlocal Symmetry and Integrable Classes of Abel Equation

We suggest an approach for description of integrable cases of the Abel equations using the procedure of increasing the order and equivalence transformations for the induced secondorder equations. A diversity of methods were developed to date for finding solutions of nonlinear ordinary differential equations (ODE). Everybody who encounters integration of a particular ODE uses, as a rule, the accumulated databases (or reference books) of the classes of ODE and methods for their integration (e.g. [12, 19]). But if an ODE does not belong to any of the described classes then it does not mean that there is no approaches for finding solutions of this ODE in the closed form. The symmetry approach is one of the most algorithmic approaches for integration and lowering of the order of ODE that admit a certain nontrivial symmetry (see e.g. Lie’s book [13], the books [10, 17, 18] and review papers [10, 25]). In the framework of the symmetry approach (and its modifications) it is possible to obtain many of the known classes of integrable ODE. However, the needs of the applications stimulate new research into development of new methods for construction of ODE solutions in the closed form. The papers [2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25] may give an idea of current developments and directions of research in the field of symmetry (algebraic) methods for investigation of ODE. In this paper we study Abel equations of the first and the second kind [1, 12, 19] ṗ = p3f4(y) + p 2f3(y) + pf2(y) + f1(y), (1) ṗ(p+ f0(y)) = p 3f4(y) + p 2f3(y) + pf2(y) + f1(y), (2) where p = p(y), ṗ = dp dy , fi, i = 0, . . . , 4, are arbitrary smooth functions (with f1, f2, f3, f4 not identically vanishing simultaneously). Equations (1), (2) along with the Riccati equation are among the “simplest” nonlinear first-order ODE that have extensive applications. At the same time the problem of description of integrable classes of these equations stays within the focus of current research, and was previously considered in many papers (see e.g. [5, 6, 7, 8, 16, 19, 20, 22, 23, 24]. Note that the Abel equations of the first and the second kind (1), (2) are related with each other by a local change of variables (namely, the equation (2) can be reduced to the form (1) by means of the change of variables p = 1/v(y)− f0). Besides, the well-known Riccati equation is a partial case of equation (1). The problem of finding Lie symmetries for the first-order ODE is equivalent to finding solutions for these equations, and for this reason the direct application of the Lie method is complicated in the general case. On of the well-known approaches in the cases when for a given ODE it is not feasible (or not effective) to apply the Lie method directly, is increasing of the order of the ODE under consideration (in particular, to obtain a second-order ODE related to the respective ODE by a change of variables). For examples of utilisation of such approach we can refer to papers [2, 3, 4, 5, 6, 9, 14, 15, 16]. In such cases, if the “induced” equation of a