Interplay of symmetries, null forms, Darboux polynomials, integrating factors and Jacobi multipliers in integrable second-order differential equations (Q2830144)

This paper deals with second-order ODE's of the form NEWLINE\[NEWLINE \ddot{x}=\frac{P}{Q} NEWLINE\]NEWLINE where \(P,Q \in \mathbb{C}[t,x,\dot{x}]\). and the over dot denotes differentiation with respect to the independent variable \(t\). The authors are interested in the integrability of such an equation, that is, the existence of a first integral \(I(t,x,\dot{x})\) which is a non-locally constant functions such that NEWLINE\[NEWLINE \frac{\partial I}{\partial t} + \frac{\partial I}{\partial x} \dot{x} + + \frac{\partial I}{\partial \dot{x}} \frac{P}{Q}\equiv 0. NEWLINE\]NEWLINE There are several methods in order to show that a second-order ODE is integrable. The present paper establishes some interconnections among these different methods. In particular, it provides relations between the extended Prelle-Singer method with several well-known methods such as Lie symmetries, \(\lambda\)-symmetries, Darboux polynomials, Jacobi last multiplier and adjoint symmetries methods. The illustrate these assertions with the specific example of the modified Emden equation: NEWLINE\[NEWLINE \ddot{x}=-3x \dot{x} - x^3. NEWLINE\]NEWLINE They extended Prelle-Singer method was previously stated by the authors in several previous works.