# COURSE OUTLINE: MH3110

Course Title

Course Code

### MH3110

Offered Study Year 3, Semester 2
Course Coordinator Fedor Duzhin (Dr) fduzhin@ntu.edu.sg 6513 7469
Pre-requisites MH2100, CY1602
AU 4
Contact hours Tutorials: 39, Technology-enhanced Learning: 13, Lectures: 2
Approved for delivery from AY 2018/19 semester 2
Last revised 12 Feb 2019, 15:16

### Course Aims

Building on Calculus, Linear Algebra, and Algorithms and Computing course sequences, this course aims to equip you with technical knowledge of the theory of ordinary differential questions and communication skills required to work as a team on modelling a real life phenomenon with a system of ordinary differential equations.

### Intended Learning Outcomes

Upon successfully completing this course, you should be able to:

1. Solve certain classes of ordinary differential equations (separable, first order linear, second order linear with constant coefficients, linear systems with constant coefficients).
2. Apply basics of qualitative theory of ordinary differential equations, e.g., interpret phase portraits, derive existence and uniqueness of solutions, analyse stability of critical points of linear and nonlinear systems of differential equations without solving them.
3. Present mathematical ideas the form of a report written in LaTeX at the appropriate level for the intended audience.
4. Work in a team, i.e., effectively communicate on distribution of roles and work tasks and help other students with their tasks.
5. Learn independently from research papers and online tutorials.
6. Model a real life phenomenon with a system of ordinary differential equations.

### Course Content

First order ordinary differential equations, integrating factors, separable equations, linear homogeneous and non-homogeneous equations, variation of parameters, Principle of superposition

Second order linear differential equations, Wronskian, Abel's formula, variation of parameters.

First order linear systems, Wronskian, systems with constant coefficients, types of critical points at the origin, qualitative analysis of phase portraits.

First order nonlinear systems, initial value problem. Use of systems of ODE in modeling problems.

### Assessment

Component Course ILOs tested SPMS-MAS Graduate Attributes tested Weighting Team / Individual Assessment Rubrics
Continuous Assessment
Technology-enhanced Learning
Activities 1, 2, 3, 4 1. a, b, c, d
20 both See Appendix for rubric
Tutorials
Project 2, 3, 4, 5, 6 1. a, b, c, d
2. a
3. a, b
40 both See Appendix for rubric
Mid-semester Quiz
Short Answer Questions 1 1, 2 1. b, c, d
20 individual See Appendix for rubric
Short Answer Questions 2 1, 2 1. b, c, d
20 individual See Appendix for rubric
Total 100%

These are the relevant SPMS-MAS Graduate Attributes.

### 1. Competence

a. Independently process and interpret mathematical theories and methodologies, and apply them to solve problems

b. Formulate mathematical statements precisely using rigorous mathematical language

c. Discover patterns by abstraction from examples

d. Use computer technology to solve problems, and to communicate mathematical ideas

### 2. Creativity

a. Critically assess the applicability of mathematical tools in the workplace

### 3. Communication

a. Present mathematics ideas logically and coherently at the appropriate level for the intended audience

b. Work in teams on complicated projects that require applications of mathematics, and communicate the results verbally and in written form

### Formative Feedback

There will be 5 team application exercises for homework every week. You are supposed to type solutions in Overleaf and submit to the tutor. The tutor will immediately grade your work and provide feedback that you can use to improve and re-submit. You will have unlimited attempts. Note that this is a team activity and while one student will actually type a solution, the rest should verify it, help to phrase it properly, and read the tutor's comments.

• LaTeX tutorials: https://www.sharelatex.com/blog/latex-guides/beginners-tutorial.html

• Sample Homework with feedback: https://www.overleaf.com/read/ttcfptkdksvg

### Learning and Teaching Approach

 Tutorials (39 hours) Note that the actual number of tutorials hours is 36. Below are learning and teaching approaches under tutorials: 1) Team-based learning Team-based learning serves two purposes: a) It forces students to watch online lectures before coming to class because material is tested – ILO1 and ILO2. b) It helps students to know their teammates better and to build a working relationship with them – ILO4. 2) Online homework Online homework with unlimited attempts enables intructors’ feedback to help students to learn how to type in LaTeX and how to explain mathematics – ILO3. 3) Peer evaluation evaluation serves the following purposes that are linked to ILO4: a) Giving students more motivation for teamwork because in the end of the course they will not only get a letter grade that doesn’t say much about their skills and strengths, but a set or specific peer reviews. b) Ensuring that final grades are psychometrically reliable c) Discouraging free riding 4) Team project This is the main feature of the course. Students are supposed to work together as a team on a project that is too big for one person to complete within one team. Hence students will have to communicate and to distribute tasks between them (ILO4). The topic of the project is modelling a real-life phenomenon with a system of differential equations (ILO6). To do it, students will have to read research papers (ILO5) and produce a report (ILO3). Technology-enhanced Learning (13 hours) 5) Online lectures In lectures, we just explain material to students, i.e., cover course content (ILO1 and ILO2). The fact that lectures are online rather than face-to-face enables team-based learning in classroom, which is eventually linked to ILO4. 6) Online forums Since there are no lectures in class, online forums provide opportunity for students to interact with the lecturer and to clarify their doubts on course material. This helps to achieve ILO1 and ILO2. Lectures (2 hours) Note that the actual number of lecture hours is 2 (one introductory lecture in the first week). Below are learning and teaching approaches that qualify as lectures:

Textbook: Boyce, William E., Richard C. DiPrima, and Douglas B. Meade. Elementary differential equations and boundary value problems. Vol. 9. New York: Wiley, 1992.
ISBN 0-471-31999-6

References:

Amel’kin, Vladimir Vasil’evič. Differential equations in applications. Mir, 1990.
ISBN 5-03-005521-8

Zhang, Wei-Bin. Differential equations, bifurcations, and chaos in economics. Vol. 68. World Scientific Publishing Company, 2005.
ISBN 981-256-333-4

Shone, Ronald. Economic Dynamics: Phase diagrams and their economic application. Cambridge University Press, 2002.
ISBN-13 978-0-511-07676-3

Brock, William A., and Anastasios G. Malliaris. Differential equations, stability and chaos in dynamic economics. No. 90A16 BROd. 1989.
ISBN 0-444-70500-7

Cresswell, Clio. Mathematics and sex. Allen & Unwin, 2003.
ISBN 1 74114 159 1

Gottman, John Mordechai. The mathematics of marriage: Dynamic nonlinear models. MIT Press, 2005.
ISBN 0-262-07226-2

Clark, Colin W. The worldwide crisis in fisheries: economic models and human behavior. Cambridge University Press, 2006.
ISBN-13 978-0-511-33516-7

Plus a number of journal papers and student reports from previous years.

### Course Policies and Student Responsibilities

(1) General

You are expected to complete all assigned pre-class readings and activities, attend all seminar classes punctually and take all scheduled assignments and tests by due dates. You are expected to take responsibility to follow up with course notes, assignments and course related announcements for seminar sessions they have missed. You are expected to participate in all discussions and activities.

(2) Absenteeism

TBL requires you to be in class to contribute to team work. In-class activities make up a significant portion of your course grade. Absence from class without a valid reason will affect your overall course grade. Valid reasons include falling sick supported by a medical certificate and participation in NTU’s approved activities supported by an excuse letter from the relevant bodies. There will be no make-up opportunities for in-class activities.

If you miss a seminar session, you must inform your team members and the course instructor via email (see course info on NTULearn) prior to the start of the class. If you miss I-RAT or T-RAT without a valid reason, you will earn nothing for that session of absence. For I-RAT and T-RAT scores, we will consider the average of all the sessions that you were present in. This method will take care of instances where you miss classes with valid reasons.

If you miss a midterm test, a make-up test will be arranged for you.

If you miss your team’s project presentation without a valid reason, your teammates will be in their right to reflect this fact in peer evaluation. If you miss the project presentation because of a valid reason, you must warn your teammates and the course instructor via email. In that case, no penalty will be imposed on you.

(3) Online Homework

You are encouraged to submit online homework typed in LaTeX. You have unlimited attempts and each time you submit a solution, your instructor will give you immediate feedback on logic, phrasing, and typesetting of your solution. The score will not be a part of official assessment but we will give “model answers” only to teams who have completely solved at last 3 out of 5 questions. Besides, teams with higher homework scores will get feedback on the draft of their final report earlier and hence will have more time to update their final report according to feedback.

Good academic work depends on honesty and ethical behaviour. The quality of your work as a student relies on adhering to the principles of academic integrity and to the NTU Honour Code, a set of values shared by the whole university community. Truth, Trust and Justice are at the core of NTU’s shared values.

### Course Instructors

Instructor Office Location Phone Email
Fedor Duzhin (Dr) SPMS-MAS-05-23 6513 7469 fduzhin@ntu.edu.sg

### Planned Weekly Schedule

Week Topic Course ILO Readings/ Activities
1

Concept of ODE. Direction fields.

2

Introductory lecture, textbook chapter 1

2

Explicitly solvable 1st order ODE: separable and linear equations.

1, 2, 3, 4, 5

TBL on lecture 1, textbook sections 2.1, 2.2, research (browsing the library)

3

Modelling with 1st order ODE

1, 2, 3, 4, 5

TBL on lecture 2, textbook sections 2.3, 2.5, research (browsing the library)

4

Existence and uniqueness Theorem.

1, 2, 3, 4, 5

TBL on lecture 3, textbook sections 2.4, 2.7, 2.8, research (browsing the library)

5

2nd order linear ODE

1, 2, 3, 4, 5

TBL on lecture 4, textbook sections 3.2, 3.3 research (browsing the library), midterm test 1

6

2nd order linear ODE: explicit solution

1, 2, 3, 4, 5, 6

TBL on lecture 5, textbook sections 3.1, 3.4, 3.5 research (deciding the topic)

7

Modelling with 2nd order ODE

1, 2, 3, 4, 5, 6

TBL on lecture 6, textbook sections 3.7, 3.8, 3.9 research (data collection)

8

Higher order linear ODE and Laplace transform - just basics.

1, 2, 3, 4, 5, 6

TBL on lecture 7, textbook sections 4.1, 4.2, 6.1, 6.2, modelling

9

Systems of 1st order ODE

1, 2, 3, 4

TBL on lecture 8, textbook sections 7.1, 7.4, 7.5, midterm test 2

10

Systems of 1st order ODE

1, 2, 3, 4, 5, 6

TBL on lecture 9, textbook sections 7.6, 9.1, computer simulations

11

Nonlinear equations and stability

1, 2, 3, 4, 5, 6

TBL on lecture 10, textbook sections 7.7, 9.2, 9.3, 9.4, 9.5, fitting the mode into data

### Appendix 1: Assessment Rubrics

#### Rubric for Technology-enhanced Learning: Activities (20%)

This is team-based learning. It is assessed by multiple choice questions.

#### Rubric for Mid-semester Quiz: Short Answer Questions 1 (20%)

This is a traditional math exam consisting of a mixture of multiple choice and mini-essay questions. It is assessed by a marking scheme rather than a rubric.

Below is an example of a question and a marking scheme:

Question (15 points). Solve the following initial value problem:

$\frac{dy}{dt}=y^2\cdot\cos t,\quad y(0)=1$

Marking scheme: 5 points for separating the variables and integrating, 5 points for finding y from the resulting expression, and 5 points for finding the constant and solving the initial value problem.

#### Rubric for Mid-semester Quiz: Short Answer Questions 2 (20%)

This is a traditional math exam consisting of a mixture of multiple choice and mini-essay questions. It is assessed by a marking scheme rather than a rubric.

Below is an example of a question and a marking scheme:

Question (15 points). Can the function

$y=\sin\left(t^2 \right )$

be a solution of an initial value problem

$y''+p(t)y'+q(t)y=0,\quad y'(0)=y(0)=0$

with coefficients continuous on some interval containing 0?

Marking scheme: 15 points for a fully correct answer, 10 for a correct explanation with missing details, and 5 for mentioning a relevant concept.