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在常微分方程中,标准的Clairaut方程只要用常数c代替方程中的y′就可以得到其通解形式,并且必有参数形式的奇解。本文将标准的Clairaut方程进行推广,给出它们的通解和奇解(如果存在的话)的表达式,所推导的命题是文献中某些结论的深化与拓广。通过Maple编程,得到推广此类方程的一种便捷的求解方法,并且在实例方程的源程序设计中增加了奇解和通解的图像,形象生动地展示了方程奇解的几何意义。
Abstract：In the ordinary differential equation,the general solution of the standard Clairaut equation can be got by substituting differential with constant in the equation,and there must be a singular solution of the parameter form. In this paper,the standard Clairaut equation is promoted,and the expression of formula of their general solution and singular solution(if any) is given. The proposition deduced is the deepening and broadening of some conclusions in the literature. Through the Maple programming,the promotion of a convenient solution of this generalized equation is obtained and at the same time the programming of the image of the singular solution and general solution appeared in the examples. The image vividly displayed the geometric meaning of singular integral equation.
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基本信息:
DOI：10.19573/j.issn2095-0926.201801011
中图分类号:O175.1
引用信息:
[1]吕晓静,赵向东.Maple在求解推广的Clairaut型方程中的应用[J].天津职业技术师范大学学报,2018,28(01):47-51.DOI:10.19573/j.issn2095-0926.201801011.
基金信息:
国家自然科学基金资助项目(11601391);; 天津职业技术师范大学出国研修资助项目(J10011060318);; 天津职业技术师范大学教学改革与质量建设研究重点项目(JGZ2015-08)