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首页> 外文期刊>International Journal of Theoretical and Mathematical Physics >On the Superiority of the Method of Integration by Parts for Solving Exact and Semi-Exact Equations (With Physical Applications)
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On the Superiority of the Method of Integration by Parts for Solving Exact and Semi-Exact Equations (With Physical Applications)

机译：求解精确方程和半精确方程的零件积分方法的优越性（在物理应用中）

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Exact equations: two tests for exactness and three methods of solution : (a) In reviewing the present status of the topic of “exact equations”, one ought to evaluate the “test for exactness”, and the “methods of solution” currently applied in the literature, and investigate whether other more time-saving test and method can be introduced as alternatives. Below are three methods of which the first two are already in application, and the third is introduced here-under. The three methods will be contrasted in this work for determining which of them is the most economic one in terms of the space, time, and mental effort expended in the solution of the given equations. As will be shown later, an exact equation is at the same time an “equation with separable variables”, an “equation integrable in total differentials”, and an “equation integrable by parts”. The methods of solution of exact equations will then be: (1) solving “exact equations not tested for exactness” by “separation of the variables by substitution followed by term-by-term integration by the basic formulas”, (2) integration of exact equations in total differentials, (3) solving exact equations by integration by parts and by term-by-term integration in one and the same step. (b) The current test for exactness based on partial differentiation is performed by establishing the pairwise equality of the partial derivatives of the given equation and is confined to the second method of solution only. This test is: (1) totally overlooked in (and not required for) the first method of solution. Note that even with this somewhat lengthy test excluded from the solution, whether by omission or deliberately, the first method of solution will still remain comparatively prolonged because of the substitution and back substitution of the variables, (2) applied in the second rather lengthy method and contributes with it to the solution becoming more prolonged and laborious, (3) replaced in the third method with a much shorter and less laborious “test-by-sight” which involves no written steps or computations. As will be shown, this second test is performed at a glance by ascertaining the integrability of the given equation by parts. The solution is obtained by this newly introduced third method in only one, or two steps instead of half or full page of solution by the first two methods. It is actually these findings what constitute the subject-matter of this paper. (c) For the purposes of this work we quoted from some text-books six examples together with their prolonged solutions by the first two methods, and then reworked the examples by the third method for contrast. We stress that such quoting and contrasting constitute no “proving by selected instances ” but reflect various aspects of the topic of exact equations generally, with many other aspects still having to be revealed in future treatments. Brevity of solutions of problems means less space, time, and mental effort to be expended, and more simplicity, clarity, and accuracy to be achieved. Beside being a considerable abbreviation in solving exact equations, integration by parts is but a starting point in a series of similar abbreviations in solving problems of other related topics. These will be dealt with in our next papers in succession.
机译：精确方程：两种准确性检验和三种解决方法：（a）在回顾“精确方程”主题的现状时，应评估“准确性检验”，以及目前文献中使用的“解决方法”，并研究是否可以引入其他更省时的测试和方法作为替代方法。下面是三种方法，其中前两种已经在应用中，下面介绍第三种方法。在确定给定方程式所花费的时间，空间和精力方面，这三种方法将在本工作中进行对比，以确定哪种方法最经济。如稍后将显示的，精确的方程式同时是“具有可分离变量的方程”，“可在总微分中积分的方程”和“在各部分可积分的方程”。那么，精确方程的求解方法将是：（1）通过“用替换来分离变量，然后用基本公式进行逐项积分”来求解“未经精确性检验的精确方程”。 >（2）精确积分在总微分中的积分，（3）通过零件积分和逐项积分在同一步骤中求解精确方程。 （b）当前的基于偏微分的准确性检验是通过建立给定方程的偏导数的成对相等进行的，并且仅限于第二种求解方法。该测试是：（1）完全 在第一种解决方法中被忽略了（不是必需的）。请注意，即使从解决方案中排除了这个冗长的测试，无论是通过遗漏还是有意进行，由于变量（2） < b>在第二种相当冗长的方法中应用，并导致解决方案变得更长且更费力; （3） 在第三种方法中以更短，更省力的“ test-by”代替-sight”，不涉及任何书面步骤或计算。如将显示的那样，通过确定给定方程的可积分性，一目了然地进行了第二次测试。该解决方案是通过这种新引入的第三种方法而仅用一个或两个步骤而不是前两种方法的一半或整页解决方案来获得的。正是这些发现构成了本文的主题。（c）为了这项工作，我们从一些教科书中引用了六个示例以及前两种方法的长期解决方案，然后进行了重新设计例子通过第三种方法进行对比。我们强调，这样的 引用和对比并不构成“通过选定的实例证明”，而是反映了精确方程主题的各个方面，而在未来仍需揭示许多其他方面治疗。简短的问题解决方案意味着更少的空间，时间和精力可以用于工作，并且可以实现更多的简单性，清晰度和准确性。除了是求解精确方程式的相当大的缩写外，各部分的积分只是解决其他相关主题问题的一系列类似缩写中的一个起点。这些将在我们的下一篇论文中陆续处理。

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