When I first took ordinary differential equations, the pattern was always the same. Week 1 turns into a speedrun of methods: separation of variables, integrating factors, variation of parameters, Bernoulli, exact equations… and by Week 2 or 3 the course has quietly degenerated into hammer-picking. Spot the form, apply... the recipe, move on. Mechanical! Fuuuuck!😫😫😫😫 The problem is what you don’t walk away with. You leave with a toolkit, but without a feel for what a differential equation even is, especially geometrically. And that’s a big deal, because in real modeling the equations you meet are rarely nice enough to reward memorized recipes. So you end up trained to solve toy forms, while the actual subject...the behavior, the flow, the shape of solutions stays blurry. This is why I’m biased toward the old-timers. Their old-school way of doing things always surprises me:...they’ll spend time on one idea until it sticks, instead of sprinting through a syllabus checklist. One lecture from them and you start noticing a contrast. A lot of modern teaching feels like "finish the content,". You get marched through techniques, but you’re not left with a single thought that keeps bothering you later...the kind of thought that actually pushes you toward research-level curiosity. MIT OpenCourseWare’s Professor Arthur Mattuck did that to me in his very first ODE lecture. One lecture, and your whole relationship with dy/dx = f(x,y) changes. In this segment, Prof. Mattuck is basically saying: A first-order ODE is a slope field, and a solution is a curve that moves everywhere tangent to that field. The math breakdown Write the ODE as dy/dx = f(x,y). At each point (x,y) you attach a tiny line segment with slope f(x,y). A function y = y₁(x) is a solution exactly when its graph follows those slopes:. At every x, the slope of the curve equals the slope prescribed by the field at the point on the curve. That’s the single line that unifies both viewpoints: y₁′(x) = f(x, y₁(x)). So solving the ODE and drawing an integral curve are the same statement in two languages!👌🏻 Once you see that, you can stop obsessing over whether you can write y(x) in closed form. You can start asking the questions that matter: where do solutions flow, where do they get trapped, where do they blow up, and where does existence/uniqueness fail just because the field isn’t even defined? That’s the perspective shift I wish every ODE course forces early and it’s exactly why I keep pairing math with animation. #DifferentialEquations #ODEs #VectorFields #MathAnimation #Mathematicsshow more

The Lecture Series

The baffling reality of North American eng. students: They can solve Riccati equations blindfolded. Yet, they are completely stumped by uniqueness on a phase portrait. We are teaching them to compute, not to see.

Absolutely!

I took physics with calculus and advanced physics and electrical and mechanical engineering courses at the same time and it totally integrated into the toolkit with the necessary motivational background that allows one to automatically see how it can be used in new topics. Add to that Langrangian, Hamilton and Calculus of Variations based understanding of mechanics and the world’s oyster until you want non-Newtonian mechanics and then you need differential geometry. My master’s thesis was numerical ODE solutions on manifolds and my life trajectory was built. Marriage of applied mathematics tools with actual applications is “The Way” in my experience and opinion.

Very nice. One of the few times I've watched a math video in full.

He’s incredible. Maybe it’s just me, but you can feel the energy he brings,esp at his age. Old son of a gun is almost bouncing as he moves across the class 😁👌🏻🙌🏻 He respects the subject and the room. You can tell he genuinely loves what he’s doing, but there’s zero casualness about it. He’s prepared. Every example is chosen on purpose.🫡

…& Then you use the green one as fire-starter 🔥, use the purple one Pro Re Nata…& almost everything else is Runge-Kutta / Hybrid Gear / Bulirsch–Stoer. Life goes on.

Awesome 👌🏻

😬😉

Cool. pls share the system eqns

✅ DOI: 10.1086/109234

My Diff E class was nothing but a bag of tricks. I hated it because there was no underlying method or technique.

So cool to see Prof. Mattuck in action again. I took a differentials equations course that he taught as an undergraduate. I think that it was the same course in the video and in the same room and chalk boards. He was an extremely good lecturer on the topic.

Great post. I am not a mathematician, I am physical scientist. When I took DiffEQs, I immediately tied it to physics and physical chemistry and then to chemical engineering. I kind of love differential equations. That said the course was ridiculously easy because you marched through Boyce and DiPrama learning recipes, algorithms. I know you didn't like that but it gave me time to study other things closer to my major. However, it was only later, when I for example I took quantum mechanics that ideas like the time rate of change is proportional to the curvature in space, did a deeper understanding, a more intuitive and geo-spatial understanding begin to form in my mind. When I then had to teach, this became even more important to me. I wanted my students to know what the math was telling them about the system and to have some feeling before solving the equation for its parametric sensitivity, its tendency to go to a steady state and its likely solution shape in space and time.

When I taught ODE I showed near the end of the semester that almost all of the techniques presented in the course (everything except power series) were special cases of Lie's use of symmetries to solve DEs.

I’m not familiar with Lie algebra yet. When you say Lie symmetries for an ODE, do you mean noticing a way to relabel the picture (like re shifting, scaling, or rotating) that doesn’t actually change the equation?

Id appreciate an example

i remember that. we had the solve the random forms thing but also i remember the slope field thing. but true, in general, i found the course REALLY boring. and i don't remember much of it.

Really? What were you studying?

it was just beginning diffeq at the community college, forgot the name of the textbook. i was doing some kind of generic engineering path. i learnt heavier maths later.

Yes, My first pass through differential equations was just learning to apply the many techniques. What you've said here did not dawn upon me until many years later when I realized most of nature could be explained through different equations, fields and potentials.

I have noticed that when any problem is explained differently from your first approach is more appreciated as introduces a new point of view. I loved theoretical physics from Landau and Feynman mechanics but I am not sure if starting with them i would have mastered the subject.

I agree. I jus feel we should spend real time teaching geometry as much as teaching techniques. Simply because humans think in pictures.

1. The funny thing is: i was automatically mapping to euclidean geometry in R2 linear algebra, norm and hilbert spaces. When infinite dimensions came I was drawing lines on a plane, which would make no sense, understanding before studying why close and limited is not compact.

2. I am convinced that starting geometrically was natural as it is how our experience works. But logically everything could have been done the other way around. I know people who really start from algebra and deduct the experience: some are smart! we are just wired differently.

Yeah. So, we should give students the minimum tool needed to get moving, then immediately cash it out geometrically so they see what the algebra just did. Then you add the next tool with that picture already in place. Otherwise methods jus turn into button-pressing, and the geometry arrives too late to fix the terrible habits. Its often so hard to unlearn something😄

I am italian. At my old times we made mainly geometry before. Cartesian geo was fully deducted. Enrico Fermi, from whom Fermilab gets his name said that he understood math in that moment: he realized it was not a set of rules. To me was the same.

And this MIT ODR course MUST be taken after linear algebra or supplemented with it by Prof. Gilbert Strang. The amount of connections you’ll make is unreal and the intuition that will be developed is insane

True

"And that’s a big deal, because in real modeling the equations you meet are rarely nice enough to reward memorized recipes" The first lecturer I had on ODEs in his first lecture stated that in many cases you come across in practice, you'd use a computer (no closed form), but the point of the course was to extend your own expertise in a new domain, understand fundamentals, and know how and why to program those computers.

Well I always seemed to learn the n+1 time, with n being the time I needed to learn it. Didn’t really learn algebra until trigonometry, didn’t learn trigonometry until calculus, didn’t learn calculus until physics, didn’t learn physics until mechanics, etc.

Brilliant! What a wonderful thought process! (I recommend my fellow geeks and nerds to read it)

“I’m going to dedicate this first day to not solving a single differential equation” that’s how you know it’s good.

I'm not sure of others, but I have always seen any ODE or PDE as a statement about geometry. I'm surprised that others don't, if that's the case. But you can always change views, and sometimes you must For example in the simple Black Scholes equation, or its Local Vol improvement, you can see the terms in 3 different ways. You can see slopes and curvature. Or you could see convection and diffusion. Or you can you position changing in terms of delta and gamma. The latter is the essential bit in that case for any trader to imagine how his P/L will change based on whether he has left the delta unhedged or not, SC how he trades the gamma. It's the essential first step for a beginning trader and it's a better picture than (slope, curvature) or (convection, diffusion). Really you want (dynamic delta, dynamic gamma) understanding. But if you were a quant implementing numerical solutions you might want to know about (convection, diffusion) because your solver might have areas where convection dominates diffusion leaving the pde locally hyperbolic with the potential instabilities propogating all way back to the solution. Or if you were a quant doing smile modelling you might want to be aware that bad smile Interpolations could ruin your delta and gamma clocks because you had violated some constraints on implied volatility slope and curvature leading to B minus traders and non-quants like @nntaleb believing in the existence of negative probabilities. You should never force one view or the other. Switching perspectives is often the most useful approach one can adopt (My 18 year old understands maths and probability much better than him, not knowing the Wick rotation in the path integral formulation of Quantum Mechanics, but explaining to me in the most intuitive mathematics the other day why you could not work with real valued wavefunctions).

phenomenon first is systems thinking

This brought back memories

And did he see the truth by logic and reasoning of numbers? It is only the decoding symbols?

Mi libro "Calculus Without Limits" va en la misma dirección: recuperar la esencia algebraica del cálculo, sin perderse en límites y continuos.

math is crooked in platonism, reality not

See the introductory text, Differential Equations by Blanchard, Devaney, and Hall. The entire book is in reaction to the problem you describe.