# What is the right way to order the topics in a first ODEs course?

This question is long but I am asking for educated opinions on a question of math education and for this reason I'd like it not to be closed on the grounds that it invites subjective discussion. Educational practice is not a subject on which there is a single right answer, but there is such a thing as a well-reasoned answer.

I've taught a lot of ODEs, either as a dedicated course or as a unit within a linear algebra course. I never actually took one myself, though I did read Coddington's book Ordinary Differential Equations, and I would call myself an algebraist, though not the kind that hates analysis (I enjoy teaching it at this level, actually).

My feeling is that the traditional ODEs course is not very good, but it is nearly universally agreed upon by textbooks. It looks like this:

• First-order equations of all kinds, both linear and nonlinear, with techniques such as separation of variables, integrating factors, substitutions, and exact equations (so, more integrating factors).

• Second-order equations, linear and constant-coefficient only, with an ad-hoc derivation of the characteristic equation and informal justification of undetermined coefficients and variation of parameters; the Wronskian appears here as a trick. Big presentation of the "three cases" for the characteristic equation. Then maybe variable-coefficient reduction of order.

• Mention of higher-order equations, with all of the above recapitulated halfheartedly.

• Some kind of discussion of linear systems, complicated by the fact that linear algebra is often not a prerequisite for the course so must be taught on the spot (or simply avoided, in which case this topic is incomplete). The three cases are discussed again. The matrix exponential may be introduced, and this entails another discussion of the three cases.

• There is often a single day on the existence and uniqueness theorems at some point, perhaps here, but since the proof is serious analysis, it's hard to make this sound like anything but a footnote.

• There is also often a small unit on numerical methods, which doesn't get to anything serious.

• Infinite series, possibly even including Frobenius series, are given a lot of attention.

• Laplace transforms get just as much attention, often, though typically only applied to inhomogeneous constant-coefficient equations.

• Nonlinear systems are somewhere in there, with a discussion of the phase plane and equilibrium points.

That's a lot of topics and some of them are a lot of work, and I have often gotten the impression that students find the subject entirely disunified. Once, I think they didn't even understand that the "three cases" for linear equations were actually the same but in different contexts, rather than somehow being a different technique for second-, third-, ...order equations, and systems. The same problem exists in a larger sense: almost all of the techniques overlap and students can confused about whether they even solve the equations in the same way.

So, my question is: what is the best way to actually structure this course so it teaches an idea, rather than "how to solve some differential equations"? For the record, my own preference is:

• Start with a few days of first-order linear equations presented in a way as to presage the general method used for systems, and avoid special-purpose tricks.

• Then, teach (or review) linear algebra and constant-coefficient systems at the same time, with emphasis on using a (generalized) eigenbasis to find the fundamental solutions.

• Introduce the matrix exponential as a unified method to solve an initial value problem directly, and from here, on to inhomogeneous systems, variation of parameters, and generalized undetermined coefficients. The Wronskian shows up here completely naturally.

• Now, talk about higher-order equations and the reduction to systems of first-order equations, and quickly derive everything about them from the preceding.

• Talk about how there aren't such highly effective techniques for most differential equations and use that to motivate the existence and uniqueness theorems. They should be explained somewhat, since the proof technique (Picard iteration) motivates both series and also the step methods for numerical solutions.

• Now you're free to talk about special-purpose tricks. I like to go back and do exact first-order equations and integrating factors, because that's the context in which the latter really make sense. I don't bother with reduction of order, or substitution, because they seem to have no serious purpose.

• Since I'm on nonlinear equations, I move onto nonlinear systems and don't skimp on the trajectory sketching: it's as much a solution method as writing down an infinite series. The connection with linear systems is obviously to be emphasized strongly.

• Logically, the next thing must be numerical solutions, since it is a fail-safe for getting some kind of a solution. It would be nice if there were a way to pull computer work into this.

• I don't like Laplace transforms because they are only helpful for finding explicit solutions for linear equations, but those are already solved. The subject can't be taught in a way consistent with its real significance.

• I don't like power series because they are a ton of work for dubious benefit. This is a shame, because Frobenius series are very interesting. There's probably time to do this subject anyway, if you drop Laplace transforms.

This obviously reflects one person's biases about the subject. I feel that it unifies the whole thing around linear equations and makes the abstraction of the existence and uniqueness theorems seem worthwhile. It compartmentalizes the strange methods in a way that makes it clear when to use them. It does decrease the "methods" aspect of the course, and I did do this in an honors class, so maybe that's undesirable in general. But what does the traditional structure have going for it?

• I do think you have a point in including iterative approximation techniques. Also, it occurred to me, you should look at Ostberg and Finney, see books.google.com/books/about/…. For me the larger point of this is, linear algebra, or at least a matrix theory course should be very early in the curriculum so we can properly use linear techniques in the ODE course. – James S. Cook Oct 4 '14 at 14:14
• @James Thanks for the link; I was going to inquire if anyone knew of a textbook that used linear algebra seriously. Unfortunately, that book is pretty unavailable. – Ryan Reich Oct 4 '14 at 16:26
• Indeed, I am fortunate to have picked it up from the "free book table" at NCSU. I think Rabenstein is also worth a look, his understanding of DEqns is hard not to appreciate. – James S. Cook Oct 4 '14 at 17:50

For a thoughtful view on the subject, see Gian-Carlo Rota, "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations." The tone of the linked text may be off-putting to some, but the ideas within are a good starting point.

I share many of his frustrations and agree with many of his conclusions. I've pasted the "ten lessons" below; please see the link for details on a justification of each. According to the website, Rota delivered the invited address on this topic at the meeting of the Mathematical Association of America at Simmons College on April 24, 1997.

• Lesson 1: Most of the material now taught in an introductory differential equations course is hopelessly obsolete
• Lesson 2: Reduce to a minimum the discussion of first order differential equations at the beginning of the course
• Lesson 3: Linear differential equations with constant coefficients are the bottom line
• Lesson 4: Teach changes of variables
• Lesson 5: Forget about existence and uniqueness of solutions
• Lesson 6: Linear systems with constant coefficients are the meat and potatoes of the course
• Lesson 7: Stay away from differentials
• Lesson 8: Avoid word problems
• Lesson 9: Motivate the Laplace transform
• Lesson 10: Teach concepts, not tricks

In particular, I've reduced my discussion of existence and uniqueness a little more each time I cover it, and I do believe that next time I will not present it at all.

• Wow, you would think I'd read that essay before writing my question; he anticipated all of my criticisms. I disagree with the uselessness of existence and uniqueness, but I agree with his opinion of it: I just don't think that a "psychological theorem" is without value. Also, his presentation of exact equations and integrating factors is brilliant. Too bad he has no idea what to do with Laplace transforms either. – Ryan Reich Oct 3 '14 at 22:54

I agree that the standard textbook approach is not too exciting from a pure math perspective. I would say the order of the topics is largely based on an attempt to keep the subject from getting too abstract too soon. Also, I would wager the applications which are solved by the methods also benefit from the parsing of topics.

For example, the first order problems (even nonlinear) represent interesting physics problems perhaps involving friction. These we could not solve before the clever first order techniques are given. There are fairly simple physical problems which necessitate the use of the Ricatti equation or Bernoulli equation. Really, pretty much every weird first order problem is tied to some obscure example in physics, biology or perhaps chemistry.

Then, the second order problems have much direct physical relevance. Harmonic oscillators are everywhere. The trichotomy of solutions have disparate and interesting application to the separate applications in question. Mostly I mean RLC-circuits and damped spring mass oscillators. Furthermore, the "forcing function" is an honest to goodness force at this juncture. So, discussions of superposition are really direct discussions of how the sum of the forces cause a sum of responses which eventually lead to the response of the net-force.

That said, I whole-heartedly agree the recapitulation of the constant coefficient case at higher order does not seem to be a good economy of presentation. Moreover, applications at higher order are harder to come by... this is probably a lot of the reason this material gets less air time in many courses. No application = no interest. I don't agree, I just observe this for the purpose of understanding.

Let me just briefly explain how I structure the course:

• week 1: terminology ODE vs. PDE solution, implicit vs. explicit. First order ODEs, I look at the three main methods: separation of variables, integrating factor technique, exact equations.
• week 2: geometry of DEqns, maybe play with some pplane, applications
• week 3: uniqueness, existence (by now they appreciate the question)
• week 4: theory of higher order ODEs, focus on structure of solution sets, Wronskian
• week 5: develop theory of smooth differential operators. Build solution set from kernels of your basic factors. (generalized e-vectors abound, but, no matrix exponential in sight). Discussion of complexification of problems is given.
• week 6: solve $n$-th order ODEs. All of them. We own them now.
• week 7: method of annihilators solves easy nonhomogenenous problems. Then I tackle the $n$-th order variation of parameters. Make a point to emphasize it is a general method and we get to make nice connections with the theory from week 4.
• week 8: systems of ODEs introduced, theory from week 4 in some sense repeated, but it is a bit different, see eigenvector solutions
• week 9: complex eigenvectors, again see complexification idea in play. Of course, e-vectors are not enough so we discover matrix exponential and I show them the basics of how to find real Jordan form ( I don't tell them this directly usually) for arbitrary solutions of $x'=Ax$. I make a point at the end of it to emphasize the use of our method to linearize more general problems which are not merely linear.
• week 10, 11, 12 series, singular points, Frobenius. This is hard, but they actually do pretty well. They're much worse at finding eigenvectors. (go figure)
• week 13, 14 Laplace Transforms. Here we finally find a good method to treat discontinuity. And, we get a glimpse at the dark art of distributions. Interesting connections with Green's functions can be made here and the general theory of linear systems can give joy to the purest of mathematicians. It's just hard to find in the current books.

I recommend Rabenstein. Or, for a pale echo, see my notes: http://www.supermath.info/DEqns2014.pdf Keep in mind, my main goal is computation. I do not have an audience which appreciates the difference between $\Leftrightarrow$ and $\Rightarrow$.

• with all due respect James, I started to read the PDF notes, and I found some of it quite confusing, and I understand diffeques. EG: "Thinking in terms of functions alone would be a severely limiting perspective in this course. Forbexample, if you implicitly differentiate xy3 + y2 = sin(x) + 3 then it is easy to see the equation xy3 + y2 = sin(x) + 3 defines a solution of y^3 + 3xy2 dy^dx + 2y^dy/dx − cos(x) = 0. I would rather not find the solution as a function of x in that example." You don't say why you don't want a solution, so i was left shrugging my shoulders. (sorry for formatting) – rbp Oct 3 '14 at 16:09
• @rbp I'm sure there are errors and unfortunate grammatical choices. But, I'm not sure what you're saying about that paragraph. I'm merely trying to warn there are two main types of solutions: implicit or explicit. The point of that example is that to solve for $y$ it would not be easy, yet, the equation $xy^3+y^2=\sin x+3$ serves as a solution of $y^3+(3xy^2+2y)\frac{dy}{dx}-\cos x=0$. (this is from page 9) – James S. Cook Oct 3 '14 at 16:29
• sorry if I was unclear about what was unclear! I don't understand the statement: "I would rather not find the solution as a function of x in that example" and there is no further explanation. Maybe you mean "I would rather find a solution for y than for x?" or "I'd rather have an equation that satisfies the DE, rather than a solution to the DE"? Anyway, if its not important, there's no need to respond. – rbp Oct 3 '14 at 16:37
• @rbp "I'd rather have an equation that satisfies the DE, rather than a solution to the DE" would be ok, except, I include equations the term "solutions". $y=f(x)$ or $F(x,y)=0$ both constitute "solutions" in my terminology. I realize this differs with some who insist only functions (of the form $y=f(x)$) are "solutions". – James S. Cook Oct 3 '14 at 16:41

This is not a complete answer, just a few pointers of what has worked for me.

Different students react differently to different "methods" of solving linear equations. Should one learn to find the homogeneous solution first, or do the exponential-of-the-unmotivated-integral, or...? Other solvable families have the same problem. Moreover, I want to get to the fun part (read dynamical) part of the course. If we spend time teaching methods of solution, the students get burned by mid-semester.

In my first ODE class I did a daring thing. On the first day of class the students received a 100 equations homework sheet. These 20 are linear; solve them. These 20 are separable; solve them. These are homogeneous, these are exact... The instructions were to use any method in any book that they liked. They could work together and ask me questions. The result was better than I had hoped for. After a week they could all solve equations on the spot (each using the recipes they liked), and we could get started with the really interesting portions of the course. Basically they did not think that "solution methods" was a laborious part of the course, but rather an quick application of Calculus.

I would also suggest that you take a look at the concepts of fences and funnels introduced by Hubbard and West. There is a nice article in a 1994 issue of the MAA's College Math Journal which was devoted to DE teaching. This is an awesome way to motivate a geometric understanding of DEs, and helps explain existence/uniqueness and other concepts.

• Your first two paragraphs look almost identical. It's a bit confusing. – Mark Fantini Oct 3 '14 at 12:15
• @Mark: thanks... editing mistake. – Rodrigo A. Pérez Oct 3 '14 at 14:36

Your desire for some beautiful integrated thematic math person version of course shows not thinking about two important things:

1. Your students and what they need.

2. How people learn best.

For number 1, they are mostly engineering and physics students. They benefit from exposure to the tricks of the trade so that when they have derivations and problems in their courses (fluids, EE, etc.) they have some mathematical familiarity with the tools they are using in the course.

For number 2, the beautiful sleek presentation is not always the easiest way to learn complex and new abstract material. It tends to make more sense for a second exposure than a first. Or to someone who is a professor than a student.

In terms of practical needs: basic linear first order dffy qs with integrating factors and then the second order equation for harmonic oscillator is what people need the most.

The hard thing about diffyqs is that it basically should be close to a full year course (even at a beginner level). And I don't mean making it long with Sobolev spaces and other advanced concepts. I just mean even the basic book just can't be completed in a semester. This is very different than calc 1/2/3 or any previous math courses where the course pretty much involves going over the whole text (I mean stereotypical texts).

Students hate having the feeling of not covering a whole text. Professors don't care. But students do.

But if you want a reasonably short and sweet good ODEs course, look at the first 5 chapters of Kreyszig Advanced Engineering Mathematics.

The good thing about Kreysig is that it is reasonably short. That it still covers a lot of topics. That it selects those of most importance to an engineer or physicist. Also, the order of the topics seemed to make sense. Pretty much the classical order but with efficient coverage.

Other than that, I worked through it and just felt it very easy to grok. Sorry, I am straining brain, but that is the best I can explain. Common book and worth a look. I had 5th edition FWIW.

• But if you want a reasonably short and sweet good ODEs course, look at the first 5 chapters of Kreyszig Advanced Engineering Mathematics. The good thing about Kreysig is that it is reasonably short. That it still covers a lot of topics. That it selects those of most importance to an engineer or physicist. Also, the order of the topics seemed to make sense. Pretty much the classical order but with efficient coverage. Other than that, I worked through it and just felt it very easy to grok. Sorry, I am straining brain, but that is the best I can explain. Common book and worth a look. I had 5th ed – person Jun 5 '17 at 4:00
• A service class need not be subservient: although the math department must serve the needs of its non-mathematical students, it should not compromise its mission to teach mathematics. A course called "ODEs for Engineers" may appropriately teach only a bag of tricks, though I understand that engineers are not as simple as all that, but a regular class should aim higher. Students who take a class should expect to learn the subject on its own terms, not on theirs. – Ryan Reich Jun 5 '17 at 6:47
• I also wonder what you mean by the apparently sarcastic phrase "some beautiful integrated thematic math person version". Do you not consider yourself a "math person", or do you perhaps believe yourself to represent the interests of your non-math-person counterparts? If the former, please remove the chip from your shoulder. If the latter, I appreciate the concern, which you can see that I share, since I specifically addressed the problem of students not best learning the material in my question. But do not set me and "professors" in general against students in general. – Ryan Reich Jun 5 '17 at 6:57
• I merged all your contributions into this one answer. Using the site is more convenient with a registered account: see matheducators.stackexchange.com/help/creating-accounts If you want to merge your accounts see matheducators.stackexchange.com/help/merging-accounts – quid Jun 5 '17 at 10:29
• "Students hate having the feeling of not covering a whole text. Professors don't care. But students do." Why say that? It seems to be the nature of the beast that STEM texts are almost never covered in their entirety. I've never once had a student who complained on a course evaluation that they wish I had finished the whole text. – John Coleman Jun 5 '17 at 15:36