# Why do we study ordinary differential equations?

What is a good answer to the question: Why should one study ordinary differential equations?

I would give the answer: ODEs are used in many models to determine how the state of this model is changing (regarding time or another variable). Thus, ODEs are important for many scientific fields because they arise whenever a relation is given for the change of a model/system.

This is one application of ODEs (though a very important one). Am I missing another application or are there other reasons for studying ODEs?

• As far as physics and engineering and chemistry are concerned, pretty much every "general law" is (or often is) expressed using one or more differential equations: Newton's law of cooling, Maxwell's equations, Newton's laws of motion, fluid dynamics equations, equations in plasma dynamics, equations in stellar dynamics, Hook's law, Schrödinger's equation, acoustic wave equation, equations in chemical kinetics, equations in thermodynamics, Einstein's equations for general relativity, ... The list is seemingly endless, both in length and in variety! – Dave L Renfro Jul 28 '16 at 14:38
• (2 to 3 hours later) When revisiting this site I happened to notice the question asked about ordinary differential equations, something I missed when I quickly wrote down the applications in my earlier comment. Many of the examples in that earlier comment are partial differential equations. However, there are still plenty of other examples of ordinary differential equations they can be replaced with, and besides (this might be cheating), one of the common elementary ways of solving partial differential equations is by separation of variables, which takes you to ordinary diff. equations. – Dave L Renfro Jul 28 '16 at 17:32
• @DaveLRenfro: Since my motivation is to write an introductory article what differential equations are and why they are studied your comment really helped me... :-) – Stephan Kulla Jul 28 '16 at 17:38
• I have added the (differential-equations) tag; I also wondered about removing the (calculus) and/or (mathematical-analysis) tags, but hesitated to do so, as differential equations are often introduced in calculus and do fall under mathematical analysis, even if not necessarily taught in a course with that name. – J W Aug 12 '16 at 15:20

ODEs are used in many models to determine how the state of this model is changing (regarding time or another variable).

[…]

Am I missing another application […]?

This may be somewhat pedantic, but I think that your answer is so generic that it covers every possible application per construction:

• The fundamental formulation of an ODE has the change of the state as the unknown component (except if you flip it and want to seek an ODE that has a given solution) – so every application has to be about this.

• Every application of mathematics is inevitably based on models (i.e., mathematical theories of reality). Of course, some of these models may be so universally accepted or have become their own fields that we are not used to treating them as such anymore.

So, apart from this, there can only be inner-mathematical applications, such as being a background for more complex differential equations (PDEs, DDEs, …). As those are probably not the kind of motivation that appeals most to your students, I suggest to address the topic of ODE models in more details:

• Give some specific examples from different fields such as equations of motion (physics), predator–prey models (ecology), neuron models (biology/medicine), and so on.

• Mention the field of dynamical systems, large parts of which are about understanding and classifying the solutions to ODEs.

• Mention that physicists have developed an entire arsenal of formalisms aimed at obtaining (not solving) ODEs.

Of course I agree that one motivation for studying ODEs is that they have applications. But it might be useful to also point out another fact that students do not always think about: ODEs are often the most succinct way of defining functions.

You could use the exponential function as an example, because your students will have seen it already. Ask them to give you a precise description of the exponential function. They might start by telling you about the shape of its graph, but under further questioning they will quickly realise that they would need to give you an infinite amount of information to specify the graph exactly, until they hit upon the idea of saying that at every point its slope equals its value, so they formulate their first ODE. At that point students often get excited and wonder "what if the slope is equal to the square of the value?", i.e., they start looking at other ODEs by themselves. Unfortunately it is then sometimes a bit difficult to tell them to take it slow and stick with linear ODEs for a while.

• +1 I always point out the "slope equals value" property of $e^x$ but have never thought to use that as a way to talk about ODEs. I'll be using this from now on. – Brendan W. Sullivan Aug 1 '16 at 19:38

Echoing many of the points made in the other excellent answers... but just to focus on one (to me very significant) aspect: differential equations characterize functions in (physically?) operationally (as it turns out) meaningful ways. Other answers have noted the "dynamical" aspect, and @gustav noted in particular that some things are best described by being solutions to a differential equation.

That characterization is certainly true in an immediate sense, based on physics and mechanics. But, to-me-surprisingly, the same definitiveness of characterization is important in "pure mathematics" (for me, simply "mathematics", in the same way that "applied language" is just "language", whether it is poetic or ugly or not-to-the-point).

I'd tend to claim that the next level of "description" of functions after thinking (after Lagrange and many other luminaries) of functions as inevitably lying in some completion (not in their terms) of polynomials, was to tell the ODE they satisfied. Manifestly, this is the case for many of the classical special functions.

At the same time, due to the subtly structure situations in which those special functions arose, they did also have significant features related to their various origins. Among other very, very weighty manifestations, the Casselman-Wallach subrepresentation theorem is a (fairly stunning) generalization of the early 20th century (but highly non-trivial) asymptotics results.

So, as a fairly fancy example, much of the contemporary theory of automorphic/modular forms relies upon such things. Which is to say that much modern number theory relies...

That is, "description (of an important thing)" is quite often "by a differential equation".

• Continuing your theme, there are algebraically transcendental functions (satisfy algebraic ODE's, and greatly expand the notion of "elementary function" in symbolic integration theory) and transcendentally transcendental functions (don't satisfy an algebraic ODE), which I would imagine you know about but others might not. See my answer at Expanded concept of elementary function?. – Dave L Renfro Aug 3 '16 at 18:37
• @DaveLRenfro, indeed. For that matter, even more immediately, and maybe not known to absolutely everyone, Liouville's and others' results about (non-) integrability in elementary terms show that even the trivial differential question $u'=f$ with $f$ elementary leads outside the usual. – paul garrett Aug 3 '16 at 19:46

Models, as you mention, are a huge source of applications. One can mention some in physics (free fall, radioactivity, pendulum, ...), in biology (cell growth), in chemistry (kinetics) among other. The beauty of the subject, tying together several branches of maths that student tend to consider separate, is of course also a strong motivation.

Let me stress three further points: a particular example in economics, the educational virtues of ODE, and its applications in mathematics.

### The second law of capitalism

An example I like a lot (and used in my calculus lectures) is from Economics; the reason I like it is that it is a simple, static law which is better understood thanks to an ODE model. In Picketty's Capital in the XXIth century (where I read it) is stated as the second law of capitalism the relation $$\beta =\frac{s}{g}$$ where $$\beta$$ is the ratio of the stock of capital $$C$$ to the annual income $$R$$ produced in the considered economy, $$g=R'/R$$ is the economic growth and $$s$$ is the proportion of income which is saved, defined by $$C'=sR$$.

Now, this law is a limiting law, which may only hold true at equilibrium. But what if $$g=0$$? What if $$g<0$$? It then makes no sense whatsoever!

Now, one can consider where this law comes from. Assuming $$s$$ and $$g$$ are constant, since $$\beta := C/R$$, we get $$\beta' = \frac{C'}{R}-\frac{CR'}{R^2} = s-\frac{gC}{R} = s-g\beta$$ Now, studying the ODE $$\beta'=s-g\beta$$, one sees that the only stationary solution is $$\beta=s/g$$, the second law of capitalism. But studying it in detail, one sees how the above absurd cases now make sense: if $$g=0$$, then there is no stationary solution, and $$\beta$$ increases linearly. If $$g<0$$, then there is no positive stationary solution and $$\beta$$ increases exponentially fast. Then, one can also study the speed at which the stationary solution is approached, which economically is important to decide whether the stationarity of $$s$$ an $$g$$ can be assumed (their characteristic variation time should be much larger than the characteristic time of $$\beta$$).

### Functions as mathematical objects

The notion of function is a difficult one to grasp for student, and a central one. They have mainly learned to use them as processes, but at university we insist that they consider them as mathematical objects on their own. They have already applied processes to functions (e.g. differentiation), but we go much further (defining and using properties of function, operations on functions, relations on functions such as asymptotic comparison, etc.). ODEs are an important step in this process: we now ask them to consider functions as unknowns. Of course, other equations can do that (functional equations) but only few of them can be dealt with. (The very nice and short book by Alcock and Simpson was important for me to figure that out, I warmly recommend it). This is an important step on the path to further mathematical abstraction.

### Applications in matheamtics

One should not forget that ODE are important not only on their own, but also as a crucial framework for much other mathematics. I am a differential geometer, so I need them:

• to construct diffeomorphisms from vector fields,

• to construct the exponential map of a Riemannian manifold,

• to relate curvature and volume, ...

Mention Newton's Second Law. $F = ma$ is an ODE (a trivial one if $F$ is constant, but nontrivial when $F$ or $m$ depend on position or velocity). Since one of the fundamental laws of nature is a differential equation, it makes sense that you should study differential equations if you want to understand nature. I don't know who said it first but I've always liked the quote "The laws of nature are expressed as differential equations"

• Actually I used exactly your example for the introduction of ODEs :-) – Stephan Kulla Jul 28 '16 at 17:41

We should study Ordinary Differential Equations because it is beautiful mathematics which clearly illustrates the wondrous connection between analysis and algebra. Linear algebra, or perhaps matrix theory, when combined with calculus provides abstractions of ordinary functions which behave in ways similar yet fantastically different than ordinary functions. The matrix exponential in particular, we find simply from combining natural questions in calculus with those which arise from structure theory in linear algebra. Working out the modification for the laws of exponentiation leads straight to the Baker-Campbell-Hausdorff relation and hence the rudiments of Lie Algebras and Lie Groups.

Yes, there are applications. But, ODEs is a natural topic which is interesting on its own.

• Thanks, that is really a new view on ODEs I haven't considered before... – Stephan Kulla Jul 28 '16 at 17:42
• Perhaps a good idea to expand BCH to Baker-Campbell-Hausdorff. I had to think for a moment to recall it. – J W Jul 28 '16 at 17:52
• @JW agreed and edited. – James S. Cook Jul 28 '16 at 19:38

I'm going to take a more pragmatic view:

We study ordinary differential equations, because we can study ordinary differential equations.

That is, it is a subject that is well amenable to study, a fairly accessible theory, and a wealth of methods of exact calculation.

And then, once we understand ordinary differential equations, we can then start look for ordinary differential equations — like the old adage goes, if you have a hammer, everything looks like a nail. While some problems might naturally take the form of an ordinary differential equation, we solve other problems by finding ordinary differential equations that give us information about them.

My slant is more tactical than the general idea of modeling. An ODE course is useful for students in engineering and physics because they will need the concepts to follow derivations and work homework problems in their majors courses. In mechanics, E&M, circuits, heat transfer, and many, many more.

Also a good idea but not as vital for chemists (helps with p chem class). I'm sure someone will point out the pretty predator prey problem of biology, but this is a sidelight IN BIOLOGY. Biology has some systems to it, but is much more descriptive. Obviously English majors and the like don't need it!

By "need it", I mean in a very utilitarian manner. Not "my life was enriched", but "if I didn't have this, it would be hard to do my majors courses".

The benefit of having a course on the math involved in a topic you are learning (say in physics or engineering) is that you can concentrate more on the topic and less on the math. It can become tough to learn a new topic, and new math (and perhaps new notation) all at the same time.

In addition, math courses tend to teach more with x's and y's, which is just a less complicated fashion than learning math with a bunch of constants and variables signifying particular quantities (energy, force, etc.) The ability to move between different ways of looking at the same thing is helpful for pedagogy and practice. So being able to understand something in terms of just x and y equation solving as well as in the context of (say) thermodynamics is helpful in coming to grasp with the thing.

A simpler (than ODE) example would be knowing basic algebra through the quadratic and logs and exponentials prior to chemistry course where you work stoichiometry, rate, and equilibrium problems. Those problems are bad enough with all the unit analysis, chemical concepts, quantities (many not simple numbers). You don't want to be having to learn the math itself at the same time.

I would also note that some engineering profs really don't want you pushing a lot of modeling at the kids. That can be done fine in the engineering courses themselves. But get the basic concepts, tricks, etc. across to the kids in terms of x and y, so they have a better time of surviving their engineering courses (which can be tough on the kids).

Oh...and nail the second order linear constant coefficient (homogenous and nonhomo). That thing comes up everywhere. As the quadratic equation is to "ICE" chemical equilibrium problems, so is the harmonic oscillator for basic engineering and physics.

• I think the way to address this is some very practical list of (1) course pre-reqs and (2) applications (many of them within major courses of (1). Many of these are even covered to an extent in standard ODE books, but some are additional. I don't have the whole formed list but am really surprised that someone working on a textbook doesn't have at least an initial draft/view on this, but asks about it so de novo. – guest Apr 8 '18 at 3:03

After going through several integration techniques and standard forms of integrals, students wonder "Why are these needed?". Once answer to this question may be provided by showing real applications using ODE. The book "Chases and Escapes" by Paul Nahin contains many applications of ODE that would convince the student of the utility of studying ODE.