## COMO204 Differential Equations

Second Semester |

### Notices

The Sophia Michelle McMillan Crestani Memorial Scholarship is open for applications for 2nd year female students taking a Mathematics major or minor in 2021. To learn more about the scholarship please use the link:

https://www.otago.ac.nz/study/scholarships/database/search/otago747688.html

Closing date for applications is 23rd March 2021.

### Course Information

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. At the completion of the paper 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 details

The principal focus of COMO 204 is to develop mathematical skills for working 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 straight-forward solutions can be obtained. We discuss issues that arise when modelling with differential equations, and introduce powerful tools for analysing differential equations using computers.

In real situations it may be necessary to model sudden changes, e.g. when a switch is turned on or off, or a wind suddenly blows, or a guitar string is plucked; we learn how to do this using the Laplace transform, which conveniently turns a differential equation into an algebraic one that can easily be solved.

### Potential students

This paper is strongly recommended for all mathematics and physics students. Differential equations appear in diverse fields such as commerce, engineering and 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.

### Prerequisites

MATH170

COMO101 is recommended but not required

### Main topics

- First order differential equations
- Linear differential equations of higher order
- Systems of linear differential equations
- Analytical and numerical solutions to differential equations
- Stability
- Introduction to nonlinear systems and chaos
- Non-homogeneous linear ODEs
- Laplace transforms

### Course text (recommended)

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 also recommend the following freely available textbook:

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

### Further reading

The following texts are also useful references

- Edwards, C.H. & David E. Penney D.E., 2007,
*Differential Equations and Boundary Value Problems: Computing and Modelling*, 4th ed., Prentice Hall. - Boyce, W.E. and DiPrima, R.C., 2001,
*Elementary Differential Equations and Boundary Value Problems*, 7th ed., John Wiley and Sons.

### Lecturer

Dr. Fabien Montiel, Room 514

### Lectures

Mondays, Wednesdays and alternate Fridays, 1-2 pm

### Computer Labs

**Two hour long computer labs on Tuesdays or Thursday, 3-5 pm.**

You are expected to attend the computer lab session allocated to you and completion of computer lab exercises contributes towards internal assessment.

- Matlab resources

Matlab software is available on Mathematics and Statistics Computer Labs and also through Student Desktop.

See the following links for introduction to Matlab as well as resources:

Matlab Language Primer Booklet

### Internal Assessment

The internal assessment mark is made up of the following three components (A, L, T), see the formula for Final mark (F) below.

- Assignments

There are four marked assignments which make up your Assignment mark (A) which in turn is worth 15% of your Final mark (F).

- Computer Labs

The ten computer lab exercises make up your computer Lab mark (L) which in turn is worth 15% of your Final mark (F).

- Midterm Test

The midterm test makes up your Test mark (T) which in turn is worth 15% of your Final mark (F).

### 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 55% of your Final mark (F).

### Final mark

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

**F = max(0.55E + 0.15A + 0.15T + 0.15L, 0.7E + 0.15A + 0.15L)**

where:

- E is the Exam mark
- A is the Assignments mark
- T is the Tests mark
- L is the Labs 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.