What are Differential Equations and how do they work?

In this video I explain what differential equations are, go through two simple examples, explain the relevance of initial conditions and how differential equations generally work, and then discuss what this means to the question whether the future is determined already. This video will be continued next week by a video about what this means for free will.

#physics #science #mathematics

0:00 Motivation and Content Summary
0:55 Example Disease Spread
3:25 Example Newton's Law
5:18 Initial Values
6:15 What are Differential Equations used for?
7:08 How Differential Equations determine the Future

[hesitantjaguar]

Science is so much more interesting when explained by somebody who obviously loves knowledge and is dedicated to spreading it.

[danev]

Dr. Hossenfelder, I was great at working out Differential Equations in my youth (I'm now 78). I stumbled on your video and decided to see if I have enough brain left to follow it through. I do, thank you. I love your passion of math and science, I have never lost my own. I look forward to whatever else you can reawaken in me.

[mm_]

I struggled so much last semester attempting to understand why we were solving these differential equations and what the outcomes meant. I was often told "If you see this, do this and you'll do fine." But "Why!!?" Thank you Dr. Hossenfelder, it fills me with joy, I finally understand why.
*Opens differential equations textbook*

[mad_gamer]

Every time I watch Sabine's videos, I realize how little I know and that I should have applied myself more in mathematics and the sciences.

[williammorton]

That is the fastest summary of the use of differential equations I have ever encountered. How did you do it all in one breath? It is also the most concise and cogent explication of the fact that differential and partial differential equations are not esoteric but powerful tools for everyday use.
Thank you Sabina. I shall make an offering at your shrine.

[remlatzargonix]

I took several courses in differential equations, many decades ago and I really enjoyed using them to model various behaviours. Partial differential equations and even stochastic differential equations I found even more interesting. I have forgotten almost all of that stuff in the time since.

[jimlynch]

I wish I had a “teacher” that could actually explain this subject as Sabine has done.👍

[powelllucas]

I am mathematically impaired. I am incapable of grasping even the most basic mathematical concepts. I can understand everything in videos of this type up to and until the program starts to produce the mathematical reasoning behind every event in science. My grasp of numbers stopped at simple arithmetic. I wish it wasn't so, but at 81 I'm too old and set in my ways for anything to change me now. But, I do comprehend everything she is trying to explain. Sabine is a great explainer in a renowned history of great explainers.

[PeekPost]

Student: What is the meaning of life?
Dr.Hossenfelder: Differential Equations! 😂😂

[jimgolab]

For the sake of some readers, I will mention that you might even say that the word "initial" in "initial conditions" refers to things you know at the start of solving the problem. You can also call initial conditions "boundary conditions" or "constraints", which simply tell you some extra information about the system, and these help you pick which one of the many possible solutions best fits the particular case you are considering.

[toddwasson]

"Differential equations."

Dammit, now I can't stop saying it.

Great video as usual!

[thegirlsquad]

Many people don’t know when buying shocks for their cars in reality they buying just the dumping coefficient of solution of a differential equation of an oscillating system created by car weight, spring and a hole in the ground.

[fesimco]

I've been subscribed to Sabine for just over a year now an she's definitely the most charismatic science popularisers. When is she getting her own PBS or Netflix spot? I'm ready for it.

[nathanokun]

Interestingly enough, this topic hit me on the head when I started my first upper-division class in Physics at UCLA in 1968. I had taken all of the required math classes, such as calculus and so forth, for a Physics Major and I went into my first day of MECHANICS 101, the baseline Physics class for essentially everything else in my Physics Major to get my Bachelor Degree. I sit down and the class starts. The professor puts the HAMILTONIAN Differential Equation on the blackboard, one of the fundamental formulae for solving a LOT of Physics problems and absolutely essential to understand for this and all later Physics classes. A small problem, though: I had not taken ANY differential equations classes because they were not on the required list! I said to the professor: "That is a differential equation. I have not had any classes on how to solve them." He looked at me and said, with no exaggeration, "Then you had better learn fast!" I was so angry that I stomped all the way to the secretary of the Director of the Physics Department of UCLA (now combined with Astronomy) and stated that they have to change their requirements to warn students that they need to take Differential Equations BEFORE getting into upper-division classes. She said she would talk to the Director about this and I left. I do not know if it ever happened, though. What did happen for me was taking two classes in this end-to-end by the best professor I ever had, Dr. David Sanchez, and deciding to take a few more math classes to get a double bachelor degree in Math and Physics, so it did have a good result in the end. But I still get mad when I think about it...

[GaryFerrao]

Thanks for your deep insights in your YouTube videos.

[marshallartz]

If I hadn’t essentially failed Algebra 1 twice back in high school, I’m sure I’d enjoy your videos even more.

Your presentations are the best, and I feel smarter for having watched them. 🧬🧪👩‍🔬

[kevinarturourrutiaalvarez]

I liked the part where you said "Differential Equations"

[rv]

Perhaps it should be also emphasized that, contrary to what a student may be lead to think from the calculus experience of solving explicitly differential equations via a bunch of tricks, *in real life science one rarely solves a differential equation* in that sense. In fact, it is feasible to find explicit solutions only for very restricted classes of such equations.
Even in the very theory of (ordinary or partial) differential equations in Mathematical Analysis, one is almost never able to find explicit solutions, and finding the latter is not the goal anyway: the goal is to understand the properties of the solutions (individually or of the whole space they form).
And even in the case in which it is possible to "write down" a solution (say by expressing it via a Fourier expansion), the expression is seldom useful in order to understand its properties.

As for applied mathematics and physics, again, one typically doesn't solve DEs explicitely (physicists say "analytically"). Methods of numerical analysis are used instead, that allow one to get a numerical approximation (typically by using an extremely powerful computer).

[aresmars]

YES! Initial conditions, like weather conditions (or a double pendulum for a simple system) have feedback loops between multiple variables, and small errors in initial conditions completely dominate outcomes after some period of time.

[grantjones]

Sabine is a gift to us thinking people. I have enjoyed many of her short physics videos, and now this one really brings those ancient studies back to life with new vision. Thank you so much.