New & interesting uses of Differential equations for undergraduates?

Question

I'm teaching an elementary DE's module to some engineering students. Now, every book out there, and every set of online notes, trots out two things:

1. DE's are super-important, vital, can't live without 'em, applications in every possible branch of applied mathematics & the sciences etc etc
2. Applications: population growth (exponential & logistic), cooling, mixing problems, occasionally a circuit problem or a springs problem. Oh - and orthogonal trajectories, so that you can justify teaching non-linear exact equations.

I can't believe that these same applications are still all that educators use for examples. Surely there must some interesting, new applications, which can be explained at (or simplified to) an elementary level? Interestingly, most of these "applications" are separable. Where are the linear non-separable equations; the linear systems?

I've been searching online for some time now, and remarkably enough there's very little out there. So either educators are completely stuck for good examples, or all the modern uses are simply too difficult and abstruse to be simplified down to beginners level.

However - if there are any interesting new & modern uses of DE's, explainable at an elementary level, I'd love to know about them.

Answer 1

A novel application can be found in 'When zombies attack! Mathematical modelling of an outbreak of zombie infection' by Munz, Hudea, Imad, and Smith? - you can find this paper here, together with several other zombie-themed papers.

The mathematics used doesn't go beyond systems of linear differential equations and numerical methods. Even if the students can't understand how to solve the system, you can use this example to demonstrate the qualitative nature of modelling with DE's. In addition, you can then demonstrate the need for numerical methods - even people doing research can't always solve every system of DE's they come across.

I've only ever taught differential equations once, and that was at a very elementary level, but I showed them this paper just for fun; they seemed interested enough. I pointed out things that appeared in a genuine research paper which looked incredibly similar to things we had been dealing with earlier in the lecture. The reference list is also good for a laugh.

Answer 2

Mass action in chemistry and enzyme kinetics could be interesting. Pattern formation models in biology can also rely on DEs (often PDEs). Biology makes use of many ODE models, and their qualitative behaviour can be related to observable phenomenon (which is a great motivator, in my opinion).

Calculus of variations is also a fountain of useful and interesting differential equations, and the basic EL equation can be derived with a bit of elementary calculus (integration by parts).

Answer 3

Here are some items to explore.

• Tire Modeling
• Biological Modeling
• Recent Advances on Methods and Applications of Nonlinear Differential Equations
• Mechanical Vibration Analysis
• Fluid Dynamics and Heat Transfer
• Earthquake Analysis
• Recent-advances-in-differential-equations-and-mathematical-physics
• Differential Equations and Control Theory

Answer 4

There's lots of good stuff in Taubes, "Modeling Differential Equations in Biology" http://books.google.com/books?id=Y464SAAACAAJ

Answer 5

You may find this interesting that the ODE theory is getting involved well in studying Avalanches. See here, here and here for example.

Answer 6

I have the same problem. Until now, old books have been my best sources of interesting problems. The most interesting applications to diff. eq. I have found are:

• Time of death of a corpse (a heat transfer problem in disguise). Found in Boyce & Diprima, 4th edition, this problem is interesting, but your students need to master solving the equation y' + ay = b.

• Alcohol level in the blood; both when drinking and during the hangover. Google has lots of examples. (Again, you need to be able to solve y' + ay = b)

• How fast a rumour spreads. The same equation as a logistic diff. equation.

Hope this helps.