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 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.
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!
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. 😁
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.
@@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!
@@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.
@@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.
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.
This video is for physics students, but math students or anyone with an eye for math might be interested in some of the technicalities. For the first proof, although it is easy to verify that sine and cosine functions solve the equation, it might not be obvious how we know that a combination of a sine and a cosine with the same phase is guaranteed to give the _general_ solution; that is, it might not be obvious that every solution to the differential equation has that form. But remember that the equation is _linear_ with continuous coefficients, and so the uniqueness theorem for initial value problems for nth order linear ODEs (which seems not to have a name) ensures that the solution is unique. The two coefficients A and B account for the two initial values. We know sinusoids solve the general equation, so a specific solution must be a combination of sinusoids, which just turns out to be another sinusoid. So the general solution is a sinusoid, with an amplitude and phase shift determined by the initial values. You can write this as A cos(ωt+φ) or as A sin(ωt) + B sin(ωt), which you should remember from trig or precalc as an identity of sine and cosine. Here, ω is fixed by the differential equation, but A, B, and φ are pairwise independent and depend on the initial values. Also, you may see this equation applied to pendulums, but keep in mind that this relies on the small-angle approximation sin θ = θ and so is only a good approximation when the pendulum makes a small angle to the normal. As a final note, the nature of sinusoids is such that you will typically only see solutions like this for second-order ODEs, because these functions have a period with respect to differentiation of 2 up to a constant and correspondingly have just two degrees of freedom (like an exponential, of which they are special cases). For the second example, this is a purely mathematical consequence of Newton's laws, as the video says, but I don't have time to explain it. Technically, it is a consequence of the work-energy theorem. One way you might get insight is from the kinematic equations (which themselves are purely mathematical), one of which is (v²-u²)/2=aΔx. Multiplying by mass and defining F = ma and T = ½mv², we get ΔT := T₁ - Tₒ = FΔx := W, which in this loosey-goosey world means that work equals the change in kinetic energy. From this, we define potential energy U for conservative forces such that the difference in U between two positions equals the work done by going from one to the other. Then it is simply necessary, by definition, that energy be conserved. It's slightly more complicated for nonconservative forces, but in the end, it is always possible to define potential energy in this way. That's what teachers mean when they say potential energy is the "ability to do work": it is literally defined as the work done to go from one state to another. For some people, this might demystify potential energy a little; it's not some ethereal, nondescript substance, just a property of a state defined by what happens when you change it, much like temperature or stiffness. For the third example, you may know that not every function equals its Taylor series at every point. First, the function must have derivatives of all orders at that point for the Taylor series to even be defined. Second, a Taylor series will typically only converge on some neighborhood of the point, so you have to pick a close enough reference point. Third, in pathological cases, the Taylor series will be converge at a point but not equal the value of the function there. And fourth, even when the Taylor Series does equal the value of the function at a single point, it might fail to equal it on any neighborhood of the point (i.e. given any open set containing the point, the function's Taylor series will either fail to be defined, will diverge, or will converge to a different value than the function at at least one point in that open set). In these cases, the function is said not to be "analytic" at that point, and this method will not apply. Elementary functions (functions created by composing complex numbers, +, -, ×, ÷, exp, and log) are all analytic on their respective domains. But other functions don't necessarily have these properties, so you cannot assume this approach will work for every function you come across. In physics, however, functions are almost always at least piecewise analytic, so this is rarely an obstacle. For the fourth example, the condition is far milder. A Laplace transform will always exist when the function in question is locally integrable, i.e., whenever its absolute value is Lebesgue integrable over any compact set. Essentially, if you force the function to be always positive, but the integral around any point is still finite, then the function is locally integrable. This is a weaker condition than L₁, which requires that the integral of the entire function be finite; some functions have finite integrals over finite parts but the integral over the whole function is still infinite (e.g. f(x) = x). But also, even if the integral converges only conditionally (i.e. the Lebesgue integrals over the positive and negative parts both diverge, but the appropriate conditionally convergent integral has a finite value), the equation still holds as expected. The inverse still exists and the formula remains correct. This is the most general method of them all. (Of course, the inverse Laplace transform won't always be elementary, so you might not be able to simplify at the end, and even if you can, the simplification might be far from obvious.) The fifth and final example is the narrowest and the most physically-inclined. This method only applies to systems satisfying Hamilton's equations, which were specially designed for Newtonian mechanics (but which are also applicable in an extended form to quantum mechanics). The method will work precisely when these equations hold, which is to say, precisely when they describe a general sort of dynamical system. It is not a general fact of mathematics that this is the case, but it is simply the case for physical systems. There are various "deeper" reasons one can provide for this relating to symmetry and Noether's theorem.
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!
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.
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 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.
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 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!
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.
I struggled mightily through this stuff in college. Not only was that before UA-cam but it was before electronic calculators. This is so much easier to understand.
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.
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
Great video. As an electrical engineer, Laplace transform is the way. Or we like writing down the characteristic equation of the ODE. But I understand that they are basically the same thing = guess and validate your guess.
The last method missing, which is closely related to the final two you covered, is the method of Green's functions. All the diff eq you solved here were homogenous but most diff-eq in real-life physics applications are inhomogeneous.
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.
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.
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.
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.
Method 1 is brilliantly explained so that high school students (at least those interested in math or physics) can easily understand it. I think those less interested or not interested at all could also understand it if they had the patience.
I’m fairly new to all of this and i am still stuck on the part where he chose a sinusoidal function as his “guess”, i want to know the thought process of choosing a function to solve the differential equation.
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.
As an engineer, I took a differential equations course that I found extremely frustrating. The methods seem arbitrary and there was nothing that tied things together. If I had seen this as the intro to that course, I would have had a much better appreciation of the methods I learned. Thank you! I hope this prevents the suffering of countless engineers like me :)
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!
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.
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
As a german, who wrote his bachelor thesis in english, I wondered why the word "Ansatz" was used in english papers. It could be easily translated to "approach" or "assumption", but in the end I also used "Ansatz" in my thesis. :D
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!!!
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
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.
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
I'm glad this is the case too, I've been exposed to all of this kind of material since I was a child and now that I'm in this courses is super easy to understand the courses.
I teach DiffEq. I'll be sending my students to this video probably year after year for a summary of our entire class from a broad perspective. Thank you.
The exponential form.would be my first guess looking only at the formula. I tried working it out with Ae^bt but that failed on the conditions, then I tried with Ae^bt +Ce^dt and that resulted in A = C = x0/2, b = ik/m and d = -ik/m.... or in other words a cosine, so maybe just starting a first try with that is best 😀
Good video. Pity that in my Uni times - many years ago I did not watch such video. Impression was that lecturer wants to show how smart he is and same impression about authors of handbooks :-)
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'.
Thank you for the wonderful lectures, but specific examples with solutions using numbers would be a very good addition. Half the success in understanding a given problem is a well-selected specific example with numbers and a solution.
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.
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.
I am German, we use "Ansatz" in maths a little different. I can still totally see how non-German-speakers think Ansatz is related to guessing, concerning differential equations, but the word's meaning itself is completely different. It rather means "first step" when tackling a problem, or choosing a suitable method to solve it. For example when I have my 9th grade students calculate the surface area of a cylinder: Before getting into it head over heels, I demand to write down the Ansatz first, in this case: A_{Cyl} = 2 * A_{Circ} + A_{Coat}. Then they have a free head to think about the formulas fo A_{Circ} etc. And only in the end, after doing all the algebra I have them put in the number values. So "Finding an Ansatz" is rather a method of abstracting a problem into mathamtical language first and then solve it.
My professors called the substitution solution "solution by experience" because at that point we'd solved SHO's (or more general HO's) a ton of times and you kinda just know what works!
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
The last method was so crazy. Somehow when I saw the e to the power of a matrix, while I thought it was crazy, I couldn't help but think of the Taylor series. Physics is so beautiful. Linear algebra is so beautiful. When I barely learned it, I remember I had a physics final in mechanics. I was solving a momentum problem in 2d, billiard balls, and whatnot. I could solve it instantly in my calculator without further writing when I thought I could put the information in a matrix and invert it. That sensation was so nice because no one before had told me I could do that
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.
@@xane256 i know them.. Eigenchris videos are so good and I'm one of those who were waiting them before they get uploaded.. But Dr Elliot's method will be a lot more good
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 !
These techniques are dydt = f(t,y) but there is also dydt = f(t,y,u) where u is an action. For hamiltonian flow, one is guided by a tangent T(x) at a point x in M on a manifold and so one thinks of symplectic manifolds. But when playing chess, actions like chess moves are done which also create a trajectory (the game moves), u1,u2,u3,...,un of actions or controls. One has u = policy(x) and x(t+dt) = transition(x,u) instead of an update like x(t+h) = x(t) + x'(t)*t. Also in formal languages, an alphabet Sigma and words using symbols is like a trajectory as well and choosing the next symbol to a word or word to a sentence is making decisions. Also a Markov Decision Process (MDP) relates.
What a nice compendium of the tools used to solve differential equations. Applying each of them to the same, basic problem really ties it all together. This is up there with your exposition on the magical Taylor series.
wow man, it's a great video that i was looking for so long. Thank you! What book you can suggest to start learning about diff equations? or online course etc
I love differential equations. There's just something so enticing about the interplay of sinple ideas yielding a complex idea that is super satisfying.
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.
Thank you so much Michael!
Totally agree
@@PhysicswithElliot you are the Morgan Freeman for Physics!
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
@@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.
6 - Sturm-Liouville 7 - Green's function 8 - Hypergeometric Functions 9 - Lie symmetry method and similarity invariant 10 - Advanced Perturbation Methods
bro this is an introductory ODE course level video
Please do a Video on These aswell!
I popped my brain😂😂
Green's function as in greens theorem that is expanded to stokes theorem?
You my friend, is beyond amazing
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!
Thank you dear Dr Schneider 🙏💚
Animations look amazing! Very smooth, love it
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. 😁
Great video, thanks. 3B1B is excellent!
@@orsoncart802 I see the flow going the right way, I’m pretty sure just depends which way u look at it
This channel is going to blow up in the future.
Thanks Bruh!
@@PhysicswithElliot I ain't your (alge-)bruh
Going over an E&M course, and the boundary conditions cannot be undervalued. Good stuff! Glad to see this content on UA-cam!
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.
@@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!
@@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.
@@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.
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.
This video is for physics students, but math students or anyone with an eye for math might be interested in some of the technicalities. For the first proof, although it is easy to verify that sine and cosine functions solve the equation, it might not be obvious how we know that a combination of a sine and a cosine with the same phase is guaranteed to give the _general_ solution; that is, it might not be obvious that every solution to the differential equation has that form. But remember that the equation is _linear_ with continuous coefficients, and so the uniqueness theorem for initial value problems for nth order linear ODEs (which seems not to have a name) ensures that the solution is unique. The two coefficients A and B account for the two initial values. We know sinusoids solve the general equation, so a specific solution must be a combination of sinusoids, which just turns out to be another sinusoid. So the general solution is a sinusoid, with an amplitude and phase shift determined by the initial values. You can write this as A cos(ωt+φ) or as A sin(ωt) + B sin(ωt), which you should remember from trig or precalc as an identity of sine and cosine. Here, ω is fixed by the differential equation, but A, B, and φ are pairwise independent and depend on the initial values. Also, you may see this equation applied to pendulums, but keep in mind that this relies on the small-angle approximation sin θ = θ and so is only a good approximation when the pendulum makes a small angle to the normal. As a final note, the nature of sinusoids is such that you will typically only see solutions like this for second-order ODEs, because these functions have a period with respect to differentiation of 2 up to a constant and correspondingly have just two degrees of freedom (like an exponential, of which they are special cases).
For the second example, this is a purely mathematical consequence of Newton's laws, as the video says, but I don't have time to explain it. Technically, it is a consequence of the work-energy theorem. One way you might get insight is from the kinematic equations (which themselves are purely mathematical), one of which is (v²-u²)/2=aΔx. Multiplying by mass and defining F = ma and T = ½mv², we get ΔT := T₁ - Tₒ = FΔx := W, which in this loosey-goosey world means that work equals the change in kinetic energy. From this, we define potential energy U for conservative forces such that the difference in U between two positions equals the work done by going from one to the other. Then it is simply necessary, by definition, that energy be conserved. It's slightly more complicated for nonconservative forces, but in the end, it is always possible to define potential energy in this way. That's what teachers mean when they say potential energy is the "ability to do work": it is literally defined as the work done to go from one state to another. For some people, this might demystify potential energy a little; it's not some ethereal, nondescript substance, just a property of a state defined by what happens when you change it, much like temperature or stiffness.
For the third example, you may know that not every function equals its Taylor series at every point. First, the function must have derivatives of all orders at that point for the Taylor series to even be defined. Second, a Taylor series will typically only converge on some neighborhood of the point, so you have to pick a close enough reference point. Third, in pathological cases, the Taylor series will be converge at a point but not equal the value of the function there. And fourth, even when the Taylor Series does equal the value of the function at a single point, it might fail to equal it on any neighborhood of the point (i.e. given any open set containing the point, the function's Taylor series will either fail to be defined, will diverge, or will converge to a different value than the function at at least one point in that open set). In these cases, the function is said not to be "analytic" at that point, and this method will not apply. Elementary functions (functions created by composing complex numbers, +, -, ×, ÷, exp, and log) are all analytic on their respective domains. But other functions don't necessarily have these properties, so you cannot assume this approach will work for every function you come across. In physics, however, functions are almost always at least piecewise analytic, so this is rarely an obstacle.
For the fourth example, the condition is far milder. A Laplace transform will always exist when the function in question is locally integrable, i.e., whenever its absolute value is Lebesgue integrable over any compact set. Essentially, if you force the function to be always positive, but the integral around any point is still finite, then the function is locally integrable. This is a weaker condition than L₁, which requires that the integral of the entire function be finite; some functions have finite integrals over finite parts but the integral over the whole function is still infinite (e.g. f(x) = x). But also, even if the integral converges only conditionally (i.e. the Lebesgue integrals over the positive and negative parts both diverge, but the appropriate conditionally convergent integral has a finite value), the equation still holds as expected. The inverse still exists and the formula remains correct. This is the most general method of them all. (Of course, the inverse Laplace transform won't always be elementary, so you might not be able to simplify at the end, and even if you can, the simplification might be far from obvious.)
The fifth and final example is the narrowest and the most physically-inclined. This method only applies to systems satisfying Hamilton's equations, which were specially designed for Newtonian mechanics (but which are also applicable in an extended form to quantum mechanics). The method will work precisely when these equations hold, which is to say, precisely when they describe a general sort of dynamical system. It is not a general fact of mathematics that this is the case, but it is simply the case for physical systems. There are various "deeper" reasons one can provide for this relating to symmetry and Noether's theorem.
Thank you, your comment answered my questions
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!
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.
como te fue en el examen?
Very interesting! It was definitely instructive to see all 5 techniques applied to the same example.
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 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 studied physics for many years and I wish I had these videos back in the day. So clear !
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 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!
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.
I struggled mightily through this stuff in college. Not only was that before UA-cam but it was before electronic calculators. This is so much easier to understand.
Would love to see a similar video on partial differential equations :) Thank you for your content very well explained!
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'll do great with Legendre Polynomials, Laplace Transformations, and Léon Brillouin's "Wave Propagation and Group Velocity"!
@@douglasstrother6584 Too bad about Fourier, Poisson and Fresnel.
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
Great video. As an electrical engineer, Laplace transform is the way. Or we like writing down the characteristic equation of the ODE. But I understand that they are basically the same thing = guess and validate your guess.
The last method missing, which is closely related to the final two you covered, is the method of Green's functions.
All the diff eq you solved here were homogenous but most diff-eq in real-life physics applications are inhomogeneous.
Wow! No distractingly unnecessary music over your excellent narrative skills and important information??? I’m exponentially impressed!!!!👍😃
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.
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.
And the Schrödinger equation is lurking just around the corner...
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.
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.
Finally, a channel that I can watch without torturing my eyes! Show me a black text on a light background, and I’m yours! Just subscribed.
Method 1 is brilliantly explained so that high school students (at least those interested in math or physics) can easily understand it. I think those less interested or not interested at all could also understand it if they had the patience.
I’m fairly new to all of this and i am still stuck on the part where he chose a sinusoidal function as his “guess”, i want to know the thought process of choosing a function to solve the differential equation.
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.
Just came across your video. Holy, the best I have ever seen in explaining and summarizing in such concise and clear terms! Thanks!
lovely intro about not only the physics but also for the math and general engineering. Great video!
You are a terrific educator, sir. Thank you. This was superbly constructed.
I used laplace by far the most in EE, or some roughly approximate discrete/digital form
4th & 5th methods are mind blowing especially Hamilton's Flow. Thank you for sharing.
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 🙏🏼
Amazing video. I saw this topics before but this video really makes me enjoy what I couldnt while taking these classes...
Oh Dear Lord ...where was that video when i was in college ??? super fantastic ...well done young man.
As an engineer, I took a differential equations course that I found extremely frustrating. The methods seem arbitrary and there was nothing that tied things together. If I had seen this as the intro to that course, I would have had a much better appreciation of the methods I learned. Thank you! I hope this prevents the suffering of countless engineers like me :)
The way in which he explains gives people clarity and the animations are really cool ✨
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!
First time I understand what a Laplace Transform a Hamiltonian are! Very clear explanation. Thank you.
Method 0: use Mathematica
Method 0: go to mit open courseware
Ask wolfram alpha
@@StuffinroundWolfram Alpha is weak compared to Mathematica (and this is also logically comprehensible)
😂
The method of the gods
Appreciate your effort and pedagogical skills
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.
Great stuff 🙂I know you already did a video on Hamiltonian mechanics, but a deeper explanation of the Legendre transform involved would be nice.
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
Love the videos! What program do you use to make such videos?
I gain so much appreciation for physics despite graduating in engineering and completely taking physic as granted
As a german, who wrote his bachelor thesis in english, I wondered why the word "Ansatz" was used in english papers. It could be easily translated to "approach" or "assumption", but in the end I also used "Ansatz" in my thesis. :D
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!!!
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
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.
Found this through UA-cam recommended, and I have to say this video is a masterpiece. Instantly subscribed and looking forward to more videos from you
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
Thank you for this! It seemed daunting initially when seeing Hamiltonian equations. But you simplified and made it fun 👍🏼
I have a degree in physics. This video is beautiful. Lucky all the students that now can find this kind of material free at anytime in a phone
I'm glad this is the case too, I've been exposed to all of this kind of material since I was a child and now that I'm in this courses is super easy to understand the courses.
I teach DiffEq. I'll be sending my students to this video probably year after year for a summary of our entire class from a broad perspective. Thank you.
Thank you so much, especially to see the Laplace transform in use was an eye-opener
The exponential form.would be my first guess looking only at the formula. I tried working it out with Ae^bt but that failed on the conditions, then I tried with Ae^bt +Ce^dt and that resulted in A = C = x0/2, b = ik/m and d = -ik/m.... or in other words a cosine, so maybe just starting a first try with that is best 😀
Good video. Pity that in my Uni times - many years ago I did not watch such video. Impression was that lecturer wants to show how smart he is and same impression about authors of handbooks :-)
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.
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'.
Thank you for the wonderful lectures, but specific examples with solutions using numbers would be a very good addition.
Half the success in understanding a given problem is a well-selected specific example with numbers and a solution.
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.
Great insight to see everything together... thanks!!!
As engineer I'll keep with Laplace but uncle Hamilton was incredible! Nice...
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.
I am German, we use "Ansatz" in maths a little different. I can still totally see how non-German-speakers think Ansatz is related to guessing, concerning differential equations, but the word's meaning itself is completely different. It rather means "first step" when tackling a problem, or choosing a suitable method to solve it.
For example when I have my 9th grade students calculate the surface area of a cylinder: Before getting into it head over heels, I demand to write down the Ansatz first, in this case: A_{Cyl} = 2 * A_{Circ} + A_{Coat}.
Then they have a free head to think about the formulas fo A_{Circ} etc.
And only in the end, after doing all the algebra I have them put in the number values.
So "Finding an Ansatz" is rather a method of abstracting a problem into mathamtical language first and then solve it.
My professors called the substitution solution "solution by experience" because at that point we'd solved SHO's (or more general HO's) a ton of times and you kinda just know what works!
Mathematics and physics are the language of the universe and a means of understanding existence. Mathematics and physics forever.
Your videos are fantastic. I am fourteen, English is my second language, and I understood everything
Great content👍👍...... wonderful explanation... thankyou very much...loved it
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
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!!
Bravo! One of the clearest and detailed lesson I have ever seen...
Reading Hamiltonian mechanics recently and this video pop up great video
The last method was so crazy. Somehow when I saw the e to the power of a matrix, while I thought it was crazy, I couldn't help but think of the Taylor series. Physics is so beautiful.
Linear algebra is so beautiful. When I barely learned it, I remember I had a physics final in mechanics. I was solving a momentum problem in 2d, billiard balls, and whatnot. I could solve it instantly in my calculator without further writing when I thought I could put the information in a matrix and invert it. That sensation was so nice because no one before had told me I could do that
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.
I am a french student, this video is very good thank you !
Now I can finally say I am enjoying Physics. Hats off to you!!!
Increadible explanation! I would like to recomended this video to my students later on. Thanks :)
I think a complete general relativity course by you will be incredibly good!
You might find these helpful:
ua-cam.com/play/PLJHszsWbB6hqlw73QjgZcFh4DrkQLSCQa.html
ua-cam.com/play/PLSuQRd4LfSUTmb_7IK7kAzxJtU2tpmEd3.html
@@xane256 i know them.. Eigenchris videos are so good and I'm one of those who were waiting them before they get uploaded.. But Dr Elliot's method will be a lot more good
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 !
Thanks for the explanation, would love to see the Poisson Equation on gravitational field on next video. It would be great!
Wow. Interpretation matrix power in terms of Taylor series has some phycisc meaning. It's blowed my mind
I really love the animation style you use in this video. Looks pretty similar to what I do, lots of PowerPoint, just…um….how to say…..better😂
Thanks doc! It's the first one I've made using Manim. It's a crazy powerful tool, but took a lot of trial and error to learn!
These techniques are dydt = f(t,y) but there is also dydt = f(t,y,u) where u is an action. For hamiltonian flow, one is guided by a tangent T(x) at a point x in M on a manifold and so one thinks of symplectic manifolds. But when playing chess, actions like chess moves are done which also create a trajectory (the game moves), u1,u2,u3,...,un of actions or controls. One has u = policy(x) and x(t+dt) = transition(x,u) instead of an update like x(t+h) = x(t) + x'(t)*t. Also in formal languages, an alphabet Sigma and words using symbols is like a trajectory as well and choosing the next symbol to a word or word to a sentence is making decisions. Also a Markov Decision Process (MDP) relates.
The hardest thing gets simplified here. Nice explanation specially series expansion and Hamiltonian flow (because this also has series expansion)😀😀
With a bit of dimensional analysis on k/m the ansatz in the first method is even more clear.
This was an exceptional clear explanation. Thanks.
You gave me a different type of thinking...so thank you so much
Fantastic! Beautiful!! Great!!! I wish all students getting initiated in to physics see this before anything else.
physics and mathematics is my passion , thank you for ligthing my mind , see you in future
What a nice simple explanation of Hamiltonian mechanics!
Best Physics teacher on this platform.
What a nice compendium of the tools used to solve differential equations. Applying each of them to the same, basic problem really ties it all together.
This is up there with your exposition on the magical Taylor series.
Thanks Glen!
You've just earned another subscriber. Brilliant and elegant.
Splendid! Nicely presented and generous in content for introducing the concepts. You have a new subscriber.
wow man, it's a great video that i was looking for so long. Thank you! What book you can suggest to start learning about diff equations? or online course etc
Thanks Tima! Riley, Hobson, Bence's Mathematical Methods for Physics have several chapters with a nice overview
Simply genius. Very impressive teacher. God bless you.
I love differential equations. There's just something so enticing about the interplay of sinple ideas yielding a complex idea that is super satisfying.