Physics Students Need to Know These 5 Methods for Differential Equations

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• 4S

  @4skyrider · 3 years ago

  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.

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• CA

  @cacor4686 · 2 years ago (edited)

  6 - Sturm–Liouville   7 - Green's function 8 - Hypergeometric Functions 9 - Lie symmetry method  and similarity invariant  10 - Advanced Perturbation Methods

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• BR

  @bruh4196 · 3 years ago

  This channel is going to blow up in the future.

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• PH

  @PhysicswithElliot · 3 years ago

  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!

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• CU

  @curiousaboutscience · 3 years ago

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

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• EE

  @EebstertheGreat · 3 years ago (edited)

  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 <i>general</i> solution; that is, it might not be obvious that every solution to the differential equation has that form. But remember that the equation is <i>linear</i> 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).<br><br>For the second example, this is a purely mathematical consequence of Newton&apos;s laws, as the video says, but I don&apos;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&apos;s slightly more complicated for nonconservative forces, but in the end, it is always possible to define potential energy in this way. That&apos;s what teachers mean when they say potential energy is the &quot;ability to do work&quot;: 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&apos;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.<br><br>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&apos;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 &quot;analytic&quot; 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&apos;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.<br><br>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&apos;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.)<br><br>The fifth and final example is the narrowest and the most physically-inclined. This method only applies to systems satisfying Hamilton&apos;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 &quot;deeper&quot; reasons one can provide for this relating to symmetry and Noether&apos;s theorem.

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• JO

  @johnchessant3012 · 3 years ago

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

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• PU

  @PunmasterSTP · 3 years ago

  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. &nbsp;Thank you so much for taking the time and effort to make these really high-quality videos and then sharing them for free!

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• WI

  @wissamkadamani · 1 year ago

  Got recommended this while leaving the differential equations exam

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• CI

  @CitizenOfTheWorld2025 · 3 years ago (edited)

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

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• GE

  @georgiosapostolides1944 · 3 years ago

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

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• HA

  @halite01 · 3 years ago

  You&apos;re my favourite physics tutor! I can&apos;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&apos;ve answered to a lot of my questions so I can&apos;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.

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