We consider the equations of the form dy/dx = y²-P(x) where P are polynomials. We characterize the possible algebraic solutions and the class of equations having such solutions. We present formulas for first integrals of rational Riccati equations with an algebraic solution. We also present a relation between the problem of algebraic solutions and the theory of random matrices.

In this paper we provide the greatest lower bound about the number of (non-infinitesimal) limit cycles surrounding a unique singular point for a planar polynomial differential system of arbitrary degree.

We consider the second order self-adjoint differential equation (1) (r(t)y’(t))’ + p(t)y(t) = 0 on an interval I, where r, p are continuous functions and r is positive on I. The aim of this paper is to show one possibility to write equation (1) in the same form but with positive coefficients, say r₁, p₁ and to derive a sufficient condition for equation (1) to be oscillatory in the case p is nonnegative and ${\int }^{\infty }[1/r\left(t\right)]dt$ converges.

In this paper, sharp a priori estimate of the periodic solutions is obtained for the discrete analogue of the continuous time ratio-dependent predator-prey system, which is governed by nonautonomous difference equations, modelling the dynamics of the $n-1$ competing preys and one predator having nonoverlapping generations. Based on more precise a priori estimate and the continuation theorem of the coincidence degree, an easily verifiable sufficient criterion of the existence of positive periodic solutions...

In this paper, we employ some new techniques to study the existence of positive periodic solution of $n$-species neutral delay system $$N_i^{\text{'}}\left(t\right)=N_i\left(t\right)\left[a_i\left(t\right)-\sum _{j=1}^n{\beta }_{ij}\left(t\right)N_j\left(t\right)-\sum _{j=1}^n{b}_{ij}\left(t\right)N_j(t-{\tau }_{ij}\left(t\right))-\sum _{j=1}^n{c}_{ij}\left(t\right)N_j^{\text{'}}(t-{\tau }_{ij}\left(t\right))\right].$$ As a corollary, we answer an open problem proposed by Y. Kuang.

We study solutions tending to nonzero constants for the third order differential equation with the damping term $${\left(a_1\left(t\right){\left(a_2\left(t\right)x^{\text{'}}\left(t\right)\right)}^{\text{'}}\right)}^{\text{'}}+q\left(t\right)x^{\text{'}}\left(t\right)+r\left(t\right)f\left(x\left(\varphi \left(t\right)\right)\right)=0$$ in the case when the corresponding second order differential equation is oscillatory.

In this paper we consider positive unbounded solutions of second order quasilinear ordinary differential equations. Our objective is to determine the asymptotic forms of unbounded solutions. An application to exterior Dirichlet problems is also given.

Sufficient conditions are given which guarantee that the linear transformation converting a given linear Hamiltonian system into another system of the same form transforms principal (antiprincipal) solutions into principal (antiprincipal) solutions.