计算一类常微分方程特解的新方法  

New Method of Finding Special Solution of a Kind of Ordinary Differential Equations

作　　者：龚东山[1] 刘岳巍[1] 贾筱景[2] 

机构地区：[1]兰州大学数学与统计学院,甘肃兰州730000 [2]甘肃政法学院计算机科学学院,甘肃兰州730070

出　　处：《河北北方学院学报（自然科学版）》2008年第6期1-3,共3页Journal of Hebei North University:Natural Science Edition

基　　金：国家自然科学基金对外交流与合作项目(40640420072);中国科学院西部之光联合学者项目(2006)

摘　　要：目的计算高阶常微分方程特解的方法有待定系数法、常数变易法、拉普拉斯变换法、积分法等,它们的计算工作量一般较大,为弥补上述方法的不足,有必要探究另一种简便实用的新方法——特征函数法.方法先定义该类高阶常微分方程的对应齐次方程的特征函数,再利用特征函数的导数,可得到非齐次项为特殊函数情形时方程的一个特解.结果只需求出特征方程的根,就可得到该类高阶常微分方程的一个特解.结论利用特征函数法可以得到一类常微分方程的一个特解,该方法使用简单,所得特解形式直观.Objective Many computations have to be done to find special solutions of higher-order ordinary differential equations with current methods, such as undetermined coefficient, variation of constant, Laplace transform, integral method, etc. So it is necessary to find a simple and practical method --eigenfunction method--to compensate for the above. Methods In the paper, we define the eigenfunction of corresponding homogenous equation of the higher-order ordinary differential equations and differen- tiate it, then a special solution is attained when the non-homogenous expression is a special function. Resuits This shows that a special solution of this kind of higher-order ordinary differential equations will be attained by the roots of eigenfunction. Conclusion The eigenfunction method can help us find a special solution of a kind of ordinary differential equations and its merits are its simple use and intuitionistie ex- pression.

关 键 词：高阶 微分方程 特征函数 非齐次 特解 

分 类 号：O175.1[理学—数学]