常微分方程(组)的高次积分因子与高次积分及其微分特征列集算法

Higher Order Integral Factors and Integrals of Ordinary Differential Equation(s) and Their Determination by Characteristic Set Method

考虑了一般微分方程(组)高次积分和其微分特征列集(吴方法)机械化确定算法.首先提出微分方程的积分因子和首次积分的推广高次积分因子与其对应的高次积分的概念.其次给出了由高次积分因子确定其对应的高次积分的计算公式,使确定高次积分的问题转化为求高次积分因子的问题.再其次对确定高次积分因子的问题,给出了微分特征列集算法.最后用给定的算法确定了二阶和三阶微分方程拥有高次积分的结构定理,并给出了具体的算例和结论.

The higher order integrals of general ordinary differential equations （ODEs） and their determination by Characteristic set method （Wu＇s method） are considered. Firstly, the concepts of higher order integral factors and the corresponding higher order integrals of ODEs, which are generalization of the concepts of usual integral factors and first integral of ODEs, are put forward. Secondly, a direct computing formula for higher order integral in term of its higher order integral factor is given, which makes the question of determining the （higher order） integrals reduce to the question of determining the （higher order） integral factors. Thirdly, the differential form Characteristic set method is suggested to solve the over-determined system satisfied by the higher order integral factors. Finally, the construction theorems for second and third order ODEs which admit the higher order integrals and several concrete illustrative examples are given. The results obtained in this article lead to a systematically mechanical algorithm to determine （higher order） integrals for a given ODEs and new application of Wu＇s method in differential fields is suggested.