## MATHS 7102 - Differential Equations

### North Terrace Campus - Semester 1 - 2014

Most "real life" systems that are described mathematically, be they physical, biological, financial or economic, are described by means of differential equations. Our ability to predict the way in which these systems evolve or behave is determined by our ability to model these systems and find solutions of the equations explicitly or approximately. Every application and differential equation presents its own challenges, but there are various classes of differential equations, and for some of these there are established approaches and methods for solving them. Topics covered are: first order ordinary differential equations (ODEs), higher order ODEs, systems of ODEs, series solutions of ODEs, interpretation of solutions, Fourier analysis and solution of linear partial differential equations using the method of separation of variables.

• General Course Information
##### Course Details
Course Code MATHS 7102 Differential Equations School of Mathematical Sciences Semester 1 Postgraduate Coursework North Terrace Campus 3 Up to 3.5 hours per week MATHS 1012 MATHS 2201 Most "real life" systems that are described mathematically, be they physical, biological, financial or economic, are described by means of differential equations. Our ability to predict the way in which these systems evolve or behave is determined by our ability to model these systems and find solutions of the equations explicitly or approximately. Every application and differential equation presents its own challenges, but there are various classes of differential equations, and for some of these there are established approaches and methods for solving them. Topics covered are: first order ordinary differential equations (ODEs), higher order ODEs, systems of ODEs, series solutions of ODEs, interpretation of solutions, Fourier analysis and solution of linear partial differential equations using the method of separation of variables.
##### Course Staff

Course Coordinator: Professor Anthony Roberts

##### Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
##### Course Learning Outcomes
On successful completion of this course students will be able to:
1. understand that physical systems can be described by differential equations
2. understand the practical importance of solving differential equations
3. understand the differences between initial value and boundary value problems (IVPs and BVPs)
4. appreciate the importance of establishing the existence and uniqueness of solutions
5. recognise an appropriate solution method for a given problem
6. classify differential equations
7. analytically solve a wide range of ordinary differential equations  (ODEs)
8. obtain approximate solutions of ODEs using graphical and  numerical techniques
9. use Fourier analysis in differential equation solution  methods
10. solve classical linear partial differential equations (PDEs)
11. solve differential equations using computer software

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute Course Learning Outcome(s)
Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised. all
The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. all
An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. all
Skills of a high order in interpersonal understanding, teamwork and communication. all
A proficiency in the appropriate use of contemporary technologies. all
A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. all
An awareness of ethical, social and cultural issues within a global context and their importance in the exercise of professional skills and responsibilities. all
• Learning Resources
##### Recommended Resources
Kreyszig, E. (2011), Advanced engineering mathematics, 10th edn, Wiley.
##### Online Learning
This course uses MyUni exclusively for providing electronic resources, such as lecture notes, assignment papers, and sample  solutions.  Students should make appropriate use of these resources.  Link to MyUni login page:  https://myuni.adelaide.edu.au/webapps/login/
• Learning & Teaching Activities
##### Learning & Teaching Modes
This course relies on lectures as the primary delivery mechanism for the material.  Tutorials supplement the lectures by providing exercises and example problems to enhance the understanding obtained through lectures.  A sequence of written assignments provides the assessment opportunities for students to gauge their progress and understanding.

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

Lectures 36 90
Tutorials 6 24
Assignments 6 42
Total 156
##### Learning Activities Summary
The course will explore and develop the following.
1. Basic definitions; Physical examples; Classification of types ODEs
2. Basic definitions; IVPs; 1st order ODEs; Separable, Linear, Exact
3. Graphical and numerical methods; directional fields and Eulers method
4. Existence and Uniqueness for 1st order ODEs; Picard's Method and Theorem
5. Existence and Uniqueness of IVPs for n-th order linear ODEs; Wronskian test
6. n-th order homogenous linear constant coefficient ODEs
7. Reduction of order
8. Non-homogenous n-th order linear constant coeffs; Method of undetermined coefficients
9. Variation of parameters
10. Modelling and interpretation
11. Linear ODEs with variable coefficients; Euler-Cauchy equation
12. Power Series, via computer algebra
13. Legendre equation and polynomials
14. Frobenius series solution and Bessels equation
15. Frobenius series solution---classification of solutions.
16. Systems ODES; modelling, eigenvalues and eigenvectors
17. Systems ODES; algebraic and geometric multiplicity
18. Periodic and odd/even functions; Generalised Fourier series
19. Piecewise continous functions
20. Fourier sine, cosine and complex Fourier series
21. Fourier Integral and Transform
22. Introduction to PDEs; modelling conservation of material
23. Wave Equation and D'Alemberts solution; car traffic; shocks
24. Separation of variables; Wave, Heat, Laplace equation
25. Vibrating Drum; Fourier Bessel series; interpretation
26. Temperature field in a sphere; Fourier Legendre Series; interpretation
• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

##### Assessment Summary

ComponentWeightingObjective assessed
Assignments 30% all
Exam 70% all
##### Assessment Related Requirements
An aggregate score of at least 50% is required to pass the course.
##### Assessment Detail

Assessment itemDistributedDue dateWeighting
Continuous assessment TBA TBA 10%
Assignment 1 week 2 week 3 4%
Assignment 2 week 4 week 5 4%
Assignment 3 week 6 week 7 4%
Assignment 4 week 8 week 9 4%
Assignment 5 week 10 week 11 4%
##### Submission

1. All written assignments are to be submitted to the designated hand-in boxes in the School of Mathematical Sciences with a signed cover sheet attached, or submitted as pdf via MyUni.
2. Late assignments will not be accepted without a medical certificate.
3. Assignments normally have a two week turn-around time for feedback to students.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M10 (Coursework Mark Scheme)
FNS   Fail No Submission
F 1-49 Fail
P 50-64 Pass
C 65-74 Credit
D 75-84 Distinction
HD 85-100 High Distinction
CN   Continuing
NFE   No Formal Examination
RP   Result Pending

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through Access Adelaide.

• Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
• Policies & Guidelines
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