Differential equations are present in almost every application of mathematics to science, engineering, social sciences and economics. Difference is about change and differential equations focus on quantitatively describing the change of quantities of interest over space or time. Therefore, solving these equations can provide us with meaningful and extremely useful information about patterns of change (e.g. the volatility of stock markets) or underlying mechanisms of phenomena such as the behaviour of predators and preys in nature.  In this module, we will look at various techniques (both analytical and numerical) that can be used to solve these equations (accurately or approximately). Concepts and techniques from the first year Calculus modules will appear throughout the module. By the end of this module you will be able to describe the main features of ordinary differential equations and model real-world problems through them.

SCHEDULE OF THE MODULE

Every week with the exception of reading week (week 7) the schedule of the module will be as follows

Lessons 1-2: Wednesday 11:00-13:00 Live Arts Two LT

Lesson 3:  Thursday 17:00-18:00 Live Arts Two LT

You will be assigned to one of the following three tutorials (check your timetable to know which applies).

Tutorial 1: Thursday 10:00-11:00 PP2

Tutorial 2: Thursday  12:00-13:00  PP2

Tutorial 3: Thursday  16:00-17:00  Graduate Ctr: GC201

Support Learning Hour: Thursday 14:00-15:00


 In 2022/2023, the assessment structure will be 2 courseworks (20% of the total mark ) and a final exam (80% of the total mark). For your final exam, the marking criteria gives credit both for (clearly explained) method and final answer. You will know what do clearly explained methods mean through the lectures during the whole semester.

Please note that since the 2018 fall term, there has been some changes in the final assessment of this module. Since 2018  questions are not just focusing on solving differential equations but also on your problem solving skills. Our goal is to train your problem solving skills so that you can learn to explain real applications and you can also learn to draft your solutions. This will be important for applying what you learned from this module in real applications and in future module you might follow. Accordingly, students are also expected to be able to write down differential equations of real systems, sketch solutions of differential equations, state basic definitions and theorems, and etc. The concrete requirements will be explained during lectures.

Differential equations frequently arise in applications of mathematics to science, engineering, social science, biology, medicine and economics. This module provides an introduction to the methods of analysis and solution of simple classes of ordinary differential equations. The topics covered will include first- and second-order differential equations, autonomous systems of differential equations and analysis of stability of their solutions.



ACADEMIC CONTENT

This module covers:

  • Simple types of first-order and second-order ordinary differential equations (ODEs).
  • Autonomous systems of first-order ODEs, finding their solutions close to equilibrium points and plot phase portraits (trajectories) of an ODE system.

DISCIPLINARY SKILLS

At the end of this module, students should be able to:

  • Identify and solve first-order ordinary differential equations (ODEs) that are separable, exact or linear, or can be reduced to the above by standard methods.
  • Identify and solve linear second-order non-homogeneous differential equations with constant coefficients and associated initial value problems.
  • Identify and investigate boundary value problems for linear second-order ODEs.
  • Identify and solve general systems of first-order linear differential equations with constant coefficients using matrix operations.
  • Explain the notions of phase space, trajectories, and equilibria for autonomous systems of two first-order differential equations.
  • Explain the notions of stability for general systems of differential equations and apply a given Lyapunov function for investigating the stability of simple nonlinear systems of differential equations.
  • Find equilibria of a given autonomous system of differential equations and describe the system's behaviour and sketch the phase portrait close to an equilibrium in linear approximation.

ATTRIBUTES

At the end of this module, students should have developed with respect to the following attributes:

  • Acquire and apply knowledge in a rigorous way.
  • Connect information and ideas within their field of study.
  • Grasp the principles and practices of their field of study.
  • Apply their analytical skills to investigate unfamiliar problems.
  • Acquire substantial bodies of new knowledge.

TEACHING ARRANGEMENTS

Each lecture is based on the week’s topic. 

Formative assessments

There are in total 10 formative assessments and 10 mock quizzes for this module. Each formative assessment and each quiz  refers to  the learning content of one week of this module. They are important for you to practice methods, find and solve learning difficulties, ask questions in tutorials or interactive online lectures.

How you engage with formative assessments, is therefore crucial to your success.

CourseWorks

There are in total 2 courseworks for this module and cumulative cout for 20% marks of the module. Approximately each coursework refers to the  learning content of the first half and the second half of the module and counted as 10% of the total mark of this module.

How you engage with courseworks, is therefore crucial to your success.

PREPARING FOR THIS MODULE

  • Read the assigned reading thoroughly. Make good notes – don’t just highlight.
  • Review lecture notes and use one of the recommended survey books to enhance your understanding of the week’s topic. 
  • Practice the courseworks independently and BRING questions  into tutorials or online lectures during the interactive sessions with you.

Tip: Make it a goal to cover the material taught each week.



There may be times during the term when you get stuck doing your homework or project. This is normal. 

Who to contact for what:

You should first find the learning difficulties you have during the lectures and courseworks, then bring those questions to tutorials or in the interactive sessions in our online lectures. We really encourage you to  use the forum to post questions, and discuss your doubts with other students.

Module Lead: g.bianconi@qmul.ac.uk




ASSIGNMENT STRUCTURE AND SUBMISSION

Each assessed coursework counts 10% towards your module mark. There will be 2 such assessed courseworks, covering materials up to week 6,11 with deadline in weeks 7,12.

Both the first and the second  assessed coursework are  quizzes.

You can attempt the quizzes (coursework 1 or 2) at  any time during an entire week, however you should complete the quiz within 48 hours from when you started.

If your attempt is still in progress at the deadline the answers you have filled in so far should now be automatically submitted (whether or not you are actively working on the quiz-coursework).  However, for your own reassurance, you are recommended to explicitly submit your attempt wherever possible.  Note that if you start an attempt, or even just look at the questions, but are unable to finish due to Extenuating Circumstances (ECs) such as illness you must email maths@qmul.ac.uk (with cc to the lecturer) before the deadline; if you do not do this, your half-finished attempt will be automatically counted and any subsequent EC claim is likely to be rejected.

 

ACADEMIC INTEGRITY


You are encouraged to discuss the course material and lecture notes with other students.   However, your quiz-coursework submission must be your own work.  In particular, this means the following.

  • You should do the quiz-coursework questions by yourself, from your own computer, and not share answers.  It's OK to say to a friend "I think I need to understand the inclusion-exclusion principle, can you explain it to me?" but it's not OK to say "What is the answer to Question 2?".


TUTORIALS


Online tutorials will take place every week starting from week 1 with the exception of week 7 (reading week). 

In the tutorials you will receive the training to complete your courseworks and you will receive general feedback on past courseworks. There will also be formative assignments whose solution you don't need to turn in, but which you should be ready to discuss in the tutorial.  There will also be an opportunity to discuss general course-related queries.

 

FEEDBACK / MARKING


Feedback on coursework 1  &2 (quiz)

As soon as the quiz-coursework deadline has closed, you will be able to see whether your answers to the quiz were right and what your marks are. Finally, complete answers to each quiz-coursework will be posted on QMplus after all the tutorials have taken place.

You can further discuss the coursework with the module organizer during tutorials and Support Learning Hours.