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Leonhard Euler, born 15 April 1707 in Basel, Switzerland, died 18 Sept 1783 in St Petersburg, Russia, the father of applied mathematics, is shown here on a Swiss bank note. He had an extraordinary output in all areas of mathematics, publishing hundreds of journal articles even when he had become totally blind. After his death from a brain haemorrhage, the St Petersburg Academy continued to publish Euler’s unpublished work for nearly 50 more years. His impact in mathematics has been phenomenal.





Sample problem

Solve the following linear differential equation in the three-dimensional vector x: This has a formal solution of
where c is some constant vector. But how do we interpret the exponential quantity?
Fortunately the matrix A has the eigenvalue 1 with multiplicity three so (A - I)3 = 0 from the Cayley-Hamilton theorem (the happy pair below). Hence the exponential has a finite expansion:


Cayley and Hamilton





The Lorenz attractor

Discovered by Edward Lorenz in 1960. Using a simple system of equations to model convection in the atmosphere, Lorenz ran headlong into “sensitivity to initial conditions” and in the process he sketched out the beginnings of the first recognized strange attractor. Lorenz used a computer that could carry out only 60 multiplications per second.

MATH262 Ordinary Differential Equations

Second Semester, 18 points
Many phenomena in the physical world can be described by differential equations: either linear ones, with solutions that may have been worked out in the distant past, or nonlinear ones, which are still a hot topic today in studies of chaos. Interest in this was sparked in the 1960s, when a meteorologist called Edward Lorenz started to look at a simple model for the atmosphere (see sidebar). He found his model produced unexpected predictions, e.g. solution curves that were almost identical at their starting point, but moved far apart after a short time — a long way from linear behaviour. While Poincare had found something similar many years earlier, without computers he had been unable to investigate it in any detail. Lorenz’s discovery led to a resurgence of interest, and the beginning of the modern theory of chaos and dynamical systems. The subject is still very much alive, and interesting discoveries continue to be made, year by year.

Although MATH 262 cannot bring you to the point of making discoveries like these for yourself, it will give you some understanding and appreciation of the issues involved in them. And if you choose to go further, in time you will reach the frontier that has been forcing us to revise many of our ideas about the world.

Paper details

The principal focus of MATH 262 is to develop skills for solving differential equations. Initially, we have to learn about some of the classical ones that were discovered centuries ago, as these methods set the scene for the more interesting analyses that follow. We learn how to solve linear differential equations of various sorts, along with a few nonlinear types, and we apply algebraic methods developed in MATH 170 to the solution of linear systems of differential equations. The matrix exponential is introduced also, as an elegant way of handling linear systems (see sidebar).

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 will appeal to students majoring in more or less any science subject, plus engineering. There are also applications to the Health Sciences and Commerce, and selected topics will have relevance to the Humanities, for example to music. The paper has wide applicability because its purpose is to introduce students to a variety of classical and contemporary mathematical methods for solving the equations that arise when the real world is modelled and to create a toolbox of such methods. A method drawn from the toolbox may help find solutions to a massive economic model with hundreds of variables, it may predict the outcome of several chemicals interacting, it may decipher the basic components of a musical sequence, it may determine the modes of oscillation of a bridge, and much more.

MATH 262 is a core paper, and is the first step along a path that continues through 300 level with MATH 362, into the honours year where elective papers are taught that develop further techniques of similar but more advanced type, and into graduate study.

Main topics

  • First order differential equations.
  • Linear differential equations of higher order.
  • Laplace Transforms.
  • Systems of linear differential equations.
  • Stability.
  • An introduction to nonlinear systems and chaos.

Prerequisites

MATH170

Required text


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

Useful references

The following 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.
  • Blanchard, P., Devaney, R. L. and Hall, G. R., 2002, Differential Equations, 2nd ed., Brooks/Cole.

Lecturer

Dr Boris Baeumer, room 213

Lectures

Tuesdays, Thursdays and alternate Fridays at 11 a.m.
The Friday lectures are in weeks 1, 3, 5, ... of the semester. It is possible to take both MATH 262 and COMO 201 at the same time. Lectures will make use of clickers for feedback and to take attendance.

Internal Assessment

The final internal assessment mark (A) is based wholly on 10 exercises.

Tutorials

There is a two hour tutorial as well as a computer lab offered to support the course and help with the exercises.

Terms

You are required to do the starred questions in the exercises as well as being present during lectures.

Exam format

A three-hour exam with a variety of questions, all of which may be answered.

Final mark

The final mark F is calculated from:
F = max { E, (2E + A)/3 }
where E (exam mark) is out of 100, A (internal assessment) is out of 100.

The “max” corresponds to plussage: if your internal assessment mark is greater than your exam mark then it is combined in the proportion shown. If it is less then it is ignored and the exam mark itself is used.

It therefore pays to strive for a good internal assessment mark, to safeguard your overall assessment against unexpected disappointment in the final exam. This means steady and diligent work right throughout the course, which at the same time will strengthen your preparation for the final exam.

Plagiarism

Students should make sure that all submitted work is their own. “Plagiarism is a form of dishonest practice. Plagiarism is defined as copying or paraphrasing another’s work and presenting it as one’s own” (University of Otago Calendar). In practice this means that plagiarism includes any attempt in any piece of submitted work (e.g. an assignment or test) to present as one’s own work the work of another (whether of another student or a published authority). Any student found to be responsible for plagiarism in any piece of work submitted for assessment shall be subject to the University’s dishonest practice regulations which may result in various penalties, including forfeiture of marks for the piece of work submitted, a zero grade for the paper, or in extreme cases exclusion from the University. The University of Otago reserves the right to use plagiarism detection tools.

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.