Course Outline

Course Overview

Why does a cup of coffee cool the way it does? Why do bridges sometimes shake themselves apart? Why does a population boom, crash, or settle into balance — and why do two competing species sometimes coexist and sometimes don’t? Every one of these questions has the same secret: we rarely know the quantity we care about directly. What we do know, almost always, is how fast it’s changing. That single idea — that the rate of change of a quantity is often easier to state than the quantity itself — is the seed from which the entire subject of differential equations grows.

This course teaches you to think and work like a mathematical modeller. Across the semester, you’ll follow the same rhythm again and again, applied to progressively richer situations:

  • Motivate — start from a real question about a real system: a cooling body, a growing population, a swinging bridge, two interacting species.
  • Formulate — translate the situation into a differential equation, stating your assumptions honestly.
  • Solve — bring the right mathematical technique to bear, moving from single first-order equations, to second-order oscillatory systems, to systems of interacting equations.
  • Interpret — bring the solution back to the real world, ask whether it’s plausible, and use it to say something true (or usefully approximate) about the system you started with.

By the end of the course you won’t just be able to solve differential equations — you’ll be able to look at a real, messy situation and see the equation hiding inside it.

Staff Members

Lecturer: Mr. Abdul Hamid Carrim

Email:

Office: T.W. Kambule Mathematical Sciences Building (MSB), Room Number 118

YouTube: Abdul Hamid Carrim

Teaching Assistant: Mr. Anesu Pisira ()

Course Coordinator: Dr Rahab Kgatle-Maseko ()

A Blended Approach to Learning

For this course, we will adopt a blended approach to learning and teaching - this means that there will be increased in-person activities with a mix of complementary online activities. This online book serves as the primary mechanism to drive this approach, and serves as a complement to the planned in-person lectures.

The book will be updated regularly with new chapters including text, videos, and other resources. There will be several quizzes and tests (see the schedule set out below). The quizzes will be made available via Moodle and all submissions should be done on Moodle, and tests will take place in-person as scheduled preliminarily below.

I believe that a very important aspect of learning is interacting with peers/tutors/lecturers, so I encourage you to interact regularly in-person and on Moodle through the various channels and forums.

Academic integrity is of absolute importance. Communication during quizzes and tests, sharing of answers, and all forms of plagiarism are taken very seriously by the University and will result in failing the course (FCM) and/or being reported to the Wits Legal Office. I will be monitoring Moodle logs closely and will be checking all submissions for cheating and plagiarism.

Learning Management System

We will use Moodle exclusively for this course. We will be using Moodle for the general construction and delivery of the course.

It is your responsibility to keep up to date with all changes made on Moodle. Log in with your student number and usual Wits password. You should already be enrolled in the course.

All course-related announcements will be done through Moodle.

Timetable

In the timetable, there is a double lecture for Mathematical Methods and Modelling each week, as well as a single tutorial that will alternate every other week with Mechanics.

  • Group A Lecture – Tuesdays: 08:00-09:45
  • Group A Lecture – Wednesdays: 10:15-12:00
  • Tutorials – Every other Thursday: 12:30-13:15

This course will be presented with a blend of a synchronous and asynchronous manner. Prerecorded videos, readings and exercises will be made available to you earlier in the week to complement the in-person mode of teaching. It is vital that you stay up to date with the weekly activities throughout the semester.

The weekly lessons including pre-recorded video lectures and other course-related material will posted every week on a Thursday on my Youtube channel and embedded in the online book. During these videos, we will be concentrating on introducing and elucidating concepts, covering course content and material. These concepts will then be expounded upon and consolidated during the live in-person lectures the following week on a Tuesday or Wednesday.

You will be expected to pre-read the relevant material made available to you before engaging with any of the lecture material for that week. You will need to complete all assigned post-class tasks before the next lesson, or by the specified submission date. You will also be required to have attempted the tutorial questions before attending the tutorial, and engage in all tutorial and lab-related activities.

The main synchronous aspect of the course will be the weekly in-person lecture sessions which will take place on-campus on Tuesdays from 08:00-09:45 or Wednesdays from 10:15-12:00, depending on your grpup.

Tutorials will take place on on-campus on Thursdays from 12:30-13:15, beginning in Week 2 of the course. You will be assigned a tut group and every week you must meet your tutor for a live in-person tutorial. While tutors will be able to answer any theory/practical questions that you may have during this session, the tuts are meant to be interactive. These tutorial sessions are compulsory and will contribute to your Satisfactory Performance (see the Course Information Booklet). There will also be quizzes during the tutorial that will contribute towards your class mark.

Communication

All official communication will be posted to the announcements forum on Moodle. Moodle will send you email digests daily, but it is still your responsibility to ensure you’re receiving these emails. You can change your email preferences on Moodle to receive a single email per post if you prefer that.

Please use the Q&A Forum on Moodle to ask and answer questions. This means that the tutors and I can answer these questions publicly and helps avoid duplication. Please do not email general course-related questions to me unless it only relates to you personally or is sensitive in some way. I am always happy to help where I can, but avoiding duplication is really important in large classes like this one. I hope that you will use this platform to ask and answer questions, engage with peers, post suggestions, and share resources.

In general, please try to use public channels to speak with me and the tutors unless it is about anything that you should not be posting publicly - it is useful for others to be able to see the answers to your questions. I will always respond to Moodle queries before emails. If you have queries directly for me, I would prefer you use the Q&A Forum on Moodle where these persistent answers to course-related questions could benefit others as well.

Consultations

Due to the number of students in this course, it is really important to book time with me if you would like a live one-on-one consultation, either in-person in my office or online on MS Teams. You can book 15 minute time slots directly into my calendar using Calendly. I will will publish my available times and slots in due course as I clear my calendar. You can book up to 30 days in advance and calendly will send you a calendar invite with your booking details, and an MS Teams link if you chose an online consultation. You will need to answer two questions when setting up a meeting. You also need to come to the consult with evidence of attempted work.

If you have bandwidth or data issues, you can tell MS Teams not to receive video. If you don’t have any of these issues, feel free to keep your video on as it is nice to chat face-to-face and still get to see some of my students!

Course Background and Purpose

Applied Mathematics exists because we want to explain and predict the real world, not just describe it after the fact. Differential equations are the language in which almost all of that prediction happens — from the decay of a radioactive isotope to the spread of a disease to the sway of a skyscraper in the wind. Once you can read and write this language, an enormous amount of the physical, biological, and social world opens up to genuine quantitative understanding.

This course builds that language from the ground up, always following the same cycle: motivate → formulate → solve → interpret. You will learn to build a mathematical description of a real, simple situation (formulate), bring the appropriate technique to solve it (solve), and then critically evaluate what the solution is actually telling you about the world (interpret) — never losing sight of the real question that motivated the mathematics in the first place.

Problem-solving skills are deliberately foregrounded throughout, with the aim of developing your intellectual self-reliance: the ability to look at an unfamiliar real-world scenario, model it mathematically, and find a meaningful answer, even when nobody has handed you a template to follow. Collaboration on tutorial problems is strongly encouraged, both because talking through a model with a peer sharpens your own understanding, and because communicating modelling arguments clearly is itself a core skill of this course.

Course Content, Goals, and Outcomes

The development of concepts and skills is progressive throughout the course, and a comprehensive grasp of early sections will enhance the students’ ability to cope with later sections.

Course Learning Outcomes

By the end of this course, you will be able to:

  • Formulate a first-order, second-order, or system-of-first-order differential equation model from a verbal or physical description of a real-world situation.
  • Classify and solve ordinary differential equations analytically using the appropriate method (direct integration, separation of variables, undetermined coefficients, or characteristic-equation methods), based on the equation’s order, linearity, and structure.
  • Determine particular solutions to differential equations and systems using initial or boundary conditions.
  • Visualize and interpret solutions qualitatively — using solution curves, direction fields, and, for systems, phase-plane diagrams — to describe the long-run behaviour of a modelled system without necessarily solving it in closed form.
  • Test the plausibility of a proposed solution against the real-world system it models, using reality checks such as physical reasoning, limiting/extreme cases, units, and equilibrium behaviour.
  • Discriminate between qualitatively different regimes of behaviour a model can exhibit (e.g. growth vs decay, oscillatory vs monotonic, stable vs unstable equilibria) and connect these regimes to the underlying mathematical conditions that produce them.
  • Communicate a full modelling argument — assumptions, derivation, solution, and real-world interpretation — in mathematically precise and coherent form.

The chapter-level outcomes below map onto these course-level outcomes and show how each block of content builds toward them.

The course is organised into four chapters. Each moves through the same rhythm — motivate → formulate → solve → interpret — and technique is now taught alongside the application that motivates it, rather than all technique first and all application afterward. This mirrors how differential equations are taught in first courses at MIT, Cambridge, and Harvard, and matches the applied-modelling emphasis of UCT’s own ODE course.

Chapter 1: An Introduction to Modelling and Differential Equations

  • Distinguish between a real-world system, a model, and a mathematical model.
  • Explain the stages of the mathematical modelling process (observation, assumption, formulation, analysis, interpretation, validation) and identify what happens at each stage in a given example.
  • Evaluate the trade-offs (fidelity, cost, flexibility) between competing modelling approaches for a given situation.
  • Explain, conceptually, why rates of change — rather than direct formulas — are the natural mathematical language for describing dynamic real-world phenomena, and how this motivates the use of differential equations.

Chapter 2: First-Order Differential Equations and Models

  • Classify a given differential equation by its order, linearity, homogeneity, and coefficients, and justify the classification.
  • Explain what it means for a function to be a solution of a differential equation, and verify by substitution whether a given function is a solution.
  • Sketch and interpret a direction field for a first-order equation, and relate it to a family of solution curves without solving the equation explicitly.
  • Solve first-order ordinary differential equations using direct integration and separation of variables, and justify why a chosen method is appropriate given the equation’s classification.
  • Apply the formal model-construction process end-to-end: translate a verbal description of a real-world situation into a first-order differential equation, stating assumptions explicitly.
  • Formulate and solve growth, decay, and Newton’s Law of Cooling/Warming models, and interpret model parameters (rate constants, equilibrium values) in terms of the real-world quantities they represent.
  • Write down appropriate initial conditions for a given problem description, and use them to obtain a specific, interpretable solution.
  • Use a model’s solution to predict and critically evaluate the future behaviour of the real-world system, including checking the model’s plausibility (physical reasoning, extreme cases, units) and identifying the model’s limitations.

Chapter ??: Second-Order Differential Equations and Oscillatory Models

  • Solve second-order linear homogeneous equations with constant coefficients via the characteristic equation, and correctly interpret the three root cases (distinct real, repeated real, complex conjugate) in terms of the qualitative behaviour of the solution.
  • Solve inhomogeneous linear equations using the method of undetermined coefficients (superposition approach).
  • Obtain and justify particular solutions to initial value problems and boundary value problems.
  • Model simple mechanical (mass–spring) and electrical (RLC circuit) oscillatory systems as second-order linear ODEs, and recognise the correspondence between the two via the mechanical–electrical analogy.
  • Classify an oscillatory system’s response as underdamped, critically damped, or overdamped, connecting each regime to the corresponding characteristic-equation root case and to physically observable behaviour.
  • Explain, conceptually, the phenomenon of resonance and why it matters in real mechanical and electrical systems.

Chapter ??: Systems of Ordinary Differential Equations

  • Formulate a system of first-order ODEs that captures the dynamics of two or more interacting quantities described in words (e.g. interacting populations, coupled compartments, coupled circuits).
  • Solve simple linear systems of first-order ODEs and determine their equilibrium points.
  • Sketch a direction field and phase portrait for a simple system by hand, and use it to classify the qualitative long-run behaviour near an equilibrium (e.g. approaching, spiralling toward/away from, or circling an equilibrium) without formal eigenvalue analysis.
  • Connect the qualitative predictions of a system model back to the real-world question that motivated it (e.g. do two species coexist, does an epidemic die out, does a coupled system reach a shared equilibrium).

Tentative Class Schedule

Block 3

Week Topic
1 (21 - 25 July) Orientation, Course Outline
2 (28 July - 1 August) Introduction to Modelling and Differential Equations
3 (4 - 8 August) First-Order ODEs: Classification, Solutions, and Direction Fields
4 (11 - 15 August) First-Order ODEs: Direct Integration and Separation of Variables
5 (18 - 22 August) First-Order Models: The Formal Modelling Process, Growth and Decay Models
6 (25 - 29 August) First-Order Models: Newton’s Law of Cooling/Warming and Initial Value Problems
7 (1 - 5 September) Test 1

Block 4

Week Topic
8 (15 - 19 September) Second-Order ODEs: The Characteristic Equation and the Three Root Cases
9 (22 - 26 September) Second-Order ODEs: Undetermined Coefficients (Superposition), Initial & Boundary Value Problems
10 (29 September - 3 October) Oscillatory Models: Mass-Spring Systems, RLC Circuits, Damping and Resonance
11 (6 - 10 October) Systems of ODEs: Formulating and Solving Simple Linear Systems
12 (13 - 17 October) Systems of ODEs: Phase Portraits, Qualitative Behaviour, and Applications
13 (20 - 24 October) Test 2

Assessment

There will be a mix of invigilated in-person formal assessments as well as inviglated online formative and continuous assessment tasks. All online assessments will take place on the Moodle platform.

Assessments will be set that will require the student to show a thorough knowledge of applying the theoretical material, and to show a conceptual and deep understanding of the course. The relevant topics to be covered in the assessments will be made known to students in a timely manner.

There will be weekly formative assessments in the form of Did You Get It? questions (DYGIT). These do not contribute towards your course mark, but are there for you to check your current conceptions and self-ascertain your understanding of the course.

The course will be assessed through two in-person timed tests (60 - 90 minutes), multiple quizzes (30 - 60 minutes), an assignment, and a final examination (120 minutes). There will aslo be tutorial submissions that will take place and must be submitted during the weekly tutorial sessions.

The structure and procedure of class tests shall be communicated to you well in advance. Both tests are compulsory and contains content that will be examined in the final exam. The table below shows a full tentative and provisional schedule of all assessment tasks scheduled for this course. The assessment will be scheduled within the specified week. Assessment open and closing times will be communicated to you in advance of the assessment.

Examinations will take place in-person during the November examination period. You will have to consult the examination timetable made available online for the relevant information.

Assessment Schedule

Assessment Week Date Venue
Quiz 1 4 TBC TBC
Quiz 2 6 TBC TBC
Test 1 8 1st September Flower Hall
Quiz 3 10 TBC TBC
Assignment 11 TBC TBC
Quiz 4 12 TBC TBC
Test 2 13 21st October Flower Hall
  • The structure and format of continuous assessments is such that it will comprise of both problem and conceptual questions, and may comprise of Multiple Choice Questions (MCQ’s), fill in the blanks, matching, short numerical answers, long calculation questions, etc.
  • The tests will take place in-person and on-campus.
  • The quizzes will take place and be submitted online on Moodle.
  • The examination is expected to take place in-person and on-campus during the November examination period. The format of the examination shall be announced before the start of the examination period. You will have to consult the examination timetable made available online for the relevant information.

Calculation of the final mark

The final course mark for APPM1026A will comprise of an average of your first semester mark and your second semester mark.

Your final second semester course mark will be calculated as follows. It will comprise of your year mark comprising of all continuous assessment tasks and your final November summative assessment mark. The year mark component will be derived from continuous assessment tasks. The final exam mark will be derived from the November summative assessment task.

The final course mark breakdown is as follows:

Item Weight
Tests 30%
Assignment 10%
Quizzes 10%
Tutorial Submissions 10%
November Assessment 40%