Physics Students Need to Know These 5 Methods for Differential Equations

Differential equations are hard! But these 5 methods will enable you to solve all kinds of equations that you'll encounter throughout your physics studies. Get the notes for free here: courses.physicswithelliot.com...
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Almost every physics problem eventually comes down to solving a differential equation. But differential equations are really hard! Fortunately, there are powerful tools for tackling them, and in this video I'll introduce you to five of them: substituting an ansatz, using energy conservation, making a series expansion, using the Laplace transform, and finally using Hamilton's equations, which give a new way to visualize the solution as what's called a flow on phase space, as well as a way to solve an equation with a matrix exponential.
We'll see how they all work using one of the most important differential equations in physics: the F=ma equation for a simple harmonic oscillator, or in other words a block attached to a spring. You certainly don't need crazy powerful tools to solve such a simple equation, but seeing how they work in a simple problem will help prepare you for the harder problems you'll inevitably meet later on in physics!
Related videos:
All about the simple harmonic oscillator, and why it's so important: • To Master Physics, Fir...
The Fourier transform, with applications to quantum mechanics: • To Understand the Four...
The math and physics of Taylor series: • The Most Important Mat...
0:00 Introduction
2:20 The equation
4:01 1: Ansatz
9:10 2: Energy conservation
14:17 3: Series expansion
18:23 4: Laplace transform
22:41 5: Hamiltonian Flow
26:48 Matrix Exponential
29:31 Wrap Up
If you find the content I’m creating valuable and would like to help make it possible for me to continue sharing more, please consider supporting me! You can make a recurring contribution at / physicswithelliot , or make a one time contribution at www.physicswithelliot.com/sup.... Thank you so much!
About me:
I’m Dr. Elliot Schneider. I love physics, and I want to help others learn (and learn to love) physics, too. Whether you’re a beginner just starting out with your physics studies, a more advanced student, or a lifelong learner, I hope you’ll find resources here that enable you to deepen your understanding of the laws of nature. For more cool physics stuff, visit me at www.physicswithelliot.com.

[@4skyrider]

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.

[@PhysicswithElliot]

Thank you so much Michael!

[@sumitpakhare9546]

Totally agree

[@Zero-ef4sc]

@@PhysicswithElliot you are the Morgan Freeman for Physics!

[@mapachepataki5013]

Yes that is the sad part " Your notes are equal to any physics book I've see" Its al dark and ambiguous as any physic would approach

[@ericephemetherson3964]

@@PhysicswithElliot This is excellent even though the pace of explanation is very tough to follow. I got lost after 12 minutes of the video even though I used to be famiIiar with the contents of the video once. I am not a mathematician in any sense. But I studied physics and took calculus a long time ago. I am still studying physics on my own at my own inspiration and times when it overwhelms me. But may I say that even in school one variable always gave me trouble to understand. And it was and still is time. Denoting time (t) we use it in many equations and mathematical formulas. But after years and years of pondering over ''time'' I cannot undestand how ''time'' is being used in mathematics without a definition of time. We know what distance or space are and we can define them in a scalar manner and use vectors or whatever else. But - excuse my coy knowledge (I've forgotten so much that I need to reread a lot of math) of math - I think ''time'' cannot be associated with clocks at all. When I see a clock or even read about atomic clocks I do not apprehend ''time'' in them. They do not show me ''time''. The idea of time flowing in some direction is an erroneous way to approach this elusive entity. Time does not flow niether has a direction. If time flowed (as you hear all over) it would have to be moving. In my opinion ''time'' is some kind of force. After all it forces us to get up in the morning to do things and live. But in the deeper sense if I one says that an hour has passed I cannot grasp that hour and adhere it to some point of reference. In your video of the example of the block oscillating you have to define the initial condition in order to perform differentiation. But I envison that with ''time'' one cannot do that. Might as well start using words like ''I did it then'' and ''I do it now''. But one cannot use these words in mathematics even if you give them symbols. Definition of ''time'' would be so much helpful in seeing the whole picture.

[@PhysicswithElliot]

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!

[@sohailtabarhossain6096]

Thank you dear Dr Schneider 🙏💚

[@dwaynep6174]

Animations look amazing! Very smooth, love it

[@orsoncart802]

Very nice. Thank you! 👍 Just one thing. The animation at ~24:45. The red ball is swimming against the flow. I’m told that phenomenon occurs only in Australian toilets. 😁

[@howwitty]

Great video, thanks. 3B1B is excellent!

[@navi4259]

@@orsoncart802 I see the flow going the right way, I’m pretty sure just depends which way u look at it

[@CeRz]

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.

[@bruh4196]

This channel is going to blow up in the future.

[@PhysicswithElliot]

Thanks Bruh!

[@johnchessant3012]

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

[@PunmasterSTP]

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!

[@curiousaboutscience]

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

[@douglasstrother6584]

Maxwell's Equations are the best; but it's all fun 'n' games until boundary conditions are imposed! After that trial, someone imposes mixed Dirichlet and Neumann boundary conditions.

[@curiousaboutscience]

@@douglasstrother6584 Very true! It's enlightening though when you finally understand the physical implications/meaning of boundary conditions. This of course applies to many fields of study. Acoustics was another fun area to see these applications!

[@douglasstrother6584]

@@curiousaboutscience E&M is my favorite Unified Field Theory; the collaboration between Faraday and Maxwell is sorely underappreciated. Learning to visualize charge and current distributions and field patterns is invaluable, even with the existence of numerous E&M computation tools. The boundaries are where most of the interesting stuff in happening.

[@curiousaboutscience]

@@douglasstrother6584 There is so much to say about the power and accuracy of this theory. My first class I didn't appreciate how much was related to the importance of the boundaries.

[@randomz5890]

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.

[@PC-ee7tz]

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

[@cringotopia8850]

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.

[@georgiosapostolides1944]

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

[@seangeoghegan]

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!

[@lord7th63]

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

[@johnhamilton7762]

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

[@clieding]

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.

[@lemadfab]

I studied physics for many years and I wish I had these videos back in the day. So clear !

[@curiousstudent7961]

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

[@laman8914]

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.

[@maus3454]

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

[@rajendramisir3530]

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.

[@vignesh2891]

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 🙏🏼

[@marquesjr.5796]

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 🇧🇷

[@StratosFair]

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

[@kristianwichmann9996]

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

[@user-tp9kq8gr8h]

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

[@ecdavek230]

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.

[@antonkot6250]

Appreciate your effort and pedagogical skills

[@soulintent4129]

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

[@vernonmeidlinger870]

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!

[@bingosunnoon9341]

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

[@CADable]

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

[@jongxina3595]

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

[@swizzbeats1212]

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

[@mylittlememes7395]

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

[@kycb101]

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!

[@ibryce6ex]

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!!!

[@danieljulian4676]

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

[@aaron_wolcott]

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

[@XZellTheBest]

So high quality! Thank you!

[@odebroqueville]

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

[@ChaineYTXF]

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

[@blackguardian89]

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

[@goliadkin830]

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!

[@seanriopel3132]

You could argue the ability to express complex ideas in a simpler manner is what defines a great teacher from a sufficient one. The ability to understand a person's abilities and limitations to such an extent that you can translate the most obscure information that your target audience can easily understand and utilize is the most important factor. It's not what you know but what you can convey to others.

[@pacificll8762]

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

[@thiagoabsc]

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

[@MadScientyst]

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!!

[@aboveskyphysics]

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.

[@yogieariana]

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

[@ikechukwuewuzie3836]

Beautiful and concise. Thanks Elliot.

[@fabiobiffcg4980]

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

[@unaimarquezsanchez9452]

What a masterpiece. Please continue with this excellent work

[@codywohlers2059]

What a nice simple explanation of Hamiltonian mechanics!

[@KevinOchola]

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

[@muhammadhabibannajar1115]

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

[@takyon24]

Man this is high quality, easy some of the best physics educational content on youtube. Do you still plan on uploading any problem sets for this video? Thanks a lot for the notes btw

[@jeromevie9156]

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 !

[@sietsebuijsman8523]

Super interesting video, as always! The quality of these videos is really great. I wonder, the Hamilton equations kind of reminds me of a cross product. Is there a relation there, or am I imagining things?

[@neyhmor]

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 😊

[@patrickBaiterMan]

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.

[@Ashley-de3tu]

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

[@michaelbaum6796]

A very excellent presentation. Thanks a lot Elliot👍

[@dylanpalmer7789]

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

[@mustafizurrahman5699]

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

[@paweborkowski6959]

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

[@GlowingMpd]

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

[@--0--0-]

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

[@gbrgaming8206]

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

[@vasumehra960]

Great work man! Got to learn a lot more today. Especially the Hamiltonian way, It was awesome

[@WalterFocke]

Brilliant lecture! Thank you!

[@LivingGuy484]

I know little to nothing about Physics, but your narration and visuals were interesting enough to get me to sub If I meet any Physics students, I'll be sure to recommend this channel

[@bzl7956]

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

[@freckhard]

Hey Elliot, I am so glad I found your channel today (subscribed!) and that you have the time and opportunity to release someone of the finest "math physics" videos on KZbin that are on the same superb level of quality like 3b1b's math videos! Please feel free to dive more into details, but easier said than done I guess as it must take quite a while to create such a high quality video and maybe I am not your main target-audience 🙂

[@maurocruz1824]

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.

[@lalitasharma6687]

Reading Hamiltonian mechanics recently and this video pop up great video

[@razex_sama5744]

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.

[@douglasstrother6584]

You'll do great with Legendre Polynomials, Laplace Transformations, and Léon Brillouin's "Wave Propagation and Group Velocity"!

[@PMA65537]

@@douglasstrother6584 Too bad about Fourier, Poisson and Fresnel.

[@narfwhals7843]

Very interesting video! At 20:30 this almost looks like an Eigenvalue equation. Which makes sense, as the exponential function is the Eigenfunction of the derivative operator. So it looks like we can not only turn a DE into an algebraic equation, but into a geometry problem as well.

[@gcewing]

And the Schrödinger equation is lurking just around the corner...

[@zunaidparker]

This was brilliant! You've gained a new subscriber!

[@kth2188]

Great explanation appreciate it

[@guliyevshahriyar]

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

[@AmrFoxy93]

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

[@taejonglee6204]

Thanks for this great visualization of the equation. It really brings Physics to the imaginations of our lives.

[@vohoangquannguyen7706]

Hi Elliot, many thanks for the video. Kudos!

[@eyloneliyahukrause7518]

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)

[@subhrapratimsharma2825]

Thank you. Enjoyed the 30 minute wholeheartedly.

[@lingarajpatnaik6514]

Fantastic! Beautiful!! Great!!! I wish all students getting initiated in to physics see this before anything else.

[@thornil2231]

I need to pose and play again a few times... sorry I am a little slow...😁

[@jumpydino3015]

*a few (thousand) times

[@user-su8ke8mr6q]

I am rooting for you. Slow but constant.

[@user-rp8ij8pm7l]

That's totally fine and normal dear

[@kyrilkaplan5011]

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 !

[@sayserloks]

That was sick! Gonna try to master these methods now

[@anilgercekci8224]

Very clear explanation, bravo!

[@automazioneindustriale]

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

[@uzdefrederic1055]

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

[@flamurbedrolli802]

Stunning explanation .

[@lecturesfromleeds614]

Good video, not many people show several different methods for the same problem. This is really good for clarity and to show the consistency of mathematics

[@deepakjanardhanan7394]

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

[@Wielorybkek]

That was fun! :D Those methods seem so easy in this video, I would love to see how and why they fail for some more complex differential equations.

[@ajn8110]

Simply genius. Very impressive teacher. God bless you.

[@TripleA_channel]

Eliot! Why did I not know of your channel before! But I am glad I do now 😊

[@hrperformance]

That was awesome. I'm going to try to find Hamilton flows of other systems and see if I can plot it on python. Hamilton was an absolute 1st class genius!

[@tapiomakinen]

I was kind of hooked, when you said guessing is a valid method to solve a differential equation. I came here to admire your animations, but it was a surprised that I could follow the math (until Laplace) . It takes a h*ll of a teacher to make me enjoy math and physics this long. Thanks.

[@haniamritdas4725]

Brilliant work as always Sir. This one is another gem.

[@PhysicswithElliot]

Thanks Hani!

[@souravmandal5514]

Absolutely love this.