The solution of the cartan equivalence problem for $\frac{{d^2 y}}{{dx^2 }} = F(x,y,\frac{{dy}}{{dx}})$under the pseudo-group $\bar x = \varphi (x),\bar y = \psi (x,y)$

Abstract

We give a complete solution to the local equivalence problem for $\frac{{d^2 y}}{{dx^2 }} = F(x,y,\frac{{dy}}{{dx}})$under the pseudo-group of coordinate transformations $\bar x = \varphi (x),\bar y = \psi (x,y)$. Applying Cartan's equivalence method, we obtain an e-structure on J¹ (ℝ, ℝ) x G, where G is a certain three-dimensional real Lie group. Vie show that except for the equivalence class of $\frac{{d^2 y}}{{dx^2 }} = 0$, the G-action can be used to reduce this {e}-structure on J¹ (ℝ, ℝ) x G to an e-structure on a lower-dimensional space J¹(ℝ, ℝ) x G₍₁₎, where the Lie group G₍₁₎ is at most one-dimensional. We then show how the invariants obtained by this procedure can be used to obtain necessary and sufficient conditions for equivalence.