First-Order Ordinary Differential Equations with Several Bounded Separate Solutions

Bounded Nonoscillatory Solutions for First Order Neutral Delay Differential Equations

This paper deals with the first order neutral delay differential equation (x(t) + a(t)x(t− τ))′ + p(t)f(x(t− α)) +q(t)g(x(t − β)) = 0, t ≥ t0, Using the Banach fixed point theorem, we show the existence of a bounded nonoscillatory positive solution for the equation. Three nontrivial examples are given to illustrate our results. Mathematics Subject Classification: 34K4

Order Ordinary Differential Equations

The DI methods for directly solving a system ofa general higher order ODEs are discussed. The convergence of the constant stepsize and constant order formulation of the DI methods is proven first before the convergencefor the variable order and stepsize case.

Existence of solutions for first order ordinary differential equations with nonlinear boundary conditions

We present an existence theorem for nonlinear ordinary differential equations of first order with nonlinear boundary conditions. The result includes, for instance, the initial value problem, the final value problem, and the antiperiodic boundary value problem. The main novelty of the method lies in that it unifies different techniques for initial or boundary conditions.

On the Number of Zeros of Bounded Nonoscillatory Solutions to Higher-order Nonlinear Ordinary Differential Equations

Rational solutions of first-order differential equations∗

We prove that degrees of rational solutions of an algebraic differential equation F (dw/dz,w, z) = 0 are bounded. For given F an upper bound for degrees can be determined explicitly. This implies that one can find all rational solutions by solving algebraic equations. Consider the differential equation F (w′, w, z) = 0, (w′ = dw/dz) (1) where F is a polynomial in three variables. Theorem 1 For ...