Embedded Files
Resources
Resources
To work through the lessons, you need to download and print the following:
Each lesson has videos and/or other resources linked individually.
Exercises marked, for example, Exercise DE.2, represent a collection of questions from Maths in Focus and Cambridge, and are not provided here because of copyright. The composition of the exercises is this:
DE.1: MIF 13.01 Q1,2; CBR 13A Q1,2,4-8,11
DE.2: MIF 13.02 Q3-8; CBR 13D Q6
DE.3: MIF 13.03 Q2-6; CBR 13C Q11-15
DE.4: na
DE.5 MIF 13.01 Q3-8 13.02 Q2 13.03 Q1; CBR Example 12 13B 5,6,9-11
DE.6 MIF 13.04 all; CBR 13E 3,5,6,12-16
Credits: most examples in the lessons come from Maths in Focus, and some come from Cambridge. Some come from other YouTube videos that form part of the lessons.
Lesson 1: Introduction; y' = f(x)
Lesson 1: Introduction; y' = f(x)
There are two videos (1a and 1b) and an exercise (DE.1) for this lesson.
Video 1a (15 minutes) introduces the topic, demonstrating what a differential equation (hereafter DE) is, and how we can use our existing skills to answer some questions.
After watching this video, complete Exercise DE.1: MIF Q1,2 and CBR Q4.
Video 1b (30 minutes) gives a deeper introduction to DEs, including an outline of what we will solve in this topic and how DEs are essential for science.
After watching this video, complete other questions from the exercise, but especially CBR Q1,2.
Lesson 2: y' = f(y)
Lesson 2: y' = f(y)
With the introduction out of the way, we focus on the new skills of the topic and the lessons are shorter.
Video 2 (35 minutes) shows you how to solve a differential equation where y' is a function of y instead of x.
After completing the lesson, you can work on Exercise DE.2.
Lesson 3: y' = f(x)g(y)
Lesson 3: y' = f(x)g(y)
Doesn't that title look unusual? Fortunately the skill is accessible once you have worked through a few examples.
Complete the lesson while watching Video 3 (30 minutes), then work on Exercise DE.3.
Lesson 4 (optional): Further notes on y' = f(x)g(y)
Lesson 4 (optional): Further notes on y' = f(x)g(y)
Video 4 (20 minutes) teaches you a skill that is not needed in the Extension 1 course called implicit differentiation. So why teach it? It gives a more complete treatment of what you learned in Lesson 3.
There is no separate exercise for this lesson, but there are challenging questions in the lesson notes.
Note: Question 1 is accessible and valuable for all students.
Consolidation
Consolidation
If you would like to consolidate what you have learned so far, these YouTube videos are succinct (each under 10 minutes) and well worth watching.
Chris Odden: Intro to differential equations · Initial value problems · Separation of variables
Lesson 5: Slope fields
Lesson 5: Slope fields
The focus here is on what differential equations look like.
We know that the solution to a DE is a family of functions, so it's not possible to graph just one function and call it done. Using a slope field, we can visualise all solutions to a first-order DE.
The textbooks include slope fields in all exercises, but we decided they should be separate.
You need to work through the lesson notes for Lesson 5 yourself and watch videos when directed. Here are the links:
The times are, respectively: 8 min, 5 min, 8 min, 9 min, 17 min.
After you have finished, there is Exercise DE.5.
Lesson 6: Exponential and logistic equations
Lesson 6: Exponential and logistic equations
Exponential growth arises from a simple first-order DE. It neatly models phenomena such as population growth, but it has time limitations because populations cannot tend to infinity. In this lesson, we will review all the exponential growth knowledge we already have, then we will see how a modification to the exponential growth DE allows a cap to be placed on population growth.
There is a lot of material in this lesson, so be prepared to take your time.
Complete two questions "Before you begin!". Answers are given; solutions are in the Year 11 textbook if required.
Attempt Question 1; watch Video 6a (8 minutes)
Attempt Question 2; watch Video 6b (6 minutes)
Attempt Question 3; watch Video 6c (10 minutes)
Attempt Question 4; watch Video 6d (8 minutes)
Complete MIF 13.04 Q1-6
Make a strong cup of tea to get through the next part
Read the text in Section 5, watch Video 6e (7 min), and then Video 6f (29 min)
Watch Video 6g (50 min) while going through Questions 6-8
Complete the textbook work (see Exercise DE.6)
Attempt Question 9 (a sample HSC question) and watch Video 6h (13 min)
Enrichment
Enrichment
If you would like to extend your knowledge of differential equations:
Many questions in Cambridge 13D and 13E will provide an interesting challenge.
A 3blue1brown YouTube series uses flow of heat through a rod to motivate an understanding of DEs. The mathematics is quite involved but the presentation is for recreational interest, not for traditional teaching and learning.
The coronavirus curve is a video that was referred to at the end of Lesson 1.
About this site
About this site
This unit of work was created by Gavin Sinclair in April-May 2020 and is available to others through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. The licence applies to this website and the videos that I created; videos made by other people will have their own terms. Some questions in the lesson notes and videos are sourced from various places, so the Creative Commons licence does not apply to the questions themselves.
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