# Workshop about ODEs and connected rates of change (A-levels)

I'm preparing a workshop about ODEs and connected rates of change for pre-college students (smart and interested in mathematics). I'd like to include some fun parts as well - even if these are standard exercises but in unexpected context. Could you share your experience in teaching these subjects?

• silly question: what's A-level ? – James S. Cook Mar 17 '17 at 22:59
• English system, right? – kcrisman Mar 18 '17 at 12:49
• Sorry, A-levels are the "final" exams in English high schools. So basically pre-college students, but only the ones interested in mathematics. – Paula Mar 18 '17 at 18:43

I would like to comment immediately below the original question, but am currently blocked from doing so. @Paula. Why do you feel the "standard" connected rates of change questions e.g. relating the rates of volume, surface area and linear dimensions of a 3D object such as a cylinder, sphere or cone where one of the rates is constant, are lacking? For example a standard question, liquid pours into an empty right cone at a fixed rate but leaks out at a rate proportional to the depth of fluid in the cone, then what is height of fluid at time = t? Does the height reached a fixed value? In what way are you looking for questions to go "beyond" this?

• There's nothing wrong with standard questions. The kids I'm teaching are exceptionally smart (they participate in a program for talented students), this is why I assume that they've done most standard questions and I'd like to challenge them a bit. – Paula Mar 27 '17 at 9:55
• How about concentrating on the modelling process where the description of the scenario gets more sophisticated and the resulting differential equations get more difficult (or impossible) to solve exactly? e.g. a mechanics free-fall scenario with no air resistance then make resistance proportional to velocity or proportional to the velocity ^(2/3). Then explore different approximate methods for solution and determine "how good" the answers are. How about: nrich.maths.org/11052, nrich.maths.org/11055. How about integral equations: nrich.maths.org/7064 – Clive Long Mar 27 '17 at 20:02
• I like the idea of starting from a simple model and going deeper, thank you, @Clive Long! – Paula Mar 31 '17 at 13:08

A problem I often given in calculus III goes roughly like this: give a car is driving on a hill with equation $x^2+3y^2+z^2=1$ if you know the speed in the $x$-direction and the speed in the $y$-direction then find the speed in the $z$-direction at a given point. Also find the speed.

The total differential gives $2xdx+6ydy+2zdz=0$ then divide by $dt$ to obtain $2x \frac{dx}{dt}+6y \frac{dy}{dt}+ 2z\frac{dz}{dt} = 0$.

On the other hand, we can use the same math to do things like find curves which line up with vector fields. First, understand $F(x,y)=c$ is a solution to $dF = \partial_x F dx+ \partial_yF dy = 0$ and $\nabla F = \langle \partial_x F , \partial_yF \rangle$ points directly away from the curve $F(x,y)=c$. Thus, to find a curve which lines-up with $\vec{G} = \langle M , N \rangle$ we want to find leve curves which have a gradient which is perpendicular to $\vec{G}$. That means, solve $Ndx- Mdy = 0$.

For example, to find integral curves to $\vec{G} = \langle y, -x \rangle$ we solve $-xdx-ydy = 0$ hence $-ydy = xdx$ or $-\frac{1}{2}y^2 = \frac{1}{2}x^2+C$ which is better expressed as $x^2+y^2 = R^2$. If you picture $\vec{G}$ the fact that circles are integral curves is expected.

The idea that you can calculate with total differentials then simply divide by $dt$ to obtain relations between rates involved is fascinating. A victory of notation. Students who understand nothing of the implicit function theorem are able to successfully implement it through little more than symbol pushing.

• Incidentally Paula, I'm sorry if I have misunderstood the type of things you're looking for. Certainly the applications of DEqns are endless... – James S. Cook Mar 20 '17 at 2:07
• thank you for a nice example. I'm just afraid that it might be too hard for high school students - but in general I'm looking for something like that. – Paula Mar 21 '17 at 12:24

Not sure the level of knowledge your students and what ODE they have had to date. I am going to assume they have had some exposure (the baby DE you get towards end of a calculus class, but not a regular survey course).'

Given that, I would do predator prey. (1) Likely to be new to them. (2) visually interesting, the graphs and all. (3) intuitively interesting problem (the eaters and the eaten), (4) is an ODE and connected rate of change problem. (5) easy example in terms of the actual manipulation

An alternate one to consider is xenon transients in nuclear reactors. It's a real system of ODEs that affects navy nuclear cores (older worse, higher recent power worse).