二阶变系数线性微分方程的一类通解  

General Solution of Second Order Linear Differential Equations with Variable Coefficients

作　　者：王慧[1] 叶永升[1] 

机构地区：[1]淮北师范大学数学科学学院,安徽淮北235000

出　　处：《淮北师范大学学报（自然科学版）》2017年第4期88-91,共4页Journal of Huaibei Normal University：Natural Sciences

基　　金：安徽省教育厅教育教学研究项目(2016jyxm0932);安徽省大规模在线开放课程(MOOC)示范项目(2015mooc053);安徽省大学数学系列课程教学团队(2015jxtd120);淮北师范大学教学研究重点项目(jy2017105)

摘　　要：文章利用待定函数法,把二阶变系数线性微分方程y″+p(x)y′+q(x)y=f(x)降为一阶线性微分方程,从而推导出二阶变系数线性微分方程的一类通解为y=(x+k)∫1/(x+k)~2e^(-∫p(x)dx) [∫(x+k)f(x)e^(∫p(x)dx)dx+C_1]dx+C_2(x+k),其中C_1,C_2为任意常数,k为常数,并证明该通解存在的充要条件是p(x)+(x+k)q(x)=0,同时还得出特殊情形的相应结果.In this paper,the second order linear differential equations with variable coefficients y″ + p(x)y′+ q(x)y = f(x) are reduced to the first order linear differential equations by using the undetermined function methods. Next,this paper derived out a sufficient and necessary condition p(x)+(x + k)q(x)= 0 for the second order linear differential equations with variable coefficients existing a general solution as the form y=(x+k)∫1/(x+k)~2e^(-∫p(x)dx) [∫(x+k)f(x)e^(∫p(x)dx)dx+C_1]dx+C_2(x+k) where C_1 and C_2 are arbitrary constants,and k is some constant.Moreover,some results of the special condi-tions for the linear equations are obtained.

关 键 词：变系数 线性 微分方程 通解 

分 类 号：O175.1[理学—数学]