# Differential Equation Examples for Calculus Students

I've been teaching calculus courses for a while now, and something always bothers me each time I teach it. Students always seem to have trouble connecting with the differential equation material for the following reason: I always tell them that they are one of the most important topics for applications of calculus (this is mainly a course for students in the sciences) and that all sorts of fields use them...and then all I have to tell them are things that are to a certain extent quite dull: exponential growth, Newton's law of cooling, the logistic equation, and a few other of the classics. While each of these is quite important and do have broad applications, I've never seen anyone be shocked to learn that populations of rabbits breeding in the wild grow approximately exponentially.

My knowledge of applied fields isn't terrible, but I'm still at a loss as to what plausible models I could teach them about where the global results are not immediately obvious, so I ask: what are some simple differential equations, simple enough for a freshman calculus class, which occur in the sciences and have behavior interesting enough the catch peoples' interest?

• I think it is best not to talk about ODE in calculus at all (i.e. to skip relevant sections of the book). The basic ODEs you are describing will be retaught in science/engineering courses anyway, and anything deeper would have to wait until a math ODE course. Jan 19, 2012 at 13:42

This worksheet from Dept. of Mathematics in University of Washington guides through two examples using differential equations: forensic mathematics and spread of rumor.

Also the internet magazine called +plus magazine has many examples of applications such as mathematics of traffic jam.

• Thanks, the +plus magazine link started me thinking and hopefully I'll be able to come up with something from that. Jan 20, 2012 at 1:02

People have been shocked precisely that breeding rabbits grow exponentially. In particular the Australians were in the 19th century. In 1859, some guy named Thomas Austin brought 24 rabbits from England so he could amuse himself by shooting at them. In a few years the population had grown to hundreds of millions of rabbits, decimating Australia's native ecology. Oops. (link: http://en.wikipedia.org/wiki/Rabbits_in_Australia)

Another simple example is money. You invest \$1,000 in a bank account earning 10% compounded continuously; how much do you have after 30 years? The answer: A whole heck of a lot of money. I think it is not all that intuitive to many people, but the calculus unambiguously proves it.

I have also covered predator-prey models (which are covered in Stewart's Calc I book.) The differential equations are simple to explain, but draw the oval in the phase plane, and to a beginner the behavior is really rather stunning: the populations oscillate! I think the most natural naive guess is that the populations' behavior would settle down to some equilibrium, and this is not what happens. Probably you can even compare the behavior of the ODE's with data from some real world example (I think Stewart has a graph of such data).

Perhaps you're teaching some kind of honors class where this is old hat to the students; I confess that I'm somewhat disagreeing with the premise of your question, so perhaps I've misunderstood the situation you're in. But in my experience, there is a lot of meat in the very simplest examples.

• The type of student I've typically encountered has had some exposure to calculus ideas before, at least enough that the power of exponential growth is no longer novel to them. Predator-prey models like the Lotka–Volterra equation are exactly the types of examples I'm seeking! The fact that natural populations oscillate is quite surprising and is precisely the type of behavior I seek (although I've always feared that introducing concepts like the phase plane to illustrate the behavior might do more to confuse than illuminate). Jan 19, 2012 at 4:28
• I think no-one can have so much exposure to the power of exponential growth that there's no novelty left in it. I know full well mathematically that, if the population of algae in a lake doubles every hour, and the lake is full at the end of the day, then it was only half full as late at 11 PM—but I'm still astounded by it. Apr 28 at 1:02

Airy equation! The optical phenomenon of the rainbow was already explained in ancient times by means of reflection and refraction of light within the spherical droplets of rain. But, after Snell's law of refraction, a complete model was available, which gives account of any quantitative aspectof the phenomenon, including the multiple rainbows. Describing the intensity near an optical caustic, led J. B. Airy to the ODE (now called Airy equation or also Stokes equation) $$\ddot u-xu=0$$ whose solutions are the special functions Ai(x) and Bi(x). Here's the original Airy's memory on the Transactions of the Cambridge Philosophical Society (Part 3, XVII).

I am not really familiar with the US system, so this might be too advanced.

Maybe it would be interesting to discuss partial differential equations that can be reduced to ordinary differential equations, if one uses symmetries. Take the hydrogen atom for example: You get a decomposition in a radial and spherical part, the second can be solved by separation of variables. So you get three ordinary differential equations and their solution will give you a description of the hydrogen atom in non relativistic quantum mechanics.

There are of course a lot of other examples from physics, where similiar reasoning is applied.