A geometric approach to integrability of Abel differential equations

Abstract

A geometric approach is used to study the Abel first order differential equation of the first kind. The approach is based on the recently developed theory of quasi-Lie systems which allows us to characterise some particular examples of integrable Abel equations. Second order Abel equations will be discussed and the inverse problem of the Lagrangian dynamics is analysed: the existence of two alternative Lagrangian formulations is proved, both Lagrangians being of a non-natural class. The study is carried out by means of the Darboux polynomials and Jacobi multipliers.

10 pages

References in corpus (9)

• Superposition rules, Lie theorem and partial differential equations
• Abel ODEs: Equivalence and Integrable Classes
• The nonlinear superposition principle and the Wei-Norman method
• Integrability of Riccati equation from a group theoretical viewpoint
• Multidimensional integrable vacuum cosmology with two curvatures
• Friedman vs Abel equations: A connection unraveled
• Higher-order Abel equations: Lagrangian formalism, first integrals and Darboux polynomials
• U(1)-invariant membranes: the geometric formulation, Abel and pendulum differential equations
• Lie systems and integrability conditions for t-dependent frequency harmonic oscillators

Cited by in corpus (4)

• Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations
• Lie--Hamilton systems: theory and applications
• Quasi-Lie families, schemes, invariants and their applications to Abel equations
• Generalized symmetries, first integrals, and exact solutions of chains of differential equations