摘要：常微分方程中只有一些典型方程能求出初等解(用初等函数表示的解)，大部分的方程是求不出初等解的。另外，有些初值问题虽然有初等解，但由于形式太复杂不便于应用。因此，有必要探讨常微分方程初值问题的数值解法。本文主要给出一阶常微分方程初值问题数值解法的几种导出途径，按照这些途径, 导出了求解微分方程初值问题的几种常见方法, 如: 欧拉法、改进欧拉法、梯形法、龙格库塔法、 阿当姆斯公式和另一种欧拉改进法，由此达到理顺并建立各种数值方法之间的内在联系。

关键词：微分方程；初值问题；数值解。

Abstract:Ordinary differential equation, only some typical equations can work out elementary solutions (with elementary function of the solution of the said), most of the equation is to ask for a solution of the primary. In addition, some of the initial problems though having elementary solution, but because the form is too complicated to application. Therefore, it is necessary to analyze differential equation of the initial value problems the numerical method. This paper mainly presents a order ordinary differential equation of numerical methods for solution of initial value problems of several export way, according to these pathways are deduced, solving differential equations of the initial value problems of common methods, such as: euler method, and improvement of the euler method, trapezoidal method, LongGe coulthard method, Adam, formula and the other's a euler improvement method, and thus to establish various numerical method to straighten out and the inner relationship between. 

Keywords: Differential equations; Initial value problem; The numerical solution.