常微分方程的解法

常微分方程的解法

常微分方程的解法

摘 要;本文主要讨论了1阶常微分方程和高阶常微分方程的相关解法问题．文章首先给出了微分方程的基本概念．在此基础上，探讨了1阶常微分方程的解法，讨论的主要类型有：变量可分离方程、可化为变量可分离方程的类型、齐次方程、1阶线性微分方程、恰当方程；在解决这些类型的1阶常微分方程时，用到的方法有：变量分离法、变换法、1阶线性方程的常数变易法以及恰当方程的直接观察法、分项组合法、积分对比法．最后讨论了高阶常微分方程的解法的问题，所讨论的解法有：非齐线性方程的常数变易法、常系数齐线性方程的欧拉待定指数法、非齐线性方程的比较系数法和拉普拉斯变换法、2阶常微分方程的幂级数法，最后还用变换法解决两个特殊的2阶常微分方程．
关键字:1阶常微分方程；高阶常微分方程；解法.

The solution of ordinary differential equations

Abstract: This paper mainly discusses some related solutions of the first-order and higher-order ordinary differential equations． This paper firstly introduces the basic concept of differential equations． on such a basis, the paper probes into the solutions of the first-order differential equations including the main types such as variable separable equation, separable variable equations which can be translated into the equation homogeneous equation, a linear differential equations and the proper equation． To solve such types of first-order differential equation, the methods can be used: variable separation, transformation, a linear equation of constant change of law and the appropriate equations direct observation, portfolio breakdown, integral contrast． Finally，it discusses the solutions of the higher-order differential equation． The solution are non-homogeneous linear equation of constant variation , Euler be determined index of constant coefficient of linear equations, nonhomogeneous linear equations comparison method and Laplace transform method． In addition, the method of transform is used to solve two special second ordinary differential equations．
Keywords: first-order ordinary differential equations; higher-order ordinary differential equations; Solution ．

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