Interplay of symmetries, null forms, Darboux polynomials, integrating factors and Jacobi multipliers in integrable second-order differential equations

Abstract

In this work, we establish a connection between the extended Prelle–Singer procedure with five other analytical methods which are widely used to identify integrable systems in the contemporary literature, especially for second-order nonlinear ordinary differential equations (ODEs). By synthesizing these methods, we bring out the interplay between Lie point symmetries, λ -symmetries, adjoint symmetries, null-forms, Darboux polynomials, integrating factors and Jacobi last multiplier in identifying the integrable systems described by second-order ODEs. We also give new perspectives to the extended Prelle–Singer procedure developed by us. We illustrate these subtle connections with the modified Emden equation as a suitable example.