Invariants and Symmetries of Second-Order Ordinary Differential Equations of Nonprojective Type

Abstract

The equivalence problem for second-order equations with respect to point changes of variables is solved. Equations whose right-hand side is not a cubic polynomial in the first derivative are considered. A basis of differential invariants of these equations in both main and degenerate cases is constructed, as well as operators of invariant differentiation and “trivial” relations that hold for the invariants of any equation. The use of the invariants of an equation in constructing its integrals, symmetries, and representation in the form of Euler–Lagrange equations is discussed. A generalization of the derived formulas to the calculation of invariants of equations unsolved for the highest derivative is proposed.