# Essay about the art and applications of differential equations?

I teach a high school calculus class. We've worked through the standard derivatives material, and I incorporated a discussion of antiderivatives throughout. I've introduced solving "area under a curve" problems as solving differential equations. Since it's easy to see the rate at which area is accumulating (the height of the function), we can write down a differential equation, take an antiderivative to find the area function, and solve for the constant.

Anyhow, I find myself wanting to share with my students what differential equations are all about. I don't know so much about them, but I have a sense that they are both a beautiful pure mathematics subject and a subject that has many, many applications.

I'm hoping to hear suggestions for an essay or a chapter from a book--something like 3 to 10 pages--that would discuss differential equations as a subject, give some interesting examples, and point to some applications. Any ideas?

• The first pages of Hirsch & Smale's "Differential Equations, Dynamical Systems & An Introduction to Chaos" might be useful. It is fairly easy and gives some motivated, applied and qualitative examples – Bruno Stonek Feb 9 '12 at 0:00
• If you don't mind internet resources, take a look at Alan Rendall's blog. He writes about mostly ordinary and partial differential equations in their applications to mathematical biology and general relativity. – Willie Wong Feb 9 '12 at 0:58
• Look at math.stackexchange.com/questions/107264/… it may be a real eye-opener. – Artes Feb 10 '12 at 1:52
• Try Arnold's essay Evolution Processes and Ordinary Differential Equations, written exactly for high school students. – Marius Kempe Sep 2 '14 at 17:45
• Here's a little big hobby project I started a few weeks ago along those lines (pun intended): slopefield.wordpress.com – I'm not sure if I'm ever going to finish building something "final" out of it, but I think you're going to find a lot of material for an essay in there, if you're still interested. – Bogdan Stăncescu Mar 9 '16 at 21:38

Differential equations is a rather immense subject. In spite of the risk of overwhelming you with the amount of information, I recommend looking in the Princeton Companion to Mathematics, from which the relevant sections are (page numbers are within parts)

• Section I.3.5.4 for an introductory overview
• Section I.4.1.5
• Section III.23 on differential equations describing fluids (including the Navier-Stokes equation which is the subject of one of the Millennium problems)
• Section III.36 especially on the heat equation and its relation to various topics in mathematical physics and finance
• Section III.51 on wave phenomenon
• Section IV.12 on partial differential equations as a branch of mathematics
• Section IV.14 on dynamical systems and ordinary differential equations
• Section V.36 on the three body problem
• Section VII.2 on mathematical biology

Some of these material may be too advanced or too detailed for your purposes. But they may on the other hand provide keywords and phrases for you to improve your search.

I strongly recommend to read a review paper with many interesting references therein :

PDE as a Unified Subject by Sergiu Klainerman.

An essay on partial differential equations written by a leading expert in the field, for anyone attemping to know more on the subject as well as to those who would like to get a grasp of interactions between Mathematics and Physics.

Another very interesting article dealing with certain aspects of differential equations (to some extent) and teaching is V.I. Arnold's On teaching mathematics (it may be worth to read if you are a teacher). A bit more detailed but still very clear is his essay “Mathematics and physics: mother and daughter or sisters?" (check another sites if you can't download it).

• I should add that Arnold's introduction to ODE is great reading (he tried to keep the examples very simple but powerful with emphasis on flow, topology, singularity and so on). – Raymond Manzoni Feb 12 '12 at 21:44
• Yes, I can say the same of his another books like "Mathematical Methods of Classical Mechanics" and "Geometrical Methods in the Theory of differential equations". – Artes Feb 12 '12 at 22:01