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审定:无
翻译:孙伟(简介并寄信)
编辑:陈盈(简介并寄信)


描述

本课程研究常微分方程(ODE's),包括物理系统建模。

学习主题包括:

  • 利用解析、图形和数值方法求一阶常微分方程的解;
  • 线性常微分方程,尤指常系数的二阶方程;
  • 不定系数和参变数;
  • 正弦和指数信号:振动、阻尼、共振;
  • 复数和幂;
  • 傅立叶级数、周期解;
  • Delta函数、卷积和拉普拉斯(拉氏)变换方法;
  • 矩阵和一阶线性系统:特征值和特征向量;
  • 非线性独立系统:临界点分析和相平面图。

必备并修和先修课程

18.02 or 18.022 or 18.023 或18.024 (corequisite), 18.0118.014 (prerequisite)

课程结构

课程采取“积极学习”的方式,通过讲课帮助你在理解、构建、解决和解释微分方程上达到专业水平。学生必须预习并积极参与讲课过程。第一次复习/实习课程时,所有学生会得到一套抽认卡,每节课都需要带上。(需要时堂上有额外抽认卡。)讲课中有时会引出问题,你将利用它们来宣布你的答案。如果有不同意见,将展开讨论。作为积极参与本课程的促进因素,你会经常在讲课结束的时候花少许时间回答一个简短的反馈问题。无论堂上人数多少,我都会倾听和回应反馈。

课本

Edwards, C. 与 D. Penney. 《含边界值问题的初等微分方程Elementary Differential Equations with Boundary Value Problems》第四版,Upper Saddle River, NJ: Prentice Hall, September 29, 1999. ISBN: 0130113018。

Polking. 《应用MATLAB®的常微分方程Ordinary Differential Equations using MATLAB®》第二版,Upper Saddle River, NJ: Prentice Hall, 1999年6月1日. ISBN: 0130113816。

学生还会收到两套笔记:Arthur Mattuck的“18.03: 微分方程”(18.03: Differential Equations)和我的"18.03补充笔记"。"

复习/实习课

这些小组每周开会两次,讨论并熟悉课堂内容。复习/实习课比讲课更需要你积极参与。请提前准备,复习/实习课导师可能以提问开始,因此知道问题的话请提前准备。他也许会在小组内分发习题交小组解答。及早并经常提出问题。复习/实习课导师也有办公时间,不要错过请教的机会。

辅导

另一个极有价值的资源就是辅导教室,由富有经验的大学生担任辅导。一小时测验前会有更多辅导人员。这是一个完成课外作业的好地方。

评分

期末成绩将由三个权重一致的部分组成:

  • 八个课外作业
  • 三次一小时测验
  • 一次期末考试

习题集方针声明

我们将舍弃得分最低的一个习题集分数,然后把其余习题集的分数加起来作为习题集总成绩。一共有8组习题集,因此习题集成绩是由7组分数组成。。

我会尽可能准确给出本学期这门课程的预期目标。你应计划真正掌握一些附件(PDF) 说明的基本技能。这是麻省理工学院开设的技能课程,你必须掌握必备先修课程18.03,讲授接下来这些课程的教员都很清楚这份清单。

课外作业

作业每个周末提交。每次课外作业都由两部分组成:一部分来自书本或笔记,还有一部分是分发的习题。两部分都紧扣讲课内容,你应养成在相关课程间完成相应作业的习惯,不要在截止日期前熬夜完成整个习题集。复习/实习课导师会在下一次复习/实习课时给出你上一次的成绩。

我鼓励在课程中相互协作,但必须坚持要诚实。如果你以小组形式完成作业,务必保证你从中获益而不是相反。你未经思考而取得作业好分数,会导致考试的低分。你必须提交独立完成的所有习题;如果你是和人合作的,请在答题纸上列出合作者的名字。由于答案会在习题集截止后立即公布,因此作业不允许延期提交。

一小时测验

学期中有三个星期五的课堂安排一小时测验,考试的教室将在课上通知。如果你因故错过考试,请提前联系大学生数学办公室另行安排补考。补考只在特殊情况才批准,例如生病、家庭紧急事故等。

期末考试

在期末考试期间有一次三小时的综合考试,考试时间和地点将另行通知。

常微分方程操作软件(“Mathlets”)

我们会使用一系列称作“Mathlets”的特编计算机玩具或示例软件。你有时候会在堂上见到使用,每个习题集包含一个或多个使用Mathlets的习题。(注意:开放式课程用户目前不能使用Mathlets)


 

Description

This course is a study of Ordinary Differential Equations (ODE's), including modeling physical systems.

Topics include:

  • Solution of first-order ODE's by analytical, graphical and numerical methods;
  • Linear ODE's, especially second order with constant coefficients;
  • Undetermined coefficients and variation of parameters;
  • Sinusoidal and exponential signals: oscillations, damping, resonance;
  • Complex numbers and exponentials;
  • Fourier series, periodic solutions;
  • Delta functions, convolution, and Laplace transform methods;
  • Matrix and first order linear systems: eigenvalues and eigenvectors; and
  • Non-linear autonomous systems: critical point analysis and phase plane diagrams.

Corequisites/Prerequisites

18.02 or 18.022 or 18.023 or 18.024 (corequisite), 18.01 or 18.014 (prerequisite)

Format

These lectures will follow an "active learning" approach. The lecture period will be used to help you gain expertise in understanding, constructing, solving, and interpreting differential equations. You must come to lecture prepared to participate actively. At the first recitation you will be given a set of flashcards. Bring these with you to each lecture. (Extras will be available in lecture in case of need.) You will use them to announce your answer to questions posed occasionally in the lecture. In case of divided opinions a discussion will follow. As a further element of your active participation in this class, you will often be asked to spend a minute responding to a short feedback question at the end of the lecture. Despite the large size of this class, I will listen and respond to this feedback.

Texts

Edwards, C., and D. Penney. Elementary Differential Equations with Boundary Value Problems. 4th ed. Upper Saddle River, NJ: Prentice Hall, September 29, 1999. ISBN: 0130113018.

Polking. Ordinary Differential Equations using MATLAB®. 2nd ed. Upper Saddle River, NJ: Prentice Hall, June 1, 1999. ISBN: 0130113816.

Students will also receive two sets of notes "18.03: Differential Equations" by Arthur Mattuck, and my "18.03 Supplementary Notes."

Recitations

These small groups will meet twice a week to discuss and gain experience with the course material. Even more than the lectures, the recitations will involve your active participation. Come prepared. Your recitation leader may begin by asking for questions, so be ready if you have them. He may then hand out problems for you to work on in small groups. Ask questions early and often. Your recitation leader will also hold office hours, a resource you should not overlook.

Tutoring

Another resource of great value is the tutoring room. This is staffed by experienced undergraduates. Extra staff is added before hour exams. This is a good place to go to work on homework.

Grading

The final grade will be based on three components of the course, which will be given equal weight:

  • Eight Homework Assignments
  • Three Hour Exams
  • One Final Exam

Problem Set Policy Statement

We will drop the problem set with the lowest per-problem average score, and multiply up the remaining problem sets to give a total problem set score. There will be 8 problem sets, therefore, seven of the eight will constitute your PS grade component.

I will try to be very precise about what I expect you to learn in the course of this semester. You should plan to achieve a real mastery of a few Essential Skills (PDF), which are spelled out in the attached document. These are the skills courses at MIT with 18.03 as a prerequisite will expect you to have down cold, and the faculty teaching these next courses are aware of this list.

Homework

Assignments will be due at the end of each week. Each homework assignment will have two parts: a first part drawn from the book or notes, and a second part consisting of problems which will be handed out. Both parts will be keyed closely to the lectures, and you should form the habit of doing the relevant problems between successive lectures and not try to do the whole set on the night before they are due. Your recitation leader should have the graded problems sets available for you at the next recitation.

I encourage collaboration in this course, but I insist on honesty about it. If you do your homework in a group, be sure it works to your advantage rather than against you. Good grades for homework you have not thought through will translate to poor grades on exams. You must turn in your own writeups of all problems, and, if you do collaborate, please write on your solution sheet the names of the students you worked with. Because the solutions will be available immediately after the problem sets are due, no extensions will be possible.

Hour Exams

Hour exams will be held during the lecture hour on three Fridays during the term. Examination rooms will be announced in lectures. If you must miss an exam, contact the Undergraduate Mathematics Office before the exam to arrange for a make-up which can be granted under certain limited circumstances such as illness or family emergency.

Final Exam

There will be a three hour comprehensive examination, during the Final Exam Period, at a time and place to be announced.

ODE Manipulatives ("Mathlets")

We will employ a series of specially written computer toys, or demonstrations, called "Mathlets." You will see them used in lecture occasionally, and each problem set will contain a problem based around one or another of them. (Note: Mathlets not currently available to OCW users.)


 
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