Equivalence problem for first-order and second-order ODEs with a quadratic restriction

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We consider the equivalence problem for first-order ordinary differential equations (ODEs) and recover the well-known result that any two first-order ODEs are equivalent under a point transformation by Cartan’s equivalence method. Curvature properties of some first-order and second-order ODEs motivate us to define the equivalence problem for such equations as the metric equivalence problem which defines the [Formula: see text]-structure on the corresponding manifolds. Accordingly, we show that there are first-order ODEs and linear second-order ODEs which may not be equivalent under a class of fiber-preserving diffeomorphisms.