﻿ 二阶变系数线性微分方程化为标准型的求解 Solution of Two Order Variable Coefficient Linear Differential Equation into Standard Form

Vol.05 No.01(2016), Article ID:17012,11 pages
10.12677/AAM.2016.51013

Solution of Two Order Variable Coefficient Linear Differential Equation into Standard Form

Xiong Chen1,2, Shiyou Lin1*, Haohan Zhang1

1School of Mathematics and Statistics, Hainan Normal University, Haikou Hainan

2Yilong County No.2 Middle School, Hainan Normal University, Sichuan Nanchong

Received: Feb. 3rd, 2016; accepted: Feb. 19th, 2016; published: Feb. 26th, 2016

ABSTRACT

This paper discusses the solution of the two order variable coefficient linear differential equation with standard type, which transforms the traditional method of reducing order. Through simplifying the original differential equation and using means of cofunction and particular integral, we can get the homogeneous and non-homogeneous solution of the standard type. Finally we can construct the general solution of the original equation.

Keywords:Two Order Variable Coefficient, Linear Differential Equation, Standard Type, Cofunction, Particular Integral, General Solution

1海南师范大学数学与统计学院，海南 海口

2仪陇县第二中学，四川 南充

1. 引言

2. 标准型

(1-1)

(1-2)

(1-3)

(1-4)

(1-5)

(1-6)

(1-7)

(1-8)

(1-9)

(1-10)

(1-11)

(1-12)

(1-13)

(1-14)

(1-15)

(1-16)

(1-17)

(1-18)

(1-19)

(1-20)

3. 标准型的求解

1) 余函数的求解

(2-1)

(2-2)

(2-3)

2) n阶微分方程特积分的求解

(2-4)

(2-5)

(2-6)

(2-7)

(2-8)

(2-9)

(2-10)

(2-11)

(2-12)

(2-13)

(2-14)

(2-15)

(2-16)

(2-17)

(2-18)

(2-19)

(2-20)

(2-21)

(2-22)

(2-23)

(2-24)

3) 二阶微分方程特积分

(2-25)

(2-26)

(2-27)

(2-28)

(2-29)

(2-30)

(2-31)

(2-32)

(2-33)

(2-34)

(2-35)

(2-36)

4. 例题

Solution of Two Order Variable Coefficient Linear Differential Equation into Standard Form[J]. 应用数学进展, 2016, 05(01): 87-97. http://dx.doi.org/10.12677/AAM.2016.51013

1. 1. 姜嵛芃. 二阶变系数线性常微分方程解法研究[J]. 金融理论与教学, 2012(4): 90-91.

2. 2. 方辉平, 叶鸣. 二阶变系数齐线性常微分方程的求解[J]. 重庆工商大学学报(自然科学版), 2011, 28(1): 14-17.

3. 3. 卢亦平, 钱椿林. 二阶常系数线性微分方程的降阶法[J]. 苏州市职业大学学报, 2014, 25(3): 49-52.

4. 4. [美]G.F.塞蒙斯, 著. 微分方程[M]. 张理京, 译. 北京: 人民教育出版社, 1981.

5. 5. 国振喜. 工程微分方程[M]. 北京: 机械工业出版社, 2004.

6. 6. 王高雄, 周之铭. 常微分方程[M]. 第3版. 北京: 高等教育出版社, 2012.

*通讯作者。