解一阶线性微分方程组的代数消元法  

Algebraic elimination method for solving first order linear differential equations with constant coefficients

作　　者：刘玉忠[1] 曹淑怡 张军 LIU Yuzhong;CAO Shuyi;ZHANG Jun(College of Mathematics and Systems Science,Shenyang Normal University,Shenyang 110034,China;Shenyang No.134 Middle School,Shenyang 110000,China)

机构地区：[1]沈阳师范大学数学与系统科学学院,沈阳110034 [2]沈阳市第一三四中学,沈阳110000

出　　处：《沈阳师范大学学报（自然科学版）》2023年第4期346-349,共4页Journal of Shenyang Normal University:Natural Science Edition

基　　金：辽宁省教育厅科学研究经费项目(LJKZ01007)。

摘　　要：一般地,一阶线性常系数微分方程组通常采用待定系数的方法求解,其原理是利用矩阵的若当标准型理论将其转化为求解一系列方程,进而求得方程组的解。这种解法需要矩阵理论和线性子空间的直和等基本知识,相对来说较难理解。针对一阶线性常系数微分方程组,给出一种类似于代数方程的更易于理解的新的解法即代数消元法。通过建立方程组的n个未知函数满足的若干个代数方程(约束方程),把含有n个变元的一阶线性微分方程组化为含有r(r<n)个变元的一阶线性非齐次微分方程组,从而获得原方程组的解。特别地,当系数矩阵相似于对角矩阵时,可以得到传统方法的经典结论。文中举例说明了代数消元法的具体应用。Generally,first order linear differential equations are solved by the method of undetermined coefficients and the problem is changed into solving a series of matrix equations based on the theory of Jordan standard type.The theory of matrix and the direct sum of linear subspace are required in the solution,therefore,this method is above comprehension.However,in this paper a new method,algebraic elimination method(AEM),which is easier to understand is given for the first order linear differential equations,and the AEM is similar to the elimination method of system of linear algebraic equation with nth unknowns.In other words,by establishing a number of algebraic equations(constraint equations)about unknown functions,the considered equations have become the inhomogeneous equations with r(r<n)unknown functions and then the solution is given for the original equations.In particular,when the coefficient matrix is similar to the diagonal matrix,the classical conclusions of the traditional methods can be obtained.An example is given to illustrated the application of the new method.

关 键 词：一阶线性微分方程组 待定系数法 代数消元法 常数变易法 

分 类 号：O175[理学—数学]