Group classification of ODE

Generating functions and existence of contact symmetries of third order scalar ordinary differential equations

“…In order to ensure that the first term remains isolated, the power (n + 2) cannot coincide with the other powers ofẍ. From (8) it is easy to see that this excludes the values n = −1, 0, 1, 2. In addition, n = 3, since otherwise the first term vanishes identically.…”

“…Lie point symmetries of scalar ordinary differential equations (ODEs) have been studied in detail by many authors, and their structure and classification types are nowadays well known ( [1][2][3][4][5][6][7][8] and references therein). For systems of ODEs, various general results also exist, although in this case the analysis is still far from being complete [9][10][11][12][13].…”

Sistema óptimo, soluciones invariantes y clasificación completa del grupo de simetrías de Lie para la ecuación de Kummer-Schwarz generalizada y su representación del álgebra de Lie

“…where k 1 , k 2 and k 3 are arbitrary constants. Therefore the group of symmetries are [18,9]. If f = a ̸ = 0 where a is an arbitrary constant.…”

“…The global dynamics of the Kummer -Schwarz differential equation was studied in [29]. Following this line, in [9,18] considered a generalization of Kummer -Schwarz differential equation for KS − 3, namely…”

“…That is, it's corresponding Riccati equation. In [9,20], was presented the group of Lie symmetries of (1), such symmetry group is (without explaining the details in their calculations)…”