Mathematics
Te Tari Pāngarau me te Tatauranga
Department of Mathematics & Statistics

## COMO204 Differential Equations

 Second Semester
18 points

### Overview

This course is an introduction to the theory and applications of ordinary differential equations (ODEs) which are fundamental tools used in modelling and solving problems in applied mathematics, economics, engineering, physical sciences and life sciences. The principal focus of COMO 204 is to develop mathematical skills for working analytically and numerically with differential equations.

We study techniques which can be applied to obtain analytical solutions of differential equations, as well as tools for working with equations for which no exact solutions can be obtained using pen and paper. We discuss issues that arise when modelling with differential equations, and introduce powerful tools for analysing differential equations using graphical and numerical approach. We also explore how to deal with systems that respond to external forcing.

COMO204 is a standard 2nd year ODE course and offers the foundational bases to go on to more advanced topics, e.g. partial differential equations, dynamical systems and control theory.

### Eligibility

This paper is strongly recommended for all mathematics and physics students. Differential equations appear in diverse fields such as commerce, engineering and the natural sciences. This paper is fundamental for any work in applied mathematics or computational modelling. Simply put this paper will be useful for anyone who wants to work with models or processes that involve changes over time.

MATH170 or MATH140 is a pre-requisite. COMO101 is recommended but not required (it can also be taken concurrently).

Fabien Montiel

### Contact

If you have any enquiries, please use the Discussion Boards on Blackboard or email Fabien at: fmontiel@maths.otago.ac.nz

### Learning Outcomes

Upon completion of COMO204 students will:

• Understand how to classify differential equations and how they arise in applications;
• Have a working knowledge of analytical techniques and theorems used to study and solve first and second order differential equations and systems of linear differential equations;
• Be familiar with numerical techniques used to solve differential equations, their strengths as well as their limitations;
• Understand techniques and theory for the qualitative analysis of differential equations, and their importance;
• Improve skills in mathematical writing and report preparation.

### Paper Structure

The paper is structured around 5 topics:

1. First order ODEs

2. Systems of linear first order ODEs

3. Nonlinear systems of first order ODEs

4. Linear second-order ODEs with sinusoidal forcing

5. The Laplace Transform method

### Course Content

A full breakdown of the course content is provided in Blackboard under "Slides and resources".

### Course text

The material presented in this course is based on the two following textbooks:

• Blanchard, P., Devaney, R. L. and Hall, G. R., 2012, Differential Equations, 4th ed., Brooks/Cole.
• Brannan, J.R and Boyce, W.E., 2011, Differential Equations: An Introduction to Modern Methods and Applications, 2e, Wiley.

These are excellent, but very expensive texts. Copies are available in the library.

I recommend the following freely available textbook, which covers most of the COMO204 topics:

• Logan, J.D., 2011, A First Course in Differential Equations, Springer (link to download).

### Teaching arrangements

COMO204 is a standard 13-week semester paper, with five 1-hour lectures every fortnight and a weekly 2-hour laboratory. The lectures will be on Tuesdays, Thursdays and alternate Fridays, 1-2 pm.

### Assessment

The paper contains the following forms of assessment with corresponding weighting towards your final grade:

• Four handwritten assignments (15%)
• Ten Computer lab activities (15%)
• Ten weekly quizzes (10%)
• A midterm test (10% plussage)
• A final exam (50% or 60%)

### Assignments

There will be 4 handwritten assignments. To submit your assignment, scan it and submit it electronically via Blackboard. The dates for the assignments are:

• Assignment 1 due on Friday 29 July 2022
• Assignment 2 due on Friday 19 August 2022
• Assignment 3 due on Friday 16 September 2022
• Assignment 4 due on Friday 7 October 2022

Your assignment mark (A) is worth 15% of your Final mark (F).

### Lab activities

Two hour long computer labs on Tuesdays or Thursday, 3-5 pm. (check your stream on your timetable).

Both attendance and completion of the lab activities contributes towards your lab mark. The lab activities will help you develop your computing skills as you will work with computing software MATLAB. If you are not familiar with MATLAB, you are asked to complete the MATLAB OnRamp course (link to couse) to learn the basics of programming with MATLAB. Your lab mark (L) is worth 15% of your Final mark (F).

### Quizzes

We will have 10 online quizzes posted at the end of each week, which must be completed by end of the following week. The top 80% of your quiz grades will be used to determine your quiz mark (Q) which in turn is worth 10% of your Final mark (F).

### What do I do if I need more time?

If you need more time to complete an assessment, contact the lecturer. You may be eligible for an exemption depending on the reason for the delay.

### Midterm Test

The midterm test will be held on Thursday 25 August at 1pm during the lecture slot. It makes up your Test mark (T) which in turn is worth 10% of your Final mark (F). Plussage applies.

### Terms

There are no terms requirement for this course.

### Exam format

A three-hour exam makes up your Exam mark (E) which in turn is worth 50% or 60% of your Final mark (F), depending on your midterm test mark.

### Final mark

Your final mark F in the paper will be calculated according to this formula:

F = max(0.5E + 0.15A + 0.1T + 0.15L + 0.1Q, 0.6E + 0.15A + 0.15L + 0.1Q)

where:

• E is the Exam mark
• A is the Assignments mark
• T is the Tests mark
• L is the Labs mark
• Q is the Quizzes mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

Academic integrity means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

Academic misconduct is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

#### Plagiarism

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

• Although not intended, unintentional plagiarism is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided.
• Intentional plagiarism is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

#### Unauthorised Collaboration

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

#### Impersonation

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

#### Falsiﬁcation

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

#### Use of Unauthorised Materials

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

#### Assisting Others to Commit Academic Misconduct

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.

Further information

While we strive to keep details as accurate and up-to-date as possible, information given here should be regarded as provisional. Individual lecturers will confirm teaching and assessment methods.