MTH2032: Differential Equations with Modelling

6 quizzes worth 2% each, 4 assignment quizzes worth 2% each, 1 midsemester exam worth 20%, 1 final exam worth 60%.

This unit was not well run. There were a lot of issues with cheating in the past and as a result of trying to combat that, we ended up with a much worse and unfair unit.
The assessments were mostly done in class. This had some perks, as it meant there was less to do outside of class. However, because classes ranged from Monday to Friday, people who took the Friday classes had 4 extra days to understand the content. The assessments also meant the classes were cut shorter and there was less time to complete the actual questions.
The assignments were very strange. We were given a prompt with some information and basically had to prepare everything we could to answer some multiple-choice questions. Once again, these quizzes were done in class so some people had a lot less time than others to complete it. The quizzes also had the problem of answers being in a certain format, and with only 5 minutes to do the questions, if you didn't use the right format, you were screwed.
There was not a lot of communication between the tutors, with some allowing notes for the assessments and others not.

In terms of the content, the ODE half was pretty interesting and at a good level. Jerome's parts of lectures were very useful and made Ngan's seem redundant, as she went through the same content(albeit a little slower and less put together), only adding a couple of examples. The PDE half was very strange to me, as it felt like there wasn't a lot of procedural content(eg how to solve equations etc), and instead we were just studying the same three equations over and over and over again. It seemed like anything we learned was only used in a specific context. Mark spent a lot of time doing long-winded examples that I didn't find particularly useful. A lot of his explanations were also very confusing to me.

Biggest advice is to read the notes and get a Friday applied class. Also, don't use Chegg or cheat in any other way(which you shouldn't be doing anyway) because Mark will hunt you down.

Ngan Le took the first 6 weeks, which were on ODEs and Mark Flegg took the last 6 weeks, which were on PDEs. There were also recordings available of parts of Jerome Droniou's previous lectures on ODEs.

Yes from 2014 and 2015.

2.5 out of 5

Yes, with screen capture

None required

Three 1 hour lectures and one 1.5 hour applied class per week

2021 semester 2(in lockdown)

94 HD

Five quizzes (in tutorials): 2% each (10%)
Three assignments: 5% each (15%)
One mid-semester test (in tutorials): 15%
Final exam: 60%

This subject revises and extends upon the concepts covered in the Differential Equations part of MTH1030, as well as providing an introduction to Partial Differential Equations, and some applications of both ODEs and PDEs.

Firstly, I'd encourage you all to go and buy the lecture notes (if available). They are really helpful in both sections of the course. For ODEs, they contain all the theory and necessary algorithms (if a bit verbose and densely-written at times), and for PDEs, they are an almost exact copy of what Rosemary writes on her transparencies. They also contain tutorial questions, two of the assignments you need to do, as well as the solutions to the tutorial questions for PDEs.

focused on ODEs. We start off with some basic terminology relating to DEs - linear, homogeneous, 1st and 2nd order etc. Then, we revisit some of the techniques you used in MTH1030 to solve 1st-order ODEs, except in a slightly more rigorous manner. For example, you will probably notice "separable" ODEs are the ones where you could go or something similar in MTH1030. "Linear" ODEs are the ones where you would use "the integrating factor", except now we cover it a bit more rigorously through understanding the "homogeneous" and "particular" solutions, and the principle of superposition. We also go through two new types of 1st-order ODEs, which are 'exact' and 'homogeneous type' ODEs. These have their own solving methods.

We then look at how we can turn higher-order ODEs into 1st order ODEs, except now with vectors replacing variables. Unfortunately, this does not make them any easier to solve (and most of us were questioning why we would even do such a thing), but it does allow us to more easily state if a solution exists. Remember Euler's method from Specialist Maths? Well, we revisit it, and learn that it is in fact a relatively poor approximation scheme. We look at Heun's method, which is more accurate, and work out a rough guide to "accuracy" of a general class of approximation schemes, which both Euler's and Heun's methods fall under.

Next, we turn to 2nd order ODEs. You probably learnt how to solve ones with constant coefficients in MTH1030; we do that again here. However, we also learn some new techniques for solving 2nd order linear ODEs, with non-constant coefficients, such as the method of "variation of parameters", and the humble "trial solutions" method. We learn how to check if solutions to an ODE are linearly independent (this is important because it is a necessary requirement for many of out solution methods), through the Wronskian matrix. Then, we learn about how to deal with ODEs where we might not have an present, or a present, even if is present (and things like that).

Finally, we learn about series solutions to ODEs. Sometimes, it is very difficult for us to see if a particular ODE has a solution. So what we can try and do is see if an infinite polynomial solution works (much like a Taylor series). There was also a section on Bessel's functions, Legendre's and Frobenius' methods, but we never got around to learning those in lectures.

Throughout this part of the course, there are applications mainly to simple harmonic motion and harmonic oscillators, but also things such as radioactive decay, Newton's Law of Cooling and curves of pursuit are covered. There are also a number of "fundamental theorems", which are basically statements that do not solve an equation, but tell you that a solution exists, or some property of the solutions. You will soon see that it is very important to know the conditions under which these apply, and to invoke them appropriately.

Ordinary Differential Equations (ODEs): Jerome Droniou, Weeks 1 - 6
Partial Differential Equations (PDEs): Rosemary Mardling, Weeks 7 - 12

Yes, two. One had answers.

4.5/5

Yes, with screen capture. For Weeks 1- 6 (Jerome's section), he wrote on a tablet and what he wrote on the tablet is the "video" for the recorded lecture. He also uploaded his writings on Moodle. For Weeks 7 - 12 (Rosemary's section), she did most of her writing on transparencies, which do not show up on the recorded lectures. However, most of what she writes is same as in the lecture notes booklet.

is focused on PDEs. The general terminology is covered again, before we then look at the ways in which a PDE is different to an ODE (for example, you no longer get arbitrary constants when you integrate, but arbitrary functions). We look at boundary and initial conditions, and learn some relatively simple methods for solving PDEs (e.g. noticing it is similar to an ODE and the method of separation of variables).

We then move on to Fourier Series. Basically, the point of this section is to show you that any periodic function (a function that repeats itself after some time), can be modelled as a (potentially infinite) sum of sine and cosine graphs. Fourier Series solutions to differential equations actually make up the majority of solutions you're likely to see in the Heat and Wave equation parts, so it's really worth your while to make sure you understand this part of the course well. We look at periodic extensions of functions with limited domain, too.

After learning about Fourier Series, we look at the Heat Equation, which models how the temperature of a rod changes over time. We go over Taylor Series in two variables again, before deriving and solving the Heat Equation, given some initial and boundary conditions. Next, we learn about the Advection equation, which models how objects might float along a stream of some kind, and about "characteristics". Finally, we look at the Wave Equation, and how to solve it. We also see an alternative method of solving the Wave Equation (the solution of d'Alembert), which represents a wave as a sum of two travelling waves (if you do Physics, this should jog your memory), which interfere with each other. If nothing else, that's pretty cool.

This unit was a reasonable unit. Unfortunately, most students found it difficult to understand what Jerome was trying to teach, probably because he didn't really explain his derivations very well, and didn't have a good grasp of when he was talking about a difficult concept that he needed to spend more time on. Either that, or he expected us to work through any difficulties we faced at home. Rosemary was a pretty clear lecturer who had fairly good explanations.

In this unit, explanations are important. It's always a good idea to state what you're doing as you're doing it (even if it seems incredibly obvious to you), because there are always marks allocated for explanations. Knowing when to invoke theorems, writing down what class of DE we have, and things like that are all easy to forget, but cost you marks in the end.

The tutorial quizzes last for 20 minutes each, and were initially at the start of the tutorial, before we asked Jerome to put them at the end (so we could actually ask our tutors for help). The ODE quizzes were reasonably challenging and had a fair bit of time pressure. The PDE quizzes were a bit easier.

The first assignment was a typical maths assignment where you answer questions from a sheet. The latter two are more of a computer modelling exercise (using Excel, MATLAB, or other computer software), where you numerically approximate Fourier Series and the Heat Equation, respectively.

The mid-semester test is on Jerome's section of the course, and goes for an hour. It's not impossible, but does test several different areas of the course. The final exam is roughly of the same difficulty of the midsemester test, just covering the whole course. There were some tricky questions in both sections.
All in all, there were areas where the unit could have been improved, but it is certainly a useful unit if you wish to do maths or science, as DEs appear almost everywhere in those fields.

E. Kreysig, Advanced Engineering Mathematics (9th edition).
Did not buy, so definitely not compulsory.

3 x 1 hour lectures per week
1 x 2 hour tutorial per week (Technically not compulsory, but you'll be attending most weeks for the quizzes and assignments).

Semester 2, 2013

Unknown at this point.

3x5% Assignments (The second two are 'reports' on excel modelling PDEs), 5x2% Quizzes, 15% Mid Semester Test, 60% exam.

I enjoyed this unit, although at times there are some small annoyances. For the first 6 weeks under Jerome you will need to be meticulous in your working, and will need to write what you're doing for each step (i.e. "We have a second order linear homogeneous ODE (Ordinart Differential Equation)", . "Using the variation of parameter method" . Missing after introducing a constant of proportionality can lose you a mark, as well as if you don't have the right keyword in your explanation. i.e. I lost a mark on tests for not stating "By superposition" when finding the particular solution for a Second Order Linear Non-Homogenous ODE with constant coefficients. So yeah, what I'm getting at is you have to be picky with everything for the first six weeks. For the second six weeks, you can be fairly lazy in this regard, as Rosemary doesn't mind as much.

You won't be able to learn enough by just going to lectures. It's better to sit down with the lecture notes and go through and work out a method for each type of question, with the overhead slides from the lecture complimenting this. With that being said, I stopped going to lectures after week 7, but that was not because of Rosemary, (she was actually pretty good for the few I went to).

The averages for the 2% tests were all over the place, with some being med-high and others being really low. They're a good trial run though, as the questions are very similar to the mid-semester test (which is only on Jerome's content). The first assignment is like any other normal maths assignment, while the latter two will require you to come to the tute to do some work on excel (or Matlab or w.e. you wan't really), to model a situation regarding PDEs. The first is just about modelling a wave with a Fourier Series. The second is on the Heat Equation, so the heat transfer through a 1-dimensional rod and how it varies through time, modelling the temperature distribution with a Fourier Series.

Most people found the exam itself quite hard, and it was above average in the end. You probably won't be short on time, but will get to some questions and go "well,.. what... where am I even meant to start?". There was a 10 mark question (out of 99 marks, so around ~10%), that not many (if any) were able to get. Your best be for preparing is to make sure you know how to apply each method to solve an ODE, and memorise the few theorems that you will encounter throughout the unit, then do past exams and you should be fine.

Topics for those who are interested.

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Weeks 1-6: Dr Jerome Droniou, Weeks 7-12: Rosemary Mardling

Yes, as per the Mathematics Faculty policy, 2 exams but only 1 with solutions.

4 Out of 5

Yes, with screen capture.

You don't need it, (but if you're that keen: E. Kreysig, Advanced Engineering Mathematics (9^th edition).

3x1 hr lectures, 1x2 hr tute per week

Semester 2 2013

82 - HD