9 ms·

Show HN: Differential Equations Explained

171 points by lewis500 11y ago

stickperson 11y ago
Cool idea, and the site looks nice. That being said, I was lost by the end of the first section.
- semi-extrinsic 11y ago
  I think calling a DE "a fact about the derivative of a function" and then speaking about "solving it" in the same sentence is a big source for confusion, since we don't usually "solve facts". EDIT: reading the whole thing, there's quite a bit of sloppiness and stuff that's flat-out wrong, like " the faster the cart goes, the faster it stops", or the statement at the end claiming that nonlinear DEs don't have solutions. In short: this needs a major cleanup.
  - stickperson 11y ago
    I think it would be helpful to define a derivative first.
  - dragonwriter 11y ago
    All equations are facts (or, more accurately, propositions) and solving an equation is deriving another proposition that is implied by that proposition (or perhaps that proposition and another proposition, such as when solve an equation in one or more unknown for a set value of one unknown.) OTOH, you are correct to the extent that we don't usually use both the "fact/proposition" language and the "solve" language at the same time when discussing equations.
    - semi-extrinsic 11y ago
      Exactly. I quite like the idea of calling a DE a fact about the derivative of a function, it's just the juxtaposition that is confusing.
  - musername 11y ago
    If it from the latter sentence means the derivative, the sentence makes sense, because with y'=f(x,y(x)) we wanna solve the equation for y.
simlevesque 11y ago
The Tupac reference was appreciated.
- Expeditus419 11y ago
  +1
JadeNB 11y ago
> Some of them don't even really have solutions (non-linear differential equations). There certainly are differential equations without solutions, but I'm almost positive that's not what's meant here. Rather, it is very likely that one can't find an explicit solution to a non-linear equation (EDIT: although there are non-linear equations that one can solve, such as $\dot y = 1/y$, whose solutions have graphs lying on straight lines through the origin); but there is a huge difference between an equation that doesn't have a solution and one whose solution we can't find. For example, away from singularities, one can approximate the solution of a differential equation, even a non-linear one, numerically to an arbitrarily high degree of accuracy, almost exactly as the author does for linear equations. EDIT: TheLoneWolfling (https://news.ycombinator.com/item?id=9654767 https://news.ycombinator.com/item?id=9654767 )'s point that this numerical simulation can get hard very quickly is well taken, and probably more important in applications than abstract existence results. Non-linear equations really are hard!
- TheLoneWolfling 11y ago
  You are correct, sort of. However, there are a number of differential equations where to simulate it requires exponential complexity w.r.t. to the given degree of accuracy or amount of "time" to simulate it for. For instance, pretty much any chaotic problem.
  - JadeNB 11y ago
    Agreed! There's a reason that non-linear equations are regarded as much harder than linear ones, and I don't mean to gloss over it. However, possibly just because I am a mathematician and not an engineer or hacker, I still find the distinction between "solution that is hard to find" and "no solution" meaningful and important. (It is the difference between "we can't predict the weather" and "there is no such thing as weather"!)
  - arielby 11y ago
    Isn't this exponential precision = polynomial time?
  - murbard2 11y ago
eranation 11y ago
Very cool. While I'm sure others found this explanation clear, I believe I should now feel dumber...
- andrepd 11y ago
  I don't think it was very clear either. If you want to learn more about this I like Arfken, Mathematical Methods for Physicists as a bible of practical mathematics, or Stewart, Calculus for calculus in general.
dragonwriter 11y ago
Win 7, IE 11 -- there's a mass of overlapping text and images.
gravity13 11y ago
Excellent stuff. I'm working on a framework right now for creating web documents like these!
- rockdoe 11y ago
  I hope yours isn't another of these "use Chrome" pages. I thought we got rid of that meme around 2000 but hey it's 15 years later and we're back to scratch.
  - gravity13 11y ago
    Huh? You mean do I currently support anything but Chrome? Actually, no. But I'm not worried about cross-browser this minute. I'm leveraging shadow-dom pretty heavily to allow for encapsulation of web components developed by third parties and Chrome is the only browser to fully support that (with the webcomponents lib). By the time my project is ready to (hopefully) be used by people, browser support will either be there or I will need to implement graceful fallback with iframes (but I think webcomponents.js will get there). Not sure how else to respond to your snarky comment, so I'll just let you know: it's not easy.
lewis500 11y ago
hey did yall get you're supposed to click and drag around the dots? it's cool to make a crazy v(t)
- amatic 11y ago
  Noup. They move by themselves. Cool page, though, keep it up!
potch 11y ago
Hovering on the graphs in Firefox, nothing happens and the console logs NaN problems.
newobj 11y ago
Thanks for explaining in literally 30 seconds what I failed to grasp in a fucking full semester of the class in college. Sigh.
deleted 11y ago
[deleted]
andrepd 11y ago
The intention is good, but without meaning to be overly negative, this is not very well written at all. There is imprecise and needlessly confusing and unclear phrasing ("To visualize derivatives, we can draw a right triangle whose hypoteneuse [sic] is tangent to a function. If the triangle's width is , then its height is the derivative." is the most confusing and unintuitive way to visualize derivatives I can think of), a poor sequence of examples for someone who is not familiar with ODEs, plenty of things left unexplained (why do I solve \dot{y} = \cos(t) by integrating?) not to mention technical problems with hovering the graphs (at least in FF). It's also way, way too short. You simply cannot go from "an ODE is like an equation with functions and derivatives" to "let's solve one" in 5 sentences. Honestly, reading the Wikipedia page would be more beneficial for someone who wants to learn ODEs. Like I said the intention is good but I would suggest throwing everything but the pictures away and starting from scratch, preferably with a good textbook by your side to help you not only with facts, but more importantly with the pedagogic aspect.
- gcb0 11y ago
  wikipedia. Why everyone keep showering ecyclopedic knowledge on blog posts instead of updating things there?
  - mmastrac 11y ago
    In fairness, wikipedia does not do well for interactive content (perhaps it could?). Khan Academy is doing much better for this, but it isn't really an encyclopaedic collection of knowledge. There really isn't a place for people to put this other than their personal blogs.
    - amatic 11y ago
      > There really isn't a place for people to put this other than their personal blogs. That is a great idea! A place like that should exist.
- fineline 11y ago
  Agree, this wasn't too enlightening. I highly recommend the differential calculus course on Khan Academy. I wanted to brush up recently after having not used this stuff for over 20 years, and it was the best resource I found.
wumbernang 11y ago
Windows phone.... http://i.imgur.com/Vo68qcm.jpg http://i.imgur.com/Vo68qcm.jpg
- amatic 11y ago
  oh, man...
graycat 11y ago
Ordinary differential equations? Okay: Just freshman calculus actually is a good start. E.g., for viral growth consider y'(t) = k y(t) ( b - y(t) ) where t is time, y(t) the size at time t, and the rate of growth y'(t) = d/dt y(t) is proportional to both the current size y(t) and the size ( b - y(t) ) yet to be achieved. So, e.g., the growth rate is proportional to the number of present customers talking y(t) and the number of potential customers listening ( b - y(t) ). We assume that the present is time t = 0 and we have the eventual size b and the current size y(0). Then, sure, we have an initial value problem (that is, we know y(0)) for a first order, linear ordinary differential equation. But all that is needed for a solution is just freshman calculus. It's just a routine exercise. Trivial? Once that equation kept FedEx from going out of business. Polished, elegant, insightful, balanced, expert, great first text: Earl A. Coddington, 'An Introduction to Ordinary Differential Equations', Prentice-Hall, Englewood Cliffs, NJ, 1961. For when you want to take some really big next steps up past nearly everyone else in mathematics: Earl A. Coddington and Norman Levinson, 'Theory of Ordinary Differential Equations', McGraw-Hill, New York, 1955. For one of the larger reasons to be interested in ordinary differential equations: Michael Athans and Peter L. Falb, 'Optimal Control: An Introduction to the Theory and Its Applications', McGraw-Hill Book Company, New York, 1966. Also E. B. Lee and L. Markus, 'Foundations of Optimal Control Theory', ISBN 0471-52263-5, John Wiley & Sons, New York, 1967. Once in graduate school I got a reading course to give a lecture a week from Coddington and Levinson, Athans and Falb, and Lee and Markus. The prof didn't show up again after the first lecture. At one time our current Fed Chair Janet Yellen indicated that she saw some potential in using control theory to help manage the economy. And a really sweetheart application of modern control theory, doing amazing things with automatic control of quadrocoptors: https://www.youtube.com/watch?v=w2itwFJCgFQ https://www.youtube.com/watch?v=w2itwFJCgFQ A broader view of some of what can be done with ordinary differential equations, and a long, gorgeous desert buffet of applied math: David G. Luenberger, 'Optimization by Vector Space Methods', John Wiley and Sons, Inc., New York, 1969. Generally, though, apparently the big glory days of ordinary differential equations were for the US DoD and NASA during the Cold War and the Space Race. Likely there are plenty of people now, with gray hair, who supply the expertise needed for current aerospace, etc. Numerical solutions? That field is also nicely developed. Partial differential equations? That's related but quite different.
princeb 11y ago
diffeqs super easy: each diffeq is like a set of directions: e.g. walk straight for 500m, turn left, walk straight 200m, turn right. if you specify a starting point or ending point (boundary/initial conditions) then the directions become a specific set of instructions to get you from let's say, the bus stop to the library (particular solution). otherwise it's just a set of directions that can be used to describe getting from a set of places to a corresponding set of other places (general solution).
mml 11y ago
I can say assuredly, this explains precisely dick to a layman.
mdturnerphys 11y ago
> Consider a cart rolling to a stop. Its motion obeys the DE v̇=−k⋅v, where v is velocity, v̇ is the rate-of-change of velocity and k is a 'coefficient of friction.' This is drag, not friction. Friction doesn't depend on velocity.
- NLips 11y ago
  Rolling resistance normally has a component proportional to speed as well as a constant component. (Aerodynamic) drag is normally modeled as proportional to v^2
  - mdturnerphys 11y ago
    Sure, there can be a component, but "coefficient of friction" usually refers to the velocity-independent term (the author even links to the Wikipedia article for it). At low speeds drag is proportional to velocity.
klunger 11y ago
The first two sections seemed fine to me. However, the "cartography" section made almost no sense. It needs to be explained more clearly how the two carts are different and why they diverge. edit: On a more general note, the animations are really slick!

171 points by lewis500 11y ago

stickperson 11y ago

Cool idea, and the site looks nice. That being said, I was lost by the end of the first section.

semi-extrinsic 11y ago
I think calling a DE "a fact about the derivative of a function" and then speaking about "solving it" in the same sentence is a big source for confusion, since we don't usually "solve facts". EDIT: reading the whole thing, there's quite a bit of sloppiness and stuff that's flat-out wrong, like " the faster the cart goes, the faster it stops", or the statement at the end claiming that nonlinear DEs don't have solutions. In short: this needs a major cleanup.
- stickperson 11y ago
  I think it would be helpful to define a derivative first.
- dragonwriter 11y ago
  All equations are facts (or, more accurately, propositions) and solving an equation is deriving another proposition that is implied by that proposition (or perhaps that proposition and another proposition, such as when solve an equation in one or more unknown for a set value of one unknown.) OTOH, you are correct to the extent that we don't usually use both the "fact/proposition" language and the "solve" language at the same time when discussing equations.
  - semi-extrinsic 11y ago
    Exactly. I quite like the idea of calling a DE a fact about the derivative of a function, it's just the juxtaposition that is confusing.
- musername 11y ago
  If it from the latter sentence means the derivative, the sentence makes sense, because with y'=f(x,y(x)) we wanna solve the equation for y.

simlevesque 11y ago

The Tupac reference was appreciated.

Expeditus419 11y ago
+1

JadeNB 11y ago

> Some of them don't even really have solutions (non-linear differential equations). There certainly are differential equations without solutions, but I'm almost positive that's not what's meant here. Rather, it is very likely that one can't find an explicit solution to a non-linear equation (EDIT: although there are non-linear equations that one can solve, such as $\dot y = 1/y$, whose solutions have graphs lying on straight lines through the origin); but there is a huge difference between an equation that doesn't have a solution and one whose solution we can't find. For example, away from singularities, one can approximate the solution of a differential equation, even a non-linear one, numerically to an arbitrarily high degree of accuracy, almost exactly as the author does for linear equations. EDIT: TheLoneWolfling (https://news.ycombinator.com/item?id=9654767 https://news.ycombinator.com/item?id=9654767 )'s point that this numerical simulation can get hard very quickly is well taken, and probably more important in applications than abstract existence results. Non-linear equations really are hard!

TheLoneWolfling 11y ago
You are correct, sort of. However, there are a number of differential equations where to simulate it requires exponential complexity w.r.t. to the given degree of accuracy or amount of "time" to simulate it for. For instance, pretty much any chaotic problem.
- JadeNB 11y ago
  Agreed! There's a reason that non-linear equations are regarded as much harder than linear ones, and I don't mean to gloss over it. However, possibly just because I am a mathematician and not an engineer or hacker, I still find the distinction between "solution that is hard to find" and "no solution" meaningful and important. (It is the difference between "we can't predict the weather" and "there is no such thing as weather"!)
- arielby 11y ago
  Isn't this exponential precision = polynomial time?
- murbard2 11y ago

eranation 11y ago

Very cool. While I'm sure others found this explanation clear, I believe I should now feel dumber...

andrepd 11y ago
I don't think it was very clear either. If you want to learn more about this I like Arfken, Mathematical Methods for Physicists as a bible of practical mathematics, or Stewart, Calculus for calculus in general.

dragonwriter 11y ago

Win 7, IE 11 -- there's a mass of overlapping text and images.

gravity13 11y ago

Excellent stuff. I'm working on a framework right now for creating web documents like these!

rockdoe 11y ago
I hope yours isn't another of these "use Chrome" pages. I thought we got rid of that meme around 2000 but hey it's 15 years later and we're back to scratch.
- gravity13 11y ago
  Huh? You mean do I currently support anything but Chrome? Actually, no. But I'm not worried about cross-browser this minute. I'm leveraging shadow-dom pretty heavily to allow for encapsulation of web components developed by third parties and Chrome is the only browser to fully support that (with the webcomponents lib). By the time my project is ready to (hopefully) be used by people, browser support will either be there or I will need to implement graceful fallback with iframes (but I think webcomponents.js will get there). Not sure how else to respond to your snarky comment, so I'll just let you know: it's not easy.

lewis500 11y ago

hey did yall get you're supposed to click and drag around the dots? it's cool to make a crazy v(t)

amatic 11y ago
Noup. They move by themselves. Cool page, though, keep it up!

potch 11y ago

Hovering on the graphs in Firefox, nothing happens and the console logs NaN problems.

newobj 11y ago

Thanks for explaining in literally 30 seconds what I failed to grasp in a fucking full semester of the class in college. Sigh.

deleted 11y ago

[deleted]

andrepd 11y ago

The intention is good, but without meaning to be overly negative, this is not very well written at all. There is imprecise and needlessly confusing and unclear phrasing ("To visualize derivatives, we can draw a right triangle whose hypoteneuse [sic] is tangent to a function. If the triangle's width is , then its height is the derivative." is the most confusing and unintuitive way to visualize derivatives I can think of), a poor sequence of examples for someone who is not familiar with ODEs, plenty of things left unexplained (why do I solve \dot{y} = \cos(t) by integrating?) not to mention technical problems with hovering the graphs (at least in FF). It's also way, way too short. You simply cannot go from "an ODE is like an equation with functions and derivatives" to "let's solve one" in 5 sentences. Honestly, reading the Wikipedia page would be more beneficial for someone who wants to learn ODEs. Like I said the intention is good but I would suggest throwing everything but the pictures away and starting from scratch, preferably with a good textbook by your side to help you not only with facts, but more importantly with the pedagogic aspect.

gcb0 11y ago
wikipedia. Why everyone keep showering ecyclopedic knowledge on blog posts instead of updating things there?
- mmastrac 11y ago
  In fairness, wikipedia does not do well for interactive content (perhaps it could?). Khan Academy is doing much better for this, but it isn't really an encyclopaedic collection of knowledge. There really isn't a place for people to put this other than their personal blogs.
  - amatic 11y ago
    > There really isn't a place for people to put this other than their personal blogs. That is a great idea! A place like that should exist.
fineline 11y ago
Agree, this wasn't too enlightening. I highly recommend the differential calculus course on Khan Academy. I wanted to brush up recently after having not used this stuff for over 20 years, and it was the best resource I found.

wumbernang 11y ago

Windows phone.... http://i.imgur.com/Vo68qcm.jpg http://i.imgur.com/Vo68qcm.jpg

amatic 11y ago
oh, man...

graycat 11y ago

Ordinary differential equations? Okay: Just freshman calculus actually is a good start. E.g., for viral growth consider y'(t) = k y(t) ( b - y(t) ) where t is time, y(t) the size at time t, and the rate of growth y'(t) = d/dt y(t) is proportional to both the current size y(t) and the size ( b - y(t) ) yet to be achieved. So, e.g., the growth rate is proportional to the number of present customers talking y(t) and the number of potential customers listening ( b - y(t) ). We assume that the present is time t = 0 and we have the eventual size b and the current size y(0). Then, sure, we have an initial value problem (that is, we know y(0)) for a first order, linear ordinary differential equation. But all that is needed for a solution is just freshman calculus. It's just a routine exercise. Trivial? Once that equation kept FedEx from going out of business. Polished, elegant, insightful, balanced, expert, great first text: Earl A. Coddington, 'An Introduction to Ordinary Differential Equations', Prentice-Hall, Englewood Cliffs, NJ, 1961. For when you want to take some really big next steps up past nearly everyone else in mathematics: Earl A. Coddington and Norman Levinson, 'Theory of Ordinary Differential Equations', McGraw-Hill, New York, 1955. For one of the larger reasons to be interested in ordinary differential equations: Michael Athans and Peter L. Falb, 'Optimal Control: An Introduction to the Theory and Its Applications', McGraw-Hill Book Company, New York, 1966. Also E. B. Lee and L. Markus, 'Foundations of Optimal Control Theory', ISBN 0471-52263-5, John Wiley & Sons, New York, 1967. Once in graduate school I got a reading course to give a lecture a week from Coddington and Levinson, Athans and Falb, and Lee and Markus. The prof didn't show up again after the first lecture. At one time our current Fed Chair Janet Yellen indicated that she saw some potential in using control theory to help manage the economy. And a really sweetheart application of modern control theory, doing amazing things with automatic control of quadrocoptors: https://www.youtube.com/watch?v=w2itwFJCgFQ https://www.youtube.com/watch?v=w2itwFJCgFQ A broader view of some of what can be done with ordinary differential equations, and a long, gorgeous desert buffet of applied math: David G. Luenberger, 'Optimization by Vector Space Methods', John Wiley and Sons, Inc., New York, 1969. Generally, though, apparently the big glory days of ordinary differential equations were for the US DoD and NASA during the Cold War and the Space Race. Likely there are plenty of people now, with gray hair, who supply the expertise needed for current aerospace, etc. Numerical solutions? That field is also nicely developed. Partial differential equations? That's related but quite different.

princeb 11y ago

diffeqs super easy: each diffeq is like a set of directions: e.g. walk straight for 500m, turn left, walk straight 200m, turn right. if you specify a starting point or ending point (boundary/initial conditions) then the directions become a specific set of instructions to get you from let's say, the bus stop to the library (particular solution). otherwise it's just a set of directions that can be used to describe getting from a set of places to a corresponding set of other places (general solution).

mml 11y ago

I can say assuredly, this explains precisely dick to a layman.

mdturnerphys 11y ago

> Consider a cart rolling to a stop. Its motion obeys the DE v̇=−k⋅v, where v is velocity, v̇ is the rate-of-change of velocity and k is a 'coefficient of friction.' This is drag, not friction. Friction doesn't depend on velocity.

NLips 11y ago
Rolling resistance normally has a component proportional to speed as well as a constant component. (Aerodynamic) drag is normally modeled as proportional to v^2
- mdturnerphys 11y ago
  Sure, there can be a component, but "coefficient of friction" usually refers to the velocity-independent term (the author even links to the Wikipedia article for it). At low speeds drag is proportional to velocity.

klunger 11y ago

The first two sections seemed fine to me. However, the "cartography" section made almost no sense. It needs to be explained more clearly how the two carts are different and why they diverge. edit: On a more general note, the animations are really slick!