Equivalence Classes of the Second Order Odes with the Constant Cartan Invariant

Abstract

Second order ordinary differential equations that possesses the constant invariant are investigated. Four basic types of these equations were found. For every type the complete list of nonequivalent equations is issued. As the examples the equivalence problem for the Painleve II equation, Painleve III equation with three zero parameters, Emden equations and for some other equations is solved.

Point classification of second order ODEs and its application to Painlevé equations

“…This classification is more general than all previous ones. The relation between the (pseudo)invariants from works [20,21] and the semiinvariants from works [4,14] (as they were presented in [2]) was shown in paper [11] and here in Section 7. Moreover, in all possible cases the set of the invariants can be broadened.…”

“…In this case N = 0 in (13), Ω = 0 in (10), (11). The pseudovectorial fields ω, ω 1 = ω 2 , ω 2 = −ω 1 from (22), (23) and α from (5) are non-collinear,…”

“…Here Λ is from (20), (21), Ω is from (10), (11), N is from (13), L is from (35), E is from (28): In the second case of intermediate degeneration the algebra of the point symmetries of equation ( 1) is 1-dimensional if and only if all invariants I 1 , I 2 , I 3 are identically constant. In other cases it is trivial.…”

“…In this case N = 0 in ( 13), M = 0 in ( 14), (15), Ω = 0 in ( 10), (11), Λ = 0 in ( 20), ( 21), K = −5/9 in (24), (25). Consider again the vectorial field ω, ω 1 = ω 2 , ω 2 = −ω 1 , from ( 22), (23).…”

“…Moreover, in all possible cases the set of the invariants can be broadened. By employing this technique, in [10], [11] and [12] the equivalence problem for some equations was solved.…”