Classification of certain class of ordinary differential equations of the first order
In a recent work, one of the authors studied the problem of classifying second order ordinary differential equations of the form \(y'' = F(x,y)\) with \(F\) a smooth function, up to equivalence under the action of (Lie) point and contact pseudogroups [the first author, Funct. Anal. Appl. 51, No. 4, 255--262 (2017; Zbl 1402.34039); translation from Funkts. Anal. Prilozh. 51, No. 4, 16--25 (2017)]. Here, the authors tackle the same problem for equation with rational right hand side; more specifically, they consider first-order equations \[ y' = \frac{A(x) \, y + B (x)}{C(x) y+ D(x)}\eqno{(*)} \] under the action of the Cremona group \texttt{Cr(2)}. They prove that the group \(G\) of symmetries for equations in the above form consists of maps \[ g:(x,y)\to\left(\frac{\phi_1 x + \phi_2}{\psi_1 x + \psi_2}, \eta y + \xi (x) \right), \] where \[ \begin{pmatrix} \phi_1 & \phi_2 \\ \psi_1 & \psi_2 \end{pmatrix} \in \mathrm{PSL} (2 , \mathbf{C}) \ , \xi (x) \in \mathbf{C} (x)\, \eta \in \mathbf{C}. \] The Lie algebra \(\mathcal{G}\) of infinitesimal symmetries of \((*)\) consists of germs of vector fields \[ X = (\alpha x^2 + \beta x + \gamma) \partial_x + (\delta y + \sigma(x)) \partial_y ,\, \alpha,\beta,\gamma,\delta \in \mathbf{C},\sigma (x) \in \mathbf{C} (x). \] Moreover they prove that the field of differential invariants on the relevant space of infinite Jets is freely generated by two differential invariants and one invariant differentiation (all of these provided in explicit form), and give explicit necessary and sufficient conditions for the equivalence -- in the sense mentioned above -- of two equations in the considered class. A number of examples are also considered in detail.
scientific article; zbMATH DE number 7043820