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;本文主要討論了1階常微分方程和高階常微分方程的相關解法問題．文章首先給出了微分方程的基本概念．在此基礎上，探討了1階常微分方程的解法，討論的主要類型有：變量可分離方程、可化為變量可分離方程的類型、齊次方程、1階線性微分方程、恰當方程；在解決這些類型的1階常微分方程時，用到的方法有：變量分離法、變換法、1階線性方程的常數變易法以及恰當方程的直接觀察法、分項組合法、積分對比法．最后討論了高階常微分方程的解法的問題，所討論的解法有：非齊線性方程的常數變易法、常系數齊線性方程的歐拉待定指數法、非齊線性方程的比較系數法和拉普拉斯變換法、2階常微分方程的冪級數法，最后還用變換法解決兩個特殊的2階常微分方程．
關鍵字:1階常微分方程；高階常微分方程；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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