Physics Students Need to Know These 5 Methods for Differential Equations 

Hope you like the animations in this one! It's the first video I've made using "manim," the programming library for math animations created by @3blue1brown for making his incredible videos, and further developed by the community of developers who work on the open source project. A huge thank you to them for their hard work!

I am extremely impressed with the high quality of your talks. It is apparent that you put much thought, and much work, into the script, the examples, the animations, and the presentations. Also, your voice is perfect for narrating videos like this -- expressive, clear, and pleasant to listen to. With this video on differential equations, you have packed a whole semester's worth of learning into a half hour. Your notes are equal to any physics book I've seen, and I appreciate that you provide them for free. I am going to increase my Patreon donation to your channel. Thank you, and best wishes. I'm so grateful for your work.

This channel is going to blow up in the future.

I had a bit of trouble following along at the end of the video, but just because the material was tough for me; the explanation was outstanding. Thank you so much for taking the time and effort to make these really high-quality videos and then sharing them for free!

I cannot express how grateful I am for these videos. Your content has single-handedly changed my outlook towards physics work, and my ability. Your easy to digest videos and worksheets talking about the mathematical rigour of such a broad range of physics is just breath-taking. And it's certainly done a lot for me. Thank you for what you do, Elliot, and I'm excited to see what's in store for the future.

Going over an E&M course, and the boundary conditions cannot be undervalued. Good stuff! Glad to see this content on RU-vid!

Very interesting! It was definitely instructive to see all 5 techniques applied to the same example.

Just came across your video. Holy, the best I have ever seen in explaining and summarizing in such concise and clear terms! Thanks!

I finished my degree about 4 years ago, and this reminded me of so much. What a great presentation! Such a clear delivery with great perspective to relatable concepts

You're my favourite physics tutor! I can't tell you how much it was painful looking for information for months and being unable to find one that make you content. But with your videos you've answered to a lot of my questions so I can't tell you sir how grateful I am. Thank you for your clear explanation and representation, and for feeding my curiosity and growing my knowledge, I owe that to you.

Would love to see a similar video on partial differential equations :) Thank you for your content very well explained!

I am just starting to learn classical mechanics and this was a great simplified bird’s eye view of all the techniques! Thank you sir 🙏🏼

Awesome work, I wish we had this around when I was studying physics and maths. This really accelerates learning and understanding. I’m envious of current students of physics having such great educational tools available!

Appreciate your effort and pedagogical skills

You are a terrific educator, sir. Thank you. This was superbly constructed.

I struggled mightily through this stuff in college. Not only was that before RU-vid but it was before electronic calculators. This is so much easier to understand.

Method 0: use Mathematica

Elliot, that was a beautiful, clear and concise presentation of these important core concepts. The time, effort and intelligence you put into your videos is very much appreciated; you are a natural born teacher.

I have studied economics and maths was part of that. This explanation really brought home some concepts I always grappled with in an easy to understand way. Thank you.

Thank you so much, especially to see the Laplace transform in use was an eye-opener

Excellent explanation of these 5 core concepts used to solve differential equations using the Manim animations. I like the whirl pool analogy and animation you used to convey a visual intuition of the Hamiltonian Flow. The matrix exponential construct is interesting. Thanks for sharing your work.

So high quality! Thank you!

This is absolutely a fantastic explanation of this subject. Many thanks for this

lovely intro about not only the physics but also for the math and general engineering. Great video!

4th & 5th methods are mind blowing especially Hamilton's Flow. Thank you for sharing.

Found this through RU-vid recommended, and I have to say this video is a masterpiece. Instantly subscribed and looking forward to more videos from you

Great stuff 🙂I know you already did a video on Hamiltonian mechanics, but a deeper explanation of the Legendre transform involved would be nice.

Bravo! One of the clearest and detailed lesson I have ever seen...

I'm so grateful for this video. I've been trying to self-study Differential Equations and kept getting stuck early on. This really helped clarify not only what to do to solve Differential Equations but WHY the methods work. Thank you!

Thank you very much! The video is gorgeous and very clear. For the first time i have connected better my knowldege about differential equations in a way i have never thought! Thank you a lot very much!!!

Amazing video. I saw this topics before but this video really makes me enjoy what I couldnt while taking these classes...

Brilliant as usual! 👍 One fun thing about the Ansatz: English-speaking world tends to solve, for example, the harmonic oscillator differential equation as A cos(omega t) + B sin(omega t), which is very sensible in from a maths point of view (you find a basis of two independent vectors in 2D vector space of solutions of this linear second order ODE and you express any solution as its decomposition on this basis). French way - for example - would be lean towards a physicist strategy and write A cos(omega t + phi), since in physics, amplitude and phase are much clearer to interpret than A and B from previous sentence. 😊 You arrive on this second writing in a very natural way with the energy reasoning, though, which is very interesting.

Here before this channel gets millions and millions of subscribers. Keep doing these animations, they are invaluable when you show the concepts. It really helps visualising the physics and the math.

I'm glad to find a high quality content explanations about basic physics, it's harder to solve cubersome problems skipping the bacics, thank you from Brazil 🇧🇷

Elliot, that was excellent and solving same problem different ways important for many different reasons from educational to checking a solution. Thanks. Have been looking at your videos on lagrangian. Again, very enjoyable and very informative. And thanks for access to "notes" .. Your students must really appreciate you.

Hi from Argentina, I am preparing for a very hard physical chemistry final exam in March, and I found this tutorial very valuable. I know a 30 minute video won't replace hours and hours of differential equation solving, but I got to say the laplace transform and hamilton parts are brilliant, because your approach has an integral view, it is perfectly edited and explained, and it shows the beauty and simplicity underlying these concepts. Too often as students we lose track of this global view because we are alienated with calculations and exercises. I found your explanation beautiful. Beauty serves as a path to a deep understanding of anything, that's my opinion. I am subscribing right now!

First time I understand what a Laplace Transform a Hamiltonian are! Very clear explanation. Thank you.

Brilliant lecture! Thank you!

Beautiful and concise. Thanks Elliot.

Extremly good video, perfect refresher for some, superb intro to others. Very, very good content. Thank you very much.

What a nice simple explanation of Hamiltonian mechanics!

I'm so glad that I found your channel I've been looking for such channel that explains physics in english. Tysm for your hard work!

Amazing stunning mesmerising. Being an electrical and electronics Engineer from the most reputed university in my country I have been struggling to fathom the inner meaning of the differential equations and its solutions. Finally I have got to understand it. Thank you awfully

I enjoyed this much more than i could, thank you a lot for your effort, this was very thoughtful, im an absolute fan

Wow! No distractingly unnecessary music over your excellent narrative skills and important information??? I’m exponentially impressed!!!!👍😃

You gave me a different type of thinking...so thank you so much

A very excellent presentation. Thanks a lot Elliot👍

Reading Hamiltonian mechanics recently and this video pop up great video

Great insight to see everything together... thanks!!! As engineer I'll keep with Laplace but uncle Hamilton was incredible! Nice...

8:59 using this method for simple harmonic motion gives the function e^iwt ( w being angular speed) because differentiating that twice would give the constant (iw)^2 i.e. -w^2 which is neat verification of euler's formula

Simply genius. Very impressive teacher. God bless you.

Thank you so much for this video, now it's really clear in my hand. I have just make tremendous progress with this video! Again thank you !

Great explanation appreciate it

Now I can finally say I am enjoying Physics. Hats off to you!!!

Splendid! Nicely presented and generous in content for introducing the concepts. You have a new subscriber.

What a wealth of knowledge!... thanks for sharing this Doc, this was truly helpful.

This is an incredibly helpful video Really helped me review some necessary content

Very clear explanation, bravo!

What a masterpiece. Please continue with this excellent work

Great video, certainly some of the best math animations and exigesis I have seen.

Thanks for the explanation, would love to see the Poisson Equation on gravitational field on next video. It would be great!

Awesome Video. Thank you very much. What I like to do in class is connecting the hamiltonian flow with the Eigenvalue Problem and find a solution in terms of Basis functions. Btw: The oscillating Block is by far my favorite example as well 😊

Increadible explanation! I would like to recomended this video to my students later on. Thanks :)

Thank you. Enjoyed the 30 minute wholeheartedly.

Incredible... This is "Quality Education". Great.... Thank you 🙏🙏🙏😊

That was sick! Gonna try to master these methods now

Nice examples! It would be interesting to do the same with a more difficult DE, too.

Excellent Work!!!!

Great content👍👍...... wonderful explanation... thankyou very much...loved it

Impressive video Elliott! I would add up that the usual solution in Matrix exponential, also in electrical circuits is using laplace transform of the matrix exponential (because it's not necessarily unitary hence Laplace and not Fourier) and then element--wise inverse laplace transform for each element. (With multiplication of b.c.s)

Hi Elliot, many thanks for the video. Kudos!

Bro, u are giving away this high level of knowledge FREE! Man I'd pay the $$ to attend your courses, the content is simply awesome!!

Absolutely love this.

Brilliant. Thank you.

Stunning explanation .

The weighted residual method is also there to solve the solution of differential equations. The popular one is the Galerkin method. We guess the solution of the differential equation that satisfies the boundary condition. And then setting the residual to zero throughout the domain.

Love the videos! What program do you use to make such videos?

Thank you very much. Good content. Greatly appreciated. Keep up the good work🎉

This is honestly fantastic

Thank you for these wonderful videos ! Are you planning one CFTs?

Beautifully explained ❤️❤️❤️

I vaguely remember doing Laplace Transformation in equations relating to electrical circuits where the equation was in time domain and we have to convert it into frequency domain by applying Laplace Transform.

Very well done. Thank you.

Thankyou so much for this precious knowledge and explanation 🙏🙏 I don't have words to express my gratitude for such an amazing lesson.

You did a great job and I like how Manin library is used.

Beautiful way explained

This is super interesting ! Never had such a bird eye view on the way to resolve such a canonical system whilst having studied the harmonic oscillator for 5 years at uni !

This is more than just math tools for the Harmonic oscillator. It's a lot about the way physics is done. Thx for the video.

This is a great video. Thanks for your nice effort 🙂

You've just earned another subscriber. Brilliant and elegant.

Thank you for this priceless video

I m actually studying physics in french language but your video is clear to understand and fun to watch I wished that I have seen you earlier. Keep your hard work sir.

You saved my brain. Thank you!

Thanks for doing this for free. I'm from India, and affording a tutor can be only possible if 10 to 15 kids combined all their savings. So mostly we just learn from one another. But with you, my peers and I could take the further step which only the rich kids had in our highschool. We owe you forever. Again Thanks.

Very good video! You've definitely won a subscriber here! I can't wait to see what's coming up next! Thank you!

Keep doing this amazing work 👌👌 You are just different and unique👏👏

The latter method or things similar to it are not only really nice (and used heavily in dynamical systems courses and books on the subject, like Strogatz's), but have also been used to completely reform calculus courses at UCLA, at least for life sciences (but the professors spearheading the changes claim their methods were very generalizable, although as a physics and math major I certainly enjoyed seeing just how much the life sciences did in fact have such a shared language with my own field I'm interested in). How the course went wasn't about rewriting things in Hamilton's equations or anything. Instead their series of calc courses starts off with the concept of 'modeling' problems, and the goal is to get the students to interact with these models as quick as possible. That means viewing differential equations as vector fields where initial conditions are where you place your 'ball' in the pool of water and see where the flow goes from there (which means learning discrete methods such as Euler's method very quickly in order to get started right away with modeling this on your computer), and learning quickly about two ways of representing diff eq's as their time series graph vs. their state space graph (the vector field). Also the ability to make analogies between different types of systems (the students start off learning how to model a 'predator-prey'/lotka-volterra problem - what he calls 'shark meets tuna', and then later on when the students get confronted with attempting to model chemical reactions (a system undergoing chemical equilibrium in other words) they learn to view it analogously to some system they've already encountered: 'shark meets tuna'. The overall idea is that more geometric methods like vector fields rather than algebraic manipulation/guessing is what made for better pedagogy, and as it turns out their students that learned this way of doing things first not only improved their grades from the regular series of courses one would take (from calculus to diff eq's), but had their students who took these reformed courses actually outperforming their own peers going the more traditional route, where learning discretization methods and visualizing your problems and methods of solving them weren't really emphasized. (The paper of interest would be 'Teaching Dynamics to Biology Undergraduates: the UCLA Experience', but also I fear I'm not doing as good of a job as I'd like in representing crucial points of their methods) I'd also search their main website that not only includes said paper outlining their methodology but also lays out a nice overall view of their course, like their list of lectures online showing the series of topics they'd go through (I'm hesitant to put links in youtube comments, because that usually screws things up): search 'modelinginbiology github' in google. The title of the hyperlink should be 'Modeling Life | UCLA Life Science Course'.

geeeeez, looking back on my all calculus courses (all 4 of them), series solutions to diff equations were just really enigmatic to me. I am an EE guy, I don’t even deal with mechanics, but thought process and the approach made me 💯percent convinced that all that complicated series forms must literally be found though looking for a solution of a physical phenomenon.

Excellent notes.

Brilliant animations and stunning video