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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?

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    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 examples2012-02-09
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    If you don't mind internet resources, take a look at [Alan Rendall's blog](http://alanrendall.wordpress.com/). He writes about mostly ordinary and partial differential equations in their applications to mathematical biology and general relativity.2012-02-09
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    Look at http://math.stackexchange.com/questions/107264/essay-about-the-art-and-applications-of-differential-equations/107686#107686 it may be a real eye-opener.2012-02-10
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    Try Arnold's essay [*Evolution Processes and Ordinary Differential Equations*](http://math.ucsd.edu/~lni/math140/ODE.pdf), written exactly for high school students.2014-09-02
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    Here's a little big hobby project I started a few weeks ago along those lines (pun intended): https://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.2016-03-09

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