微分方程解次数的界

Bounds of the number of solutions of differential equations

针对微分方程解次数界的求解问题,提出了一种用于计算二维非退化多项式微分方程代数解的算法.算法首先基于Moulin Ollagnier证明,将指数写成具有非负整数系统的奇异叶面特征比的线性组合,获得度数为1和2的丢番图方程组,从而确定多项式1形式的界.然后对算法基于的定理进行证明,最后利用实例进行验证,说明所提算法的有效性和可行性.

Aiming at solving the number bound of solutions of differential equations,an algorithm for calculating the algebraic solution of two-dimensional non-degenerate polynomial differential equations is proposed.The algorithm firstly based on the Moulin Ollagnier proof,the index is written as a linear combination of singular leaf feature ratios with non-negative integer systems,and the Diophantine equations with degrees 1 and 2 are determined to determine the bounds of the polynomial 1 form.Then the theorem based on the algorithm is proved.Finally,an example is given to demonstrate the effectiveness and feasibility of the proposed algorithm.