Invariants of a family of scalar second-order ordinary differential equations (Q2843745)

The author considers a family of differential equations of the form NEWLINE\[NEWLINE\frac{d^2 y}{dx^2}=S(x,y)\left(\frac{dy}{dx}\right)^3+3R(x,y)\left(\frac{dy}{dx}\right)^2+3Q(x,y)\frac{dy}{dx}+P(x,y). \tag{1}NEWLINE\]NEWLINE It is closed under the action of the group \(E\) of point transformations NEWLINE\[NEWLINEx\mapsto \varphi (x, y),\quad y\mapsto \psi (x, y),\quad \partial (\varphi ,\psi )/\partial (x, y)\neq 0\tag{2}NEWLINE\]NEWLINE and contains all 50 equations classified by Painlevé equations and Gambier (listed in [\textit{E. L. Ince}, Ordinary differential equations. London: Longmans, Green \& Co (1926; JFM 53.0399.07)]). Despite the very general nature of the transformations (2), the equations from the family (1) form many different equivalence classes under these transformations. Each equivalence class is characterized by a set of its invariants computed from the coefficients of equation (1), and the author gives a complete list of formulas for the calculation of these invariants. However, in some (degenerated) cases, these invariants are insufficient to describe the equivalence class and one has to use the so-called relative invariants, i.e. invariants of some proper subgroup of the group \(E\). All such cases are carefully examined in the paper, and the obtained results are applied to the invariant characterization of the Painlevé equations and finding the point transformations for equivalent equations.