Analysis of Ordinary Differential Equations (MATH2010)

Course overview

ODE's - Systems: variation of constants, fundamental matrix. Laplace transform, transform for systems, transfer function. Stability, asymptotic stability; phase plane analysis.

MATH2010 will cover the following 2 topics.ﾠ

• Topic 1. Systems of Ordinary Differential Equations (Kreyszig Chapter 4). Solutions to Homogeneous Linear Systems of ODE's with constant coeffcients using matrices. The Phase Plane for 2-Dimensional linear and some nonlinear systems. Critical Points, linearizationﾠand the stability properties of critical points. Nonhomogeneous Linear Systems.
• Topic 2. Laplace Transforms (Kreyszig Chapter 6).ﾠThe Laplace Transform and its Inverse Transform. Linearity, Shifting, Convolution and the use of Partial Fractions. Transforms of Derivatives and Integrals. Use of Laplace Transforms to solve Linear Differential Equations including those involving discontinuous functions such as the Step Function and the Dirac Delta function.

An understanding of these concepts is important for anyone who wants to apply mathematics to the real world. Differential equations are one ofﾠthe most widely used tools in mathematical modelling. They are used extensively in theoretical physics, mathematical biology and ecology, human movement studies, and all branches of engineering. In particular, they inevitably arise when we try to model anything that moves continuously in space or varies continuously in time. Students with interests in the applications of mathematics to the real world, or with the development of the tools needed for such applications, will need a good understanding of the basic ideas introduced here. This includes those interested in research careers in theoretical physics, engineering, applied mathematics, pure mathematics (analysis), and numerical analysis.

In the School of Mathematics and Physics we are committed to creating an inclusive and empowering learning environment for all students. We value and respect the diverse range of experiences our students bring to their education, and we believe that this diversity is crucial for fostering a rich culture of knowledge sharing and meaningful exploration. We hold both students and staff accountable for actively contributing to the establishment of a respectful and supportive learning environment.

Bullying, harassment, and discrimination in any form are strictly against our principles and againstﾠUQ Policy,ﾠand will not be tolerated. We have developed aﾠsuite of resourcesﾠto assist you in recognising, reporting, and addressing such behaviour. If you have any concerns about your experience in this course, we encourage you to tell a member of the course teaching team, or alternatively contact an SMP Classroom Inclusivity Champion (see Blackboard for contact details). Our Inclusivity Champions are here to listen, to understand your concerns, and to explore potential actions that can be taken to resolve them. Your well-being and a positive learning atmosphere are of utmost importance to us.

IMPORTANT:ﾠ For any administrative questions related to the course (such as switching practical groups, requesting exemptions from assessment, transferring credit, textbook requirements, etc.), please send an email toﾠ math2010@uq.edu.au. The course coordinator and lecturers will be available to discuss academic questions during consultation hours.

Course requirements

Assumed background

This course will build on foundations that you have obtained in earlier courses: differential and integral calculus, linear algebra, vector analysis,ﾠand especially differential equations.ﾠIt is your responsibility to fill in any gaps in the assumed knowledge.ﾠYou may need to undertake background reading to understand the lecture material.ﾠChaptersﾠ1, 2, 7, 8, 9, and 10 of the set textﾠ (E. Kreyszig,ﾠAdvanced Engineering Mathematics, 9th Edition) cover most of the background material.ﾠ ﾠ

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Prerequisites

You'll need to complete the following courses before enrolling in this one:

MATH1052 or MATH1072

Companion or co-requisite courses

You'll need to complete the following courses at the same time:

MATH2001or MATH2901

Incompatible

You can't enrol in this course if you've already completed the following:

MATH2100

Course contact

Course staff

Timetable

The timetable for this course is available on the UQ Public Timetable.

Additional timetable information

All classes will be conducted on campus. Consult your personal timetable for times and locations. Students are expected to attend these sessions unless they have a valid reason for being unable to attend (such as illness). Alternative arrangements will be advised on Blackboard should the campus be closed for any reason. Important: if you are ill, then do not attend any classes in person. Alternative arrangements can be organised – consult Blackboard for details.

Aims and outcomes

This course is built around two mathematical concepts: Systems of ordinary differential equations (ODEs) and Laplace transforms.

Systems of ODEs generalise the idea of an ODE, as you have seen it covered in MATH1052, for example. Now we have several unknown functions of a single independent variable, say the time t, and we have ODEs linking the unknowns together. We deal mostly with systems of two coupled first-order equations to see the sorts of things that can happen. The notion of the phase-plane is introduced, where the ODEs determine the trajectory of a representative point for the system. Basic notions of stability and instability of equilibrium (critical) points of the system are explored. Illustrative applications are described, such as predator-prey systems, an epidemic model, electrical and mechanical oscillators.

The Laplace Transform is a tool still widely used to deal with linear ODEs and PDEs (covered in MATH2011), especially in engineering and biological applications. We introduce the basic concepts, including applications to simple systems of ODEs.
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The course will also include a brief introduction into the Computer algebra software Mathematica.

Learning outcomes

After successfully completing this course you should be able to:

Assessment

Assessment summary

A hurdle is an assessment requirement that must be satisfied in order to receive a specific grade for the course. Check the assessment details for more information about hurdle requirements.

Assessment details

Assignments

Mode

Written

Category

Tutorial/ Problem Set

Weight

40%

Due date

Assignment 1 28/03/2025 4:15 pm

Assignment 2 17/04/2025 4:15 pm

Learning outcomes

L01, L02, L03, L04, L05

Task description

Submission guidelines

Assignment submission: online through Blackboard.

Deferral or extension

You may be able to apply for an extension.

The maximum extension allowed is 7 days. Extensions are given in multiples of 24 hours.

Solutions for assessment item/s will be released 7 days after the assessment is due, so an extension after 7 days will not be possible.

See ADDITIONAL ASSESSMENT INFORMATION for the extension and deferred examination information relating to this assessment item.

Late submission

A penalty of 10% of the maximum possible mark will be deducted per 24 hours from time submission is due for up to 7 days. After 7 days, you will receive a mark of 0.

You are required to submit assessable items on time. If you fail to meet the submission deadline for any assessment item, then the listed penalty will be deducted per day for up to 7 calendar days, at which point any submission will not receive any marks unless an extension has been approved. Each 24-hour block is recorded from the time the submission is due.

Final Exam

• Hurdle

Mode

Written

Category

Examination

Weight

60%

Due date

End of Semester Exam Period

7/06/2025 - 21/06/2025

Learning outcomes

L01, L02, L03, L04, L05

Task description

Hurdle requirements

Exam details

Submission guidelines

Deferral or extension

You may be able to defer this exam.

See ADDITIONAL ASSESSMENT INFORMATION for the extension and deferred examination information relating to this assessment item.

Course grading

Full criteria for each grade is available in the Assessment Procedure.

Additional course grading information

See COURSE GRADING INFORMATION for the hurdle relating to this assessment item.

Supplementary assessment

Supplementary assessment is available for this course.

Should you fail a course with a grade of 3, you may be eligible for supplementary assessment. Refer to my.UQ for information on supplementary assessment and how to apply. 

Supplementary assessment provides an additional opportunity to demonstrate you have achieved all the required learning outcomes for a course.  

If you apply and are granted supplementary assessment, the type of supplementary assessment set will consider which learning outcome(s) have not been met.  

Supplementary assessment in this course will be a 1-hour examination similar in style to the end-of-semester examination. To receive a passing grade of 3S4, you must obtain a mark of 50% or more on the supplementary assessment. 

Additional assessment information

Artificial Intelligence

Assessment tasks in this course evaluate students' abilities, skills and knowledge without the aid of generative Artificial Intelligence (AI) or Machine Translation (MT). Students are advised that the use of AI or MT technologies to develop responses is strictly prohibited and may constitute student misconduct under the Student Code of Conduct.

Applications for Extensions to Assessment Due Dates

Extension requests are submitted online via my.UQ – applying for an extension. Extension requests received in any other way will not be approved. Additional details associated with extension requests, including acceptable and unacceptable reasons, may be found at my.UQ.

Please note:

• Requests for an extension to an assessment due date must be submitted through your my.UQ portal and you must provide documentation of your circumstances, as soon as it becomes evident that an extension is needed. Your application must be submitted on or before the assessment item's due date and time.
• Applications for extension can take time to be processed so you should continue to work on your assessment item while awaiting a decision. We recommend that you submit any completed work by the due date, and this will be marked if your application is not approved. Should your application be approved, then you will be able to resubmit by the agreed revised due date.
• If an extension is approved, you will be notified via your my.UQ portal and the new date and time for submission provided. It is important that you check the revised date as it may differ from the date that you requested.
• If the basis of the application is a medical condition, applications should be accompanied by a medical certificate dated prior to the assignment due date. If you are unable to provide documentation to support your application by the due date and time you must still submit your application on time and attach a written statement (Word document) outlining why you cannot provide the documentation. You must then upload the documentation to the portal within 24 hours.
• If an extension is being sought on the basis of exceptional circumstances, it must be accompanied by supporting documentation (eg. Statutory declaration).
• For extensions based on a SAP you may be granted a maximum of 7 days (if no earlier maximum date applies). See the Extension or Deferral availability section of each assessment for details. Your SAP is all that is required as documentation to support your application. However, additional extension requests for the assessment item will require the submission of additional supporting documentation e.g., a medical certificate. All extension requests must be received by the assessment due date and time.
• Students may be asked to submit evidence of work completed to date. Lack of adequate progress on your assessment item may result in an extension being denied.
• If you have been ill or unable to attend class for more than 14 days, you are advised to carefully consider whether you are capable of successfully completing your courses this semester. You might be eligible to withdraw without academic penalty - seek advice from the Faculty that administers your program.
• There are no provisions for exemption from an assessment item within UQ rules. If you are unable to submit an assessment piece then, under special circumstances, you may be granted an exemption, but may be required to submit alternative assessment to ensure all learning outcomes are met. 

Applications to defer an exam

In certain circumstances you can apply to take a deferred examination for in-semester and end-of-semester exams. You'll need to demonstrate through supporting documentation how unavoidable circumstances prevented you from sitting your exam. If you can’t, you can apply for a one-off discretionary deferred exam.

Deferred Exam requests are submitted online via mySi-net. Requests received in any other way will not be approved. Additional details associated with deferred examinations, including acceptable and unacceptable reasons may be found at my.UQ.

Please note:

• Applications can be submitted no later than 5 calendar days after the date of the original exam.
• There are no provisions to defer a deferred exam. You need to be available to sit your deferred examination.
• Your deferred examination request(s) must have a status of "submitted" in mySI-net to be assessed.
• All applications for deferred in-semester examinations are assessed by the relevant school. Applications for deferred end-of-semester examinations are assessed by the Academic Services Division.
• You’ll receive an email to your student email account when the status of your application is updated.
• If you have a medical condition, mental health condition or disability and require alternative arrangements for your deferred exam you’ll need to complete the online alternative exam arrangements through my.UQ. This is in addition to your deferred examinations request. You need to submit this request on the same day as your request for a deferred exam or supplementary assessment. Contact Student Services if you need assistance completing your alternative exam arrangements request.

Learning resources

You'll need the following resources to successfully complete the course. We've indicated below if you need a personal copy of the reading materials or your own item.

Library resources

Find the required and recommended resources for this course on the UQ Library website.

Other course materials

Learning activities

The learning activities for this course are outlined below. Learn more about the learning outcomes that apply to this course.

Policies and procedures

University policies and procedures apply to all aspects of student life. As a UQ student, you must comply with University-wide and program-specific requirements, including the:

• Student Code of Conduct Policy
• Student Integrity and Misconduct Policy and Procedure
• Assessment Procedure
• Examinations Procedure
• Reasonable Adjustments - Students Policy and Procedure

Learn more about UQ policies on my.UQ and the Policy and Procedure Library.