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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 exampes the…

The notions of equivalence and strict equivalence for order one differential equations are introduced. The more explicit notion of strict equivalence is applied to examples and questions concerning autonomous equations and equations having…

Starting from the second Painlev\'{e} equation, we obtain Painlev\'{e} type equations of higher order by using the singular point analysis.

It is demonstrated that a certain integral equation can be solved using the Painleve equation of third kind. Inversely, a special solution of this Painleve equation can be expressed as the ratio of two infinite series of spheroidal…

Polynomials related to rational solutions of Painleve' equations satisfy certain difference equations. Conditions are given to acertain that all solutions really are polynomials.

In this paper some open problems for Painlev\'e equations are discussed. In particular the following open problems are described: (i) the Painlev\'e equivalence problem; (ii) notation for solutions of the Painlev\'e equations; (iii)…

After a brief introduction to the Painlev\'{e} property for ordinary differential equations, we present a concise review of the various methods of singularity analysis which are commonly referred to as Painlev\'{e} tests. The tests are…

The problem of Painleve classification of ordinary differential equations lasting since the end of XIX century saw significant advances for the limited equation order, however not that much for the equations of higher orders. In this work…

In this paper we \emph{explicitly} compute the transformation that maps the generic second order differential equation $y''= f(x, y, y')$ to the Painlev\'e first equation $y''=6y^2+x$ (resp. the Painlev\'e second equation ${y''=2 y^{3}+yx+…

Painleve equations belong to the class y'' + a_1 {y'}^3 + 3 a_2 {y'}^2 + 3 a_3 y' + a_4 = 0, where a_i=a_i(x,y). This class of equations is invariant under the general point transformation x=Phi(X,Y), y=Psi(X,Y) and it is therefore very…

The automation of the traditional Painleve test in Mathematica is discussed. The package PainleveTest.m allows for the testing of polynomial systems of ordinary and partial differential equations which may be parameterized by arbitrary…

We prove that under a very general setting, a system of ODE passes the Painleve test if and only if there is a good change of variable, such that the pole singularity solutions are converted to regular power series, while the converted ODE…

We present a brief overview of integrability of nonlinear ordinary and partial differential equations with a focus on the Painleve property: an ODE of second order has the Painleve property if the only movable singularities connected to…

The last decades saw growing interest across multiple disciplines in nonlinear phenomena described by partial differential equations (PDE). Integrability of such equations is tightly related with the Painleve property - solutions being free…

The method of parameter variation for linear differential equations is extended to classes of second order nonlinear differential equations. This allows to reduce the latter to first order differential equations. Known classical equations…

In in this paper we show how using D.A. it is found a simple change of variables (c.v.) that brings us to obtain differential equations simpler than the original one. In a pedagogical way (at least we try to do that) and in order to make…

The aim of this article is to provide a method to prove the irreducibility of non-linear ordinary differential equations by means of the differential Galois group of their variational equations along algebraic solutions. We show that if the…

Rational solutions for a $q$-difference analogue of the Painlev\'e III equation are considered. A Determinant formula of Jacobi-Trudi type for the solutions is constructed.

The Painlev\'e--Kovalevskaya test is applied to find three matrix versions of the Painlev\'e II equation. All these equations are interpreted as group-invariant reductions of integrable matrix evolution equations, which makes it possible to…

A new problem is studied, the concept of exactness of a second order nonlinear ordinary differential equations is established. A method is constructed to reduce this class into a first order equations. If the second order equation is not…