求非线性常微分方程线性化变换的一种系统方法：I.标量的情况
A systematic method of finding linearizing transformations for nonlinear ordinary differential equations: I. Scalar case
摘要：
在这组文件中，我们制定了一个独立的方法来获得最大数量的线性化变换的任何阶的非线性常微分方程（ODE），包括耦合的知识，较少的运动积分。所提出的算法简单、直接、有效，并且除了文献中已知的线性化变换之外，还有助于发现几种新的线性化变换。为了使我们的研究系统化，我们将我们的分析分为两个部分。在第一部分中，我们将我们的研究局限于标量常微分方程，在第二部分中，我们将我们的注意力集中在两个耦合的二阶常微分方程组。在标量常微分方程的情况下，我们考虑了二阶和三阶非线性常微分方程的详细和讨论的方法，无论它是本地或非本地类型的线性化变换的最大数量的推导和说明的基础理论与适当的例子。作为研究的一个副产品，我们发现了一种新的三阶非线性常微分方程的线性化变换。最后将研究推广到一般标量常微分方程的情形。然后，我们继续研究两个耦合的二阶非线性常微分方程的下一部分，并表明该算法带来了各种各样的线性化变换。在每种情况下的线性化变换的最大数量的提取说明与适当的例子。
英文摘要：
In this set of papers we formulate a stand alone method to derive maximal number of linearizing transformations for nonlinear ordinary differential equations (ODEs) of any order including coupled ones from a knowledge of fewer number of integrals of motion. The proposed algorithm is simple, straightforward and efficient and helps to unearth several new types of linearizing transformations besides the known ones in the literature. To make our studies systematic we divide our analysis into two parts. In the first part we confine our investigations to the scalar ODEs and in the second part we focuss our attention on a system of two coupled second order ODEs. In the case of scalar ODEs, we consider second and third order nonlinear ODEs in detail and discuss the method of deriving maximal number of linearizing transformations irrespective of whether it is local or nonlocal type and illustrate the underlying theory with suitable examples. As a by-product of this investigation we unearth a new type of linearizing transformation in third order nonlinear ODEs. Finally the study is extended to the case of general scalar ODEs. We then move on to the study of two coupled second order nonlinear ODEs in the next part and show that the algorithm brings out a wide variety of linearization transformations. The extraction of maximal number of linearizing transformations in every case is illustrated with suitable examples.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
