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The American Mathematical Monthly ( IF 0.5 ) Pub Date : 2020-07-28 , DOI: 10.1080/00029890.2020.1764825
Meredith L. Greer 1
Affiliation  

The teaching and learning of differential equations present both vast opportunities and an intensive amount of decision making. How much emphasis should one place on theory versus on application? From which fields should examples be drawn? The landscape continues to change, and resources are crucial for both instructors and students. This review considers the new book 500 Examples and Problems of Applied Differential Equations by Ravi Agarwal, Simona Hodis, and Donal O’Regan in these contexts and in relationship to other resources for studying differential equations. Just a few decades ago, first courses in differential equations followed a fairly standard path, consisting primarily of analytic solutions for well-studied types of equations and systems. Course structures usually began with first-order differential equations before moving to second-order or higher-order equations. They would then cover systems of differential equations, with separation of variables, integrating factors, power series, Laplace transforms, and other traditional closed-form solution approaches each emphasized in turn. While this type of pathway through differential equations continues to exist, the past quarter century or so has seen a growing number of instructors shifting their focus. One reason is that computing power has changed what is possible. A second, and crucial, reason is that the real-world need for differential equations far exceeds the relatively few specific equations and systems that can be analytically solved via classical methods. As a result, many courses today emphasize applications, modeling, visualization, and graphical and numerical approaches to gleaning valuable information from differential equations [2]. Such curricular changes are strongly recommended by researchers studying undergraduate mathematics education [3, 6] . Textbooks for introductory courses in differential equations have adopted these curricular updates in a variety of ways. A few remain nearly unchanged from the texts of thirty years ago. Most retain a similar structure of mathematical topics while incorporating more applications. A few pursue modeling and visualization much more fully, pointedly jettisoning most analytic techniques to make room in the curriculum for computer-centered approaches to applied questions. In all cases, instructors and students would do well to seek out additional applications of differential equations. This is a field with connections to, well, just about every aspect of our lives, not to mention a large array of student majors. A short list includes epidemiology, climate change, technology, economics, pharmacology, and engineering. We could go on and on, starting at mathematical history and not stopping before looking at systems of differential equations that represent romantic relationships. Indeed, the study of differential equations illustrates beautifully the power of mathematics to address seemingly disparate real-world phenomena using techniques that are, at their core, identical or similar.

中文翻译:

评论

微分方程的教学和学习提供了大量的机会和大量的决策。一个人应该把多少重点放在理论与应用上?应该从哪些领域抽取例子?环境在不断变化,资源对教师和学生都至关重要。本评论考虑了 Ravi Agarwal、Simona Hodis 和 Donal O'Regan 在这些背景下以及与用于研究微分方程的其他资源的关系的新书 500 个应用微分方程的例子和问题。就在几十年前,微分方程的第一门课程遵循了相当标准的路径,主要包括对经过充分研究的方程和系统类型的解析解。课程结构通常从一阶微分方程开始,然后转向二阶或更高阶方程。然后,他们将涵盖微分方程系统,并依次强调变量分离、积分因子、幂级数、拉普拉斯变换和其他传统的闭式求解方法。虽然这种通过微分方程的途径仍然存在,但在过去的 25 年左右,越来越多的教师转移了他们的注意力。原因之一是计算能力改变了一切可能。第二个也是至关重要的原因是,现实世界对微分方程的需求远远超过了可以通过经典方法解析求解的相对较少的特定方程和系统。因此,今天的许多课程都强调应用程序、建模、可视化,以及从微分方程中收集有价值信息的图形和数值方法 [2]。研究本科数学教育的研究人员强烈推荐这种课程变化[3, 6]。微分方程入门课程的教科书以多种方式采用了这些课程更新。一些与三十年前的文本几乎没有变化。大多数保留了相似的数学主题结构,同时融入了更多的应用程序。一些人更全面地追求建模和可视化,有针对性地放弃了大多数分析技术,以便在课程中为以计算机为中心的应用问题方法腾出空间。在所有情况下,教师和学生最好寻找微分方程的其他应用。这是一个与我们生活的几乎每个方面都有联系的领域,更不用说大量的学生专业了。短名单包括流行病学、气候变化、技术、经济学、药理学和工程学。我们可以继续下去,从数学历史开始,在研究代表浪漫关系的微分方程系统之前不会停下来。事实上,对微分方程的研究精美地说明了数学使用本质上相同或相似的技术来解决看似不同的现实世界现象的力量。从数学史开始,在研究代表浪漫关系的微分方程系统之前不会停止。事实上,对微分方程的研究精美地说明了数学使用本质上相同或相似的技术来解决看似不同的现实世界现象的力量。从数学史开始,在研究代表浪漫关系的微分方程系统之前不会停止。事实上,对微分方程的研究很好地说明了数学使用本质上相同或相似的技术来解决看似不同的现实世界现象的力量。
更新日期:2020-07-28
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