# Differential equations, a tourist's guide | DE1

An overview of what ODEs are all about

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Need to brush up on calculus?

Error correction: At 6:27, the upper equation should have g/L instead of L/g.

Steven Strogatz NYT article on the math of love:

Interactive visualization of the example from this video, by Ilya Perederiy:

If you’re looking for books on this topic, I’d recommend the one by Vladimir Arnold, “Ordinary Differential Equations”

Also, more Strogatz fun, you may enjoy his text “Nonlinear Dynamics And Chaos”

Curious about why it’s called a “phase space”? You might enjoy this article:

From a response on /r/3blue1brown, here are some interactives based on examples shown in the video:

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Animations made using manim, a scrappy open source python library.

If you want to check it out, I feel compelled to warn you that it’s not the most well-documented tool, and has many other quirks you might expect in a library someone wrote with only their own use in mind.

Music by Vincent Rubinetti.

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If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then “add subtitles/cc”. I really appreciate those who do this, as it helps make the lessons accessible to more people.

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Some notes on the intended use of this series. I was deliberate in using the phrase "tour of differential equations", as opposed to "introduction to" or "essence of". I think of the relationship between watching this series and taking a course as being analogous to the relationship between touring a city vs. living in it. You'll certainly see a lot less with the tour since you're spending less time overall, but the goal will be to walk around some of the most noteworthy monuments and town centers with helpful context given to you by a guide. And just as someone who lives in a city may very well have never gone to visit some of the historical sites of their town, despite living there for years, many differential equations students may not always get the chance to zoom out and appreciate the central cornerstones of the subject amidst all the computations they are learning.

I hope you enjoy the tour, but at the same time know that it is, by design, very different from taking courses on the subject.

Love the video! Thank you

9:49 Explains differential equations

I wonder if the seeming importance of ODEs and PDEs has more to do with a historical quirk. Before computers, systems that just so happen to be well modeled by 'solvable' ODEs are basically the

only typeof seemly complex systems (but actually dead simple) that humans could actually make predictions about using only chalk, blackboards, and human brains.god, i love this channel.

phase space makes me happy :^)

This is exactly how math should be taught

This man literally just derived all of introductory kinematics in under a minute lol

Your efforts are undifined…!!!!!!!🌼💜

waooo

ждём перевод на русский😴

Thank you for everything. All of our love is with you.

Thanks

I love how you have pi and tao in the relationship example 😂

I was looking for a video that describes how DEs are formulated and why math is useful as a language for describing physical phenomenon. This video is the one! Thank you so much for a quality contents!

10:01, Silly Grant, Integrals, inverse integrals. Those are just integrals and derivatives which is just a Calc 1&2 class, it can't be that hard, right?

Thanks!

“You two would be destined to an inner spiral towards mutual ambivalence, i hear wedding bells already” lol

Enlightening and beautiful.

For those not interested in maths….-> 20:45

Would you tell me please what software did you use to create this video?

Would you tell me please what software did you use to create this video?

What software do you use to animate?

the graphs are so beautiful

why is the point of lowest velocity on your pendulum graph mapped to the same moment of the point of highest velocity on your pendulum animation. Are you trying to tell us that -2 and +2 are the same value and represent the same point of maximum velocity?

Am I the only that thinks inward spiral in the love graph would mean apathy and breakup

Just incredibly beautiful and amazing as well, wonderfully picked examples of applications, I am speechless at the quality of the content, the whole time I was jumping with excitement!

He could talk about the hardest topic and I could still understand some of it. He could talk about the easiest topic and I would still learn something new about it.

Awesome video! Really sums it up and revises what is told in the university

Recently I found that a differential equation x''=F(x',x) could stand solutions which achieve a finite ending time only if the equation F(x',x) is Non-Lipschitz, so for x(t) becoming zero forever after a finite extinction time t=T by their own dynamics, uniqueness of solutions should be abandoned. This is why the pendulum equation shown in this video will be atracted to spiral towards the point (x',x)=(0,0) forever, but will never achieve it, or equivalently, the solution of the ODE will never stop moving.

As example of a ODE that indeed have a finite ending time at T=2:

x'=-sgn(x)*sqrt(|x|), x(0)=1

Could support the finite duration solution:

x(t)=1/4*(1-t/2+|1-t/2|)^2

Maybe using a different ansatz for the drag force different from Stokes' law could made it stop, as everyday intuition tells it happen (movement due random thermal noise is not derived by the pendulum dynamics – neither from these classical ODEs).

6:45 for noting that strange correlation I got a bad mark in school(

I study mathematics in the university and been trying to make my little brother be interested more in what I do. So I gave him to watch some of your videos (including this) and 2 things happened:

1. He enjoyed it a lot and asked me to help him with some of the concepts in this videos!

2. I watched the video till the end and enjoyed every second of it!!!!!!!

Your explenations are intuitive yet full of knoweledge.

Thank you very much for your work 🙂

p.s

sorry for my english

La guida è fatta senza da B a C partendo da A , da A a B ,la guida arriva anche in C , quindi nessuno punto neanche una virgola ,ma ,un interrogazione continua di che ,da parte in parte con un se un ma senza poi , quindi ,e una questione di proprietà insoluta.

Can I ask how you made the vector field animation, thx!

Beautiful

Someone help me with the geometry exercise at 7:25

Hello, what is physical significance of linear and non linear differential equation? What do they even mean?

The way you teach each math topics in a visual way is just awesome. By watching your videos I have realized how important are the visual interpretation of maths problems for deeper understanding.

By the way, I have checked your series on linear algebra, calculus, etc. But I wonder why I haven't you created such more series on other remaining topics of maths. I hope in future you make these series and I would love to watch them.

Can it be the case that Phase Diagram just represents the total energy map of the pendulum during the swing (all possible states), where theta represents time and Theta dot representing a path function. The chaotic is just the entropy?

I wish I could turn the background music off it makes it really hard to focus 🙁

cos(💜) = ?woah

A state space may be like some way through "gut checks." You can see all possible states in not many things that look not the same. You can follow the way near you one moment after another. After all, you see something big happen. After all, you see something like after you do a differential equation!

Thank you for the inspiration and the accessibility! 🙂 Great video!