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Here I mean a differential equation of the form $y'=f(x,y)$ where for some $\alpha$, we have $f(tx,ty)=t^\alpha f(x,y)$ for every $t$. I have no idea why this topic seems to appear in every ODE curriculum. Are there important applications of such equations, or is the idea perhaps simply to collect one a few cases in which we can actually solve some nonlinear equations?

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Introduction

This is some sort of “apology” for teaching homogeneous ODEs. I think there’s a certain beauty and simplicity to them. That beauty is overlooked in most textbooks, which Rota sourly criticizes. Even one of my favorite older textbooks (G. Simmons), which does a better job for conceptual and qualitative development than its contemporaries, merely frames homogeneous ODEs as “at the next level of complexity” above separable ones. While Rota gives me the impression that taking or teaching introductory differential equations is hopeless, V.I. Arnold’s textbook has a much more exciting attitude toward a reformed differential equations course, but it is at too high a level for an introductory course in most US curricula. Shirley Yap (here and here) does a good job of starting with material within reach of second or third year students.

One of the nice things about homogeneous ODEs is that it allows students to see mathematics at work. In 35 years of teaching, I have taught differential equations only about a dozen times, but I have never taught an engineering student in it. I have taught students who mainly do not become mathematicians or even mathematics majors, although in differential equation it might be mostly mathematics majors. Nonetheless, they are students who find, sometimes to their surprise, that they appreciate how mathematics works. At this level, the logic of a proof is ghostly to most students and does have any real life to it, until it can be attached to concrete examples. Richard Hamming wrote that the purpose of computing is not numbers but insight. If you show them, the student can see how mathematics works in homogeneous ODEs.

There is a beauty to mathematics that is accessible to a child. Ours brains a wired to make mathematics out of ordinary experience. That beauty can grow continuously on through a Ph.D. program and into the life of a professional mathematician. I agree with whoever says that the beauty a Ph.D. sees is difficult to translate to an average college student fresh out of high school. Such students have their own ability to see beauty in mathematics. The instructor should strive to understand that. It is more difficult than harkening back to one’s days as a teenager, because you made your choices to pursue mathematics for reasons that probably do not apply to most of your students. If you succeed in showing them the beauty, they may follow you in your wake for a little while. Eventually, one hopes, the things they are passionate about will guide them, and many will choose a path other than mathematics.

It’s time to bring this introduction to a close. It should be obvious to any teacher that they have to consider their actual students. I’ll present some things that make homogeneous equations interesting. Some of these ideas are really for the instructor and may be too much of a stretch for your students. But the instructor may understand more than the student. All the ideas are not applications, but they give reasons why homogeneous ODEs should be part of the curriculum, some of which reasons may be reasons why it is still a part of the curriculum.

Background

Before we start, let’s set up a few ways to look at homogeneous equations. The first is the ODE in derivative form $$y’=f(x,y) \tag{1}$$ where $f(tx,ty)=f(x,y)$ is homogeneous of degree zero. Taking $t=1/x$, we see $f(x,y)=f(1,y/x)={\tilde f}(y/x)$ can be reduced to a function of one variable ${\tilde f}(m)$ with $m=y/x$. We can also express the ODE in differential form $$M(x,y)\,dx + N(x,y)\,dy = 0 \,, \tag{2}$$ where $f(x,y)=-M(x,y)/N(x,y)$ and $M$ and $N$ are defined up to a common factor and homogeneous of the same degree: $$M(tx,ty)=t^a M(x,y),\quad N(tx,ty)=t^a N(x,y) \,.$$ Finally we can write an equivalent autonomous system $$\eqalign{ {dx\over dt} &= -N(x,y) \cr {dy\over dt} &= \phantom{-}M(x,y) \cr } \tag{3} $$ In this set up, the (phase) velocity field $(-N,M)$ is orthogonal to the vector field $(M,N)$ in (2), and $M$ and $N$ are likewise defined up to a common factor. Both fields are homogeneous and define tangent and normal directions to the phase curves at each non singular point.

I will call the differential equation $$y’=f(x) \tag{4}$$ the basic differential equation, which is solvable by quadrature (that is, the solution can be expressed in terms of an integral): $$y = \int f(x)\;dx + C \,.$$ Of course, the student may still be disappointed if they cannot calculate the integral. The basic equation (4) is characterized by the fact that its direction/vector field admits the symmetry given by “vertical” translation; that is, the equation is invariant under the transformation $$(x,y) \longmapsto (x, y+C) \,,$$ which translates the dependent variable $y$ and leaves the independent variable $x$ fixed. A way to say this graphically is that for all vertical lines $x=a$, each solution curve crosses the line $x=a$ with the same slope. All symbolic methods of solving ODEs come down to transforming an ODE into such a form. One takes a shortcut in separable ODEs because you can get from $dw = dy/g(y) = dx/f(x)$ to the solution without actually writing out the transformation to the variable $w$, which in some cases is hard to solve explicitly. When you can see for an ODE of the general form (1) and a new variable $u=u(x,y)$ that each solution curve crosses the curve $u=a$ with the same slope, then you have found a way to solve the ODE. In the case of separable equations, the slope varies according with the transformation to $w$ that is skipped. In the case of a homogeneous ODE, it is scaled as follows.

One of the most common variable transformations to consider is $u = \ln x$ or $u = \ln y$. It converts scaling by $t = e^C$ into translation by $C$. Translation and scaling (and perhaps its cousin, rotation) are very common symmetries in the sciences. It is also connected to the differential $du = dx/x$ representing relative growth. Note the substitution $u = \ln x$ is just one of a family of substitutions $u= \ln ty = \ln y+C$ that solve the ODE; we tend to pick a convenient one.

Remarks on homogeneous equations and scaling

First, students like solving specific problems. Having some at hand is useful for building their confidence. Homogeneous ODEs are fairly easy but nontrivial problems. However, $e^{y/x}$ is a what-the-heck-is-that sort of function bred only in the captivity of a diff. eq. book.

Students do need to understand substitutions and transformations of ODEs. And homogeneous ODEs are fairly easy but nontrivial problems in this context, too.

In physical science, if homogeneous ODEs arise, then $x$ and $y$ normally have to have the same physical dimensions. This is rare in science, but we will come back to this. One exception is that reduction of order transforms the harmonic oscillator to a first-order homogeneous ODE; see the remarks on linear equations below.

Autonomous systems from physical science in the form (3) sometimes become a homogeneous ODE of the form (1)/(2) when the dimension time is eliminated. Such a physical problem came up in the last few days on Mathematica.SE in a pair of related Q&A here and here.

In geometry, it is more common that $x$ and $y$ have the same dimensions, say, length. One can find applications in geometry. For instance, the orthogonal trajectories of a homogeneous ODE are the solutions of another homogeneous ODE; so both equations are solvable, provided one can compute the integrals that arise.

Geometric interpretation of the substitution for homogeneous equations: A homogeneous ODE in terms of $y(x)$ can be written as a separable ODE in polar coordinates $dr/r = g(\theta)\,d\theta$ in terms of the dimensionless angle $\theta$ or in the form $dx/x = g(m)\,dm$, where $m=\tan\theta$, in terms of the dimensionless slope of the line through the origin and $(x,y)$ (the standard substitution).

The scaling transformation $(x,y) \mapsto (tx,ty)$, $t>0$, is a symmetry of a homogeneous vector field, and a homogeneous ODE is invariant under scaling: $$ {d(ty) \over d(tx)} = f(tx, ty) \Longrightarrow {dy \over dx} = f(x,y)\,.$$ This means that the slope is constant along radial lines. Therefore we can use the radial lines to construct a new independent variable $z$, such as the slope $z=m$ or polar angle $z=\theta$, and the scaling symmetry implies the dependent $u$ variable will appear in the transformed ODE as $du/u = g(z)\,dz$. For instance, with $y=mx$, the ODE (1) becomes $${dx \over x} = {dm \over f(1,m) - m} \,.$$ With ${x, y} = (r \cos\theta, r \sin\theta)$, the ODE (1) becomes $${dr \over r} = {f(\cos \theta, \sin \theta) \sin \theta + \cos \theta \over f(\cos \theta, \sin \theta) \cos \theta - \sin \theta} \, d\theta\,.$$ My assertion here is that it can be seen that these substitutions will reduce the ODE (1) to quadrature in the same way the basic equation (4) , or a separable equation, can be seen to be solved by quadrature. What I mean is that students can be taught to see it. They do have to learn the ideas in order to improve their perception. The intro section to the web book, Yap, Visualizing and Utilizing Symmetries of Differential Equations, has some nice animations scaling a homogeneous ODE.

Now we come to ideas that go beyond a typical first course in ODEs in the US. The first idea, quasi-homogeneous ODEs, is accessible and is less important than homogeneous ones, about which the OP already expresses doubts. The rest provides a broader context, which the instructor may be able to use from time to time.

Quasi-homogeneous or weighted homogeneous ODES are a generalization of homogeneous ones. (I prefer “weighted” but “quasi,” while more opaque, seems more common.) An ODE is quasi-homogeneous if it is invariant under the scaling transformation $$(x,y) \longmapsto (t^a x, t^b y) \,.$$ Instead of $m = y/x$, the equation (1) is transformed into a separable ODE by $u=y^a/x^b$ or $u = x^b/y^a$. For instance, $dy/dx = (x+3y^2)/(y^3-2xy)$ is quasi-homogeneous with weights $a = 2$, $b = 1$; substitute $x = uy^2$ and one obtains a separable equation. Geometrical problems might be quasi-homogeneous if the variables have different dimensions such as length and area, but I can’t recall any.

Transformation of solutions. Symmetries of an ODE map a solution to a solution. It is sometimes possible to construct the general solution from a particular solution by applying the symmetry group.

Dimensional analysis. Despite the fact that models in science deal with quantities whose dimensions are not homogeneous, dimensional analysis can sometimes be used to find dimensionless variables. I don’t know any good examples at the intro level that are not already separable. However there is one application that might be worth mentioning, the Prandtl-Blasius problem of viscous flow over a flat plate. It involves a system of PDEs in space. Dimensional analysis can be used to reduce the problem to solving the Blasius equation $2y’’’ +y’’y=0$, which cannot be solved symbolically. However, it is quasi-homogeneous ($a=-b=1$) and has other symmetries, so that just a single numerical solution can be mapped to any other solution symbolically without any further numerical methods. (See Bluman and Anco, Symmetry and Integration Methods for Differential Equations.)

Infinitesimal generator. There is a connection between the symmetries of a differential equation and the substitution that transforms it into an ODE solvable by quadrature. First we need to write the symmetries in the form of a one-parameter group of transformations, $(x,y) \mapsto (x^*(s), y^*(s))$, with say $(x^*(0),y^*(0))=(x,y)$ being the identity transformation and the composition of the transformations for $s=s_1,s_2$ being equal to the transformation for $s=s_1+s_2$. For homogeneous scaling, we have $t=e^s$ and $(x^*(s),y^*(s))=(e^s x,e^s y)$. Then the infinitesimal generator is the differential operator $$X = \left.\left({\partial x^* \over \partial s}\, {\partial \over \partial x} + {\partial y^* \over \partial s}\, {\partial \over \partial y} \right)\right|_{s=0} \,.$$ This is the directional derivative $v_0 \cdot \nabla$ with respect to the (non-normalized) initial velocity vector $v_0$ at $(x,y)$, where the “velocity” is given by $v_0=n \left.({\partial x^* / \partial s}, {\partial y^* / \partial s})\right|_{s=0}$. Then the substitution we seek is one that satisfies $$Xu=0 \,.$$ Intuitively, this ensures that locally the curves $u=a$ cross solutions curves transversely; recall the characteristic property of the basic equation. For homogeneous scaling, $$X = x\,{\partial \over \partial x} + y\, {\partial \over \partial y} \,.$$ One can check that $u=y/x$, $u=\theta=\arctan(y/x)$, and more generally any $u=u(y/x)$ is a solution to $Xu=0$. Thus there is a beautiful theory that geometrically connects up what may seem a hodgepodge of algebraic tricks for finding solutions.

Other connections to scaling symmetries

Linear homogeneous ODEs ($f(x,y)=P(x)\,y$) are invariant under the scaling $(x,y)\mapsto(t^0 x,ty)$. The substitution with weights $a=0,b=1$ is trivial, $u=x$, because the ODE is already separable into the form $dy/y = P(x)\,dx$. For higher-order equations with constant coefficients, the substitution $u = \ln y$ or $y=e^{u(x)}$ suggested by the scaling leads to a differential equation for which $u(x) = m x + b$ “obviously” yields an algebraic equation in which $b$ does not matter. (The standard method is obvious enough, at least after trying it once or twice, but people claim it’s an unmotivated trick.) Second-order linear autonomous homogeneous ODES such as the damped harmonic oscillator equation become first-order nonlinear homogeneous ODES with the standard reduction of order transformation, $\dot x = v$, $\ddot x = v \, dv/dx$: $$a\,{d^2 x \over dt^2}+b\,{dx \over dt} +c\, x==0 \longmapsto a\,v\,{dv \over dx} + b\,v+c\,x==0 \,.$$

The Euler equidimensional equation has an added symmetry. Being linear, it is invariant under scaling $y$. But being “equidimensional,” it is also invariant under scaling $x$. Unlike a homogeneous equation, the equidimensional equation is invariant under scalings of $x$ and $y$ separately. Picking different substitutions for each, $x = e^u$ and $y=e^{ku}$, the differential equation imposes a relation on them; that is, we can plug in $y=x^k$, the standard ansatz which comes from eliminating $u$ from our two substitutions, to determine $k$ by plugging into the ODE.

Summary

Pros:

  • Substitution is an important technique. Practice is good.

  • Reduction to separable equations and finding implicit solutions are also important experiences for building confidence.

  • Homogeneity, scaling, the exponential function. Scaling symmetries are common and are connected to the importance of logarithms and exponential functions in the mathematical sciences. Homogeneous ODEs are a somewhat special case, but they illustrate the scaling symmetries in an accessible, calculable way. They are certainly the easiest example (other than the basic equation) with which to show the role of symmetry.

  • Scaling symmetries are also important throughout differential equations.

  • Homogeneous functions are important in other branches of mathematics.

Cons:

  • There are few practical applications at the intro level. Hence it is hard to motivate as an important general technique.

  • Homogeneous ODEs are a special case and perhaps not common enough to justify the time spent on them.

  • Introducing homogeneous ODEs merely for the sake of having something to solve seems stupid. Except that when people are asked to do things for seemingly stupid reasons, some of them keep asking themselves why and discover things on their own, which is good but not good enough.

  • Few textbooks support this approach.

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  • $\begingroup$ Wow, that is much more than I expected to get out of this question, thank you! I will have to try to reconstruct what you're saying the homogeneous equation as a way to motivate the group of symmetries of an ODE. Certainly a laudable goal, but it's not quite clear to me yet what is actually to be said to the students here... $\endgroup$ – Kevin Arlin Sep 16 at 4:33
  • $\begingroup$ @KevinArlin You're welcome! I know what you mean about what to say. It would help if someone wrote a textbook at the intro level, but perhaps people run into trouble wanting to give full explanations (Lie theory), which require some familiarity with non-prereq math such as PDEs. I think material at the level of Yap's intro is accessible to my students, provides opportunities for concrete calculations and, with computer assistance, graphing. $\endgroup$ – user2913 Sep 16 at 13:04
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My copy of Speigel (which has separate chapters on applications, after each theory/calculation chapter) has one problem in "geometry" (really ray optics, finding curve that gives parallel reflection of a point source) that results in a homo first order (non separable) ODE. I didn't check overall first order applictions chapter, but it seems like there were very few such applications (I didn't see any more) as compared to applications resulting in integration by an integrating factor or separation of variables.

So, yeah, I don't think it's some classic application like the 2nd order ODE with constant coefficients (control circuits, and many others). More like just a trick to learn since you can. With occasional, rare, reasons to pull it out of the toolbox.

P.s. I know that Rota's essay is engaging and often cited here, but would recommend you to take it with a grain of salt or at least think about what parts to agree/disagree with. For examples, there are many physics or engineering derivations that include solution of an ODE with an integrating factor (I know this since I just went through the applicable applications chapter or Spiegel!) On the other hand, he is dead on with the comments about 2nd order ODE with constant coefficients (which homo and nonhomo) is often included in calculus 2 texts (was when I took AP BC). On the gripping hand, he takes a sideways slice at the old Cambridge Tripos, that the winners never did anything (First this is silly to judge a set/system only by outright winners. Second Wikipedia gives a quick "Many Senior Wranglers have become world-leading figures in mathematics, physics, and other fields. They include George Airy, John Herschel, Arthur Cayley, James Inman, George Stokes, Isaac Todhunter, Morris Pell, Lord Rayleigh, Arthur Eddington, J. E. Littlewood, Jayant Narlikar, Frank Ramsey, Donald Coxeter, Jacob Bronowski, Lee Hsien Loong, Kevin Buzzard, Christopher Budd, Ben Green, and John Polkinghorne.", to which I'd argue adding Forsyth.)

Also, check out the Amazon reviews for his textbook. Very sloppy work. Maybe a little more attention to detail and less attention to lofty philosophy or being new, would have helped Rota. I suspect a cross-section of average ODE students would get more out of a Schaum's Outline than out of Rota's slipshod effort. This isn't just a slam...the point is that for pedagogical efficiency, it is important to be accurate. The more grist in the mill, the more it hiccups. This, in contrast to the constant emphasis here on debating coverage.

See also: https://mathoverflow.net/questions/235698/undergraduate-ode-textbook-following-rota

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  • $\begingroup$ Thanks. I did enjoy reading Rota for, somehow, the first time, but I agree that it has to be taken with a grain of salt. He was quite a polemicist. In particular I cannot agree with him at all on "avoiding word problems"; the point is not at all to "write hard problem sets", but to make sure students are thinking about how all the manipulations they do connect to scientific situations, even if they're a bit artificial! $\endgroup$ – Kevin Arlin Sep 10 at 21:01

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