37335 Differential Equations
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particular session, location and mode of offering is the authoritative source
of all information about the subject for that offering. Required texts, recommended texts and references in particular are likely to change. Students will be provided with a subject outline once they enrol in the subject.
Subject handbook information prior to 2024 is available in the Archives.
Credit points: 6 cp
Result type: Grade and marks
Requisite(s): (35102 Introduction to Analysis and Multivariable Calculus OR 33230 Mathematics 2 OR 33290 Statistics and Mathematics for Science OR 33401 Introductory Mathematical Methods OR 37132 Introduction to Mathematical Analysis and Modelling) AND (35212 Computational Linear Algebra OR 37233 Linear Algebra)
These requisites may not apply to students in certain courses. See access conditions.
Anti-requisite(s): 35231 Differential Equations
Description
Differential equations arise in contexts as diverse as the analysis and pricing of financial options, and the design of novel materials for telecommunications. In this subject students develop familiarity with the theory of differential equations, applications of this theory and some of the main computational techniques used in the solution of differential equations. Topics include existence and uniqueness of solutions; method of Frobenius; variation of parameters; the Taylor and Runge-Kutta methods for initial value problems; Fourier series; solving partial differential equations and boundary value problems by separation of variables, transform methods and finite difference methods.
Subject learning objectives (SLOs)
Upon successful completion of this subject students should be able to:
| 1. | demonstrate proficiency in finding solutions various types of differential equations or systems of differential equations and judge which method is applicable or most appropriate for finding the solution of a differential equation |
|---|---|
| 2. | demonstrate proficiency in the mathematical techniques which may be needed in solving differential equations. These include using variation of parameters, finding series solutions, separation of variables and Fourier series expansions and calculating and inverting Laplace transforms |
| 3. | understand and apply the various theoretical results which justify the use of the above skills |
| 4. | write an exposition on selected topics in the subject |
| 5. | summarise the main strategies of a given proof and conversely to construct a proof from verbal explanations of the methods |
| 6. | critically analyse and comment upon various stages in a given proof |
| 7. | model physical problems in terms of differential equations in other areas of mathematics and its applications |
| 8. | identify the relationship of differential equations to other areas of mathematics and its applications |
Course intended learning outcomes (CILOs)
This subject also contributes specifically to the development of following course intended learning outcomes:
- Demonstrate theoretical and technical knowledge of mathematical sciences including calculus, discrete mathematics, linear algebra, probability, statistics and quantitative management. (1.1)
- Evaluate mathematical and statistical approaches to problem solving, analysis, application, and critical thinking to make mathematical arguments, and conduct experiments based on analytical, numerical, statistical, algorithms to solve new problems. (2.1)
- Work autonomously or in teams to demonstrate professional and responsible analysis of real-life problems that require application of mathematics and statistics. (3.1)
- Design creative solutions to contemporary mathematical sciences-related issues by incorporating innovative methods, reflective practices and self-directed learning. (4.1)
- Use succinct and accurate presentation of reasoning and conclusions to communicate mathematical solutions, and their implications, to a variety of audiences, using a variety of approaches. (5.1)
Contribution to the development of graduate attributes
This subject is expected to contribute to the following graduate profile attributes:
1. Disciplinary knowledge
In this subject, you will learn use your knowledge of differential equations to solve real world problems in physics, engineering, finance and biology.
2. Research, Inquiry and Critical Thinking
There will be extensive opportunities for you to develop advanced problem solving and critical thinking skills throughout this subject. This will involve applying knowledge acquired during the lectures to solve unseen problems from a range of disciplines. This graduate attribute is embedded into all assessment tasks.
3. Professional, ethical and social responsibility
4. Reflection, Innovation, Creativity
5. Communication
Throughout this subject you will be asked to produce coherent answers to the problems presented to you. Your answers must display correct logic, appropriate structure and grammar. The ability to present your work in a clear, professional manner is key skill for all mathematicians.
Teaching and learning strategies
In this subject, you will attend lectures and tutorials. Lecture notes are available on Canvas and you are strongly encouraged to prepare for class by reading the materials and noting questions or areas you need more help with. During the lecture there will be considerable interaction between the lecturer and the students. You will have regular opportunity to answer questions in lectures and try simple calculations to help develop your understanding of the material.
In tutorials you will attempt harder problems with the assistance of your tutor to reinforce and broaden your understanding and capacity to apply the lecture materials. The tutorial session is a valuable opportunity for you to receive feedback on your progress starting in week 2. You are free to collaborate in class (and outside) with your peers to try and solve the problems on the tutorial sheets. There will also be assignments which will present aspects of the subject which depend on the material presented in lectures, but may require independent work to be fully mastered. Discussion of assignments between students is encouraged, though your final submission must be your own.
Again, feedback on the assignment will assist you to master the subject materials. Reading the recommended texts and doing exercises from the texts plays an essential part in the study of this subject. It is expected that you will spend 6 hours per week outside class in the study of the subject.
Content (topics)
- Theory of Ordinary Differential Equations, Wronskians, construction of second solutions from known solutions, series solutions, method of Frobenius, regular singular points, special functions-Bessel functions.
- Laplace Transform Methods
- Partial Differential Equations and Fourier Series
- Applications of partial differential equations to problems in science and finance.
Assessment
Assessment task 1: Assignment 1
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1. Disciplinary knowledge 5. Communication |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3 and 4 This assessment task contributes to the development of course intended learning outcome(s): 1.1 and 5.1 |
| Type: | Exercises |
| Groupwork: | Individual |
| Weight: | 10% |
| Length: | There will be five questions, each of which should take one to three pages to answer, depending upon difficulty and the level of analysis required. |
| Criteria: | Assessment will be based upon:
In summary, it must not only be correct, it must be logically and coherently presented, with steps naturally following, one from another. |
Assessment task 2: Assignment 2
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1. Disciplinary knowledge 2. Reseach, Inquiry and Critical Thinking. 3. Professional, ethical and social responsibility 5. Communication |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4, 5 and 6 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 3.1 and 5.1 |
| Type: | Exercises |
| Groupwork: | Individual |
| Weight: | 10% |
| Length: | There will be questions, each taking one to three pages to answer |
| Criteria: | Assessment will be based upon:
In summary, it must not only be correct, it must be logically and coherently presented, with steps naturally following, one from another. |
Assessment task 3: Class Test
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1. Disciplinary Knowledge 2. Research, inquiry and critical thinking 4. Reflection, Innovation, Creativity 5. Communication |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4, 5 and 6 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 4.1 and 5.1 |
| Type: | Mid-session examination |
| Groupwork: | Individual |
| Weight: | 30% |
| Length: | The test will take fifty minutes to complete. |
| Criteria: | Assessment will be based upon:
In summary, it must not only be correct, it must be logically and coherently presented, with steps naturally following, one from another. |
Assessment task 4: Final Exam
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1. Disciplinary Knowledge 2. Research, inquiry and critical thinking 4. Reflection, Innovation, Creativity 5. Communication |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4, 5, 6, 7 and 8 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 4.1 and 5.1 |
| Type: | Examination |
| Groupwork: | Individual |
| Weight: | 50% |
| Length: | The exam will be of two hours in length. |
| Criteria: | Assessment will be based upon:
Marks will be awarded for working, only if the working is correct and demonstrates genuine understanding of the material. Correct answers arrived at by sheer chance will not be awarded marks. |
Minimum requirements
A final overall mark of 50 Percent or more is required to pass the course. The final result is calculated as max(E,A), where E is the exam mark out of 100, and A=0.5E+C+B and C is the class test mark out of 30 and B is the total mark out of 20 for the two assignments.
Recommended texts
The following textbooks are useful, but not required. They are a good source of information and problems. They cover most of the material in the subject. If you wish to buy one, Boyce and DiPrima or Nagle, Saff and Snider would be good choices.
- Boyce W E, DiPrima R C, Elementary Differential Equations and Boundary Problems (10th Edn), Wiley, 2012.
- Edwards C H, Penney D E, Differential Equations and Boundary Value Problems, Computing and Modelling (4th Edn), Pearson Prentice Hall, 2008.
- Nagle R K, Saff E B, Snider A D, Fundamentals of Differential Equations and Boundary Value Problems (6th Edn), Pearson Addison-Wesley, 2012.
- Zill D G, Wright W G, Cullen M R, Differential Equations with Boundary Value Problems (8th Edn), Brooks/Cole Cengage, 2013.
- Zill D G, Cullen M R, Advanced Engineering Mathematics (3rd Edn), Jones and Bartlett, 2006.
The library has dozens of useful textbooks on this subject.
References
A set of printed lecture notes will be available online as each section is completed in lectures.