## COMO204 Differential Equations

Second Semester |

### Notices

- There are no computer labs in the first week of semester. They will start up in week 2.

### 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
- Laplace transforms
- Stability
- Introduction to nonlinear systems and chaos

### Course text (recommended)

The material presented in this course closely follows

- Blanchard, P., Devaney, R. L. and Hall, G. R., 2012,
*Differential Equations*, 4th ed., Brooks/Cole.

This is an excellent, but very expensive text. Copies are available in the library.

### Useful references

The following texts are recommended:

- 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. - Brannan, J.R and Boyce, W.E., 2011,
*Differential Equations: An Introduction to Modern Methods and Applications*, 2e, Wiley.

### Lecturer

Assoc. Prof. Boris Baeumer, Room 213, Department of Mathematics and Statistics

### Lectures

Mondays, Wednesdays and alternate Fridays, 1-2 pm

### Computer Labs

**Two hour long computer labs on Mondays, 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

### Tutorials

One hour long tutorial sessions on Tuesdays, 3-4 pm to assist you with your assignments.

### 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 with a variety of questions, all of which may be answered. The 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 endeavours at the University of Otago are built upon an essential commitment to academic integrity.**

The two most common forms of academic misconduct are *plagiarism* and *unauthorised collaboration*.

#### Academic misconduct: **Plagiarism**

Plagiarism is defined as:

- Copying or paraphrasing another person’s work and presenting it as your own.
- Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
- Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
- Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.

All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.

#### Academic misconduct: **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 student’s answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer.