Related papers: Linearisable third order ordinary differential equ…

We prove a necessary and sufficient condition for the existence of a $T$-periodic solution for the time-periodic second order differential equation $\ddot{x}+f(t,x)+p(t,x,\dot x)=0$, where $f$ grows superlinearly in $x$ uniformly in time,…

Let $k$ be a differential field of characteristic zero with an algebraically closed field of constants. In this article, we provide a classification of first order differential equations over $k$ and study the algebraic dependence of…

Some interesting (periodic!) solutions of certain systems of $4$ nonlinear Ordinary Differential Equations $dx_{n}\left( t\right) /dt=P_{2}^{\left( n\right) }\left[ x_{m}\left( t\right) \right] /\left[ x_{1}\left( t\right) +x_{2}\left(…

The problem of linearization by point transformations is solved for equations in the generalized Riccati and Abel chain of order not exceeding the fourth. It is shown in particular that nonlinear third order and fourth order equations from…

For $\Gamma_1$-structures on 3-manifolds, we give a very simple proof of Thurston's regularization theorem, first proved in \cite{thurston}, without using Mather's homology equivalence. Moreover, in the co-orientable case, the resulting…

The aim of this article is to show that systems of linear partial differential equations on filtered manifolds, which are of weighted finite type, can be canonically rewritten as first order systems of a certain type. This leads immediately…

An ordinary differential equation is said to have a superposition formula if its general solution can be expressed as a function of a finite number of particular solution. Nonlinear ODE's with superposition formulas include matrix Riccati…

In this PhD thesis we introduce a generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives, and study them using standard (indirect) and direct methods. In…

New third- and fourth-order Lagrangian hierarchies are derived in this paper. The free coefficients in the leading terms satisfy the most general differential geometric criteria currently known for the existence of a variational…

In this paper we study the general group classification of systems of linear second-order ordinary differential equations inspired from earlier works and recent results on the group classification of such systems. Some interesting results…

This paper provides necessary and sufficient conditions of optimality for variational problems that deal with a fractional derivative with respect to another function. Fractional Euler--Lagrange equations are established for the fundamental…

The normality equations for the Newtonian dynamical systems on an arbitrary Riemannian manifold of the dimension $n \geq 3$ are considered. Locally the solution of such equations reduces to three possible cases: in two of them the solution…

The paper presents a new method for finding first integrals of ordinary difference equations which do not possess Lagrangians, nor Hamiltonians. As an example we solve a third order nonlinear ordinary differential equation and its invariant…

We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…

We give a classification of all third-order nonlinear evolution equations which admit solvable Lie symmetry algebras $\mathsf{A}$ and which are not linearized. We have found that there are 48 types of equations for $\dim\mathsf{A}=3$, 88…

In this paper we discuss the first order partial differential equations resolved with any derivatives. At first, we transform the first order partial differential equation resolved with respect to a time derivative into a system of linear…

In the first part of this paper we linearize and solve the Van der Pol and Lienard equations with some additional nonlinear terms by the application of a generalized form of Cole-Hopf transformation. We then show that the same…

In this work, we consider a generalization of the nonlinear Langevin equation of fractional orders with boundary value conditions. The existence and uniqueness of solutions are studied by using results of the fixed point theory. Moreover,…

The purpose of this paper is to make a few connections among specific concepts occurring in differential geometry and the theory of differential equations with the aim of identifying an intriguing class of undetermined nonlinear ordinary…

First, we consider generalized wave and scattering operators and derive modifications of commutation relations (between scattering operators and unperturbed operators) when the corresponding deviation factors behave as $\exp\{i t {\mathcal…