The characteristic function method of finding first integrals of ODEs

The problem to prove that the system of a differential equation is “ordered” or chaotic is equally difficult and important. In general, the ordinary differential equation (ODE) which is integrable implies that its solutions are ordered, which excludes the occurrence of chaos. In this paper, the integrability of ODEs of polynomial type is studied. A effective method, namely, the characteristic function method of finding their first integrals is proposed. The method is easily carried out through computers. In addition, its properties are discussed. As applications, the first integrals of some usual systems are given.