Solution of the equivalence problem for the Painlevé IV equation

Abstract

Abstact. For an arbitrary ordinary second order differential equation a test is constructed that checks if this equation is equivalent to Painleve I, II or Painleve III with three zero parameters equations under the substitutions of variables. If it is true then in case the Painleve equations I and II an explicite change of variables is given that is written using the differential invariants of the equation.

Point equivalence of second-order ODEs: Maximal invariant classification order

“…In the latter case, the characterization obtained is a particular case of our more general result when g(x) = x. Other works devoted to the Painlevéve-I and II equations can be found in [1,6,14,15].…”

“Painlevé 34” equation: Equivalence test

“…3. I 1 = 18/5 in (14), I 9 = 0, I 21 = 0 in (17), invariant J = const in (19). Among the invariants I 3 , I 6 and I 9 from (14), (17) one can find two functionally independent.…”

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

“…The second way: note that for any equation (1) of the Type II.4 invariants J and J 6 are functionally independent. So, we can make the invariant point transformatioñ According to Theorem 1 any equation (1) of the Type III can be reduced by point transformations (2) into the canonical form: (17) y ′′ = y(ln y − 1) + t(x)y + s(x).…”

“…Invariant Theory of equation (1) goes back to the classical works of R.Liouville [19], S.Lie [18], A.Tresse [24], [23], E.Cartan [5], [22] (Late 19th-and Early 20th-Century) and continues in the works of [10], [16], [12], [4], [1], [7], [20], [21], [13] (Late 20th-Century). It remains an active research topic in the 21th-Century, see [2], [17], [14]. Background is adequately described in papers [1], [2].…”