1.0 A better way to understand Differential Equations | Nonlinear Dynamics | 1D Linear Diff Eqns

[@virtually_passed]

Hey guys, this is the first video in this mini-series. The next video is going to be on solving 2nd order linear differential equations!

[@superyoshiryan]

This explanation is fantastic! I feel I might display some ignorance here, but I see a lot of parallels with the thrust-velocity curves for SLUF (straight, level and unaccelerated flight) for conventional aircraft. Where we have both a stable and critically stable point depending on the thrust available and the trust required for our aircraft. Your explanation of this is very intuitive, on par with 3 blue one brown. Thank you!

[@snnwstt]

It shares a lot with Poincare's introduction to (mathematical) chaos theory, at first glance. Nice idea to plot dx/dt versus x, indeed.

[@rubamo5589]

This is the most useful video I've seen this year!!! Changed the way I see equation 😲 brilliant

[@virtually_passed]

glad you liked it!

[@ouvie]

You explain it well, and also you're pretty underrated. Keep up the good work

[@virtually_passed]

Thanks so much!

[@ouvie]

@@virtually_passed You're welcome, I'm gonna subscribe!

[@mohammedhamouda5986]

Amazing explanation 💞💕💞

[@chrisjudge705]

Super genius ART of Explanation👍👍👍👍👍👍

[@michaeledwardharris]

Dude, that's the best explanation of that I've ever seen! Amazing work.

[@virtually_passed]

Glad you liked it!

[@dyco486]

This video is so great, good job on explaining man!

[@virtually_passed]

Thank you! That means a lot :)

[@johnunverzagt9387]

Thank you, sir! I earned my undergrad degree in math over ten years ago. As of late my interest in the subject has been waning. Your videos on non linear dynamics have put some spark back into my mathematical interest.

[@virtually_passed]

Glad to hear that ! I've got a few more videos that are a few weeks away from release!

[@aamer5091]

Your videos are great! It really helps build intuition and can let me follow along as I probably understand up to about grade 10 math (generously). As an aside could you walk me through how you went from your fraction integration (or expansion, or how ever you solved it) from line 2 to line 3 at time 0:58. I'm brushing up on my math but haven't yet been able to solve this step as I follow along... after failing many a time.

[@virtually_passed]

Hi! Great question. Let's start at line 2: ∫ 1/(0.5*x² - 2) dx Multiply numerator and denominator by 2 2 ∫ 1/(x² - 4) dx Factor the denominator 2 ∫ 1/((x-2)(x+2)) dx Now the aim is to write the expression in this integral as the sum of two simpler fractions (this is known as the method of partial fractions) 1/((x-2)(x+2)) = a/(x-2) + b/(x+2) where 'a' and 'b' are constants that we need to find. To find 'a' and 'b', expand the RHS to have a common denominator RHS = a*(x+2)/(x-2)*(x+2) + b*(x-2)/(x+2)*(x-2) RHS = ((a+b)x + (2a-2b)) / ((x-2)(x+2)) Now notice that the RHS is in the same form as the LHS, so equate coefficients in the numerator and you'll get two equations: 1) a + b = 0 2) 2a - 2b = 1 Solving these two equations simultaneously will yield a = 0.25 b = - 0.25 Therefore the integral becomes 2 ∫ 0.25/(x-2) - 0.25/(x+2) dx Factor out the 0.25 1/2 ∫ 1/(x-2) - 1/(x+2) dx Hope that makes sense :)

[@singreed7635]

Thanks it would help me somewhere in jee preparation!

[@johnchessant3012]

Great explanation!

[@virtually_passed]

Thanks!

[@willie333b]

This is intuitive. I have never seen this in school 😮

[@virtually_passed]

Glad you liked it

[@tjalferes]

Thank you.

[@malawigw]

Very good video series, clear and concise! Would be great with some excersice for the viewer and suggestions for software

[@virtually_passed]

Thanks for the comment and feedback. I might create some examples of 1D flows in the future, but in the meantime if you want a source of some good examples I strongly suggest Steven Strogatz book "nonlinear dynamics and chaos". With regards to software I'd recommend Google'ing "phase plane plotter" and there's an online web app that can graph these things pretty well. However if you're looking for something more rubust then MATLAB is a solid option. Hope that helps!

[@RSLT]

Very Interesting Great video!

[@virtually_passed]

Thanks!

[@k.i.l.l.7935]

may the force be with you.

[@virtually_passed]

Thank you padawan

[@ToriKo_]

I’m an idiot when it comes to math and know basically nothing, so maybe ignore me, but: Can you help me understand what the x with the dot on top of it is? Is it just f’(x) if x is f(x)? I’ve never seen that notation before Also, you plot f’(x) as the y-axis and f(x) as the x-axis @ 1:24 , we usually have some intuition surrounding when we plot a variable against time t, or against another variable y. We can kind of understand that x changes as you go forward in time or that x changes as you change another variable y. But what does it mean when we plot f(x) against f’(x)? I’m struggling to have any intuitional understanding. You use the word velocity here, while I understand that velocity can be understood as the change in position over the change in time [f’(x)], a lot of differentials aren’t velocities, and aren’t talking about position or time, so what does that mean? Velocity of what actually? What does ‘pushing a particle’ mean when the equations we’re using aren’t talking about particles? 2:10 ‘Over time the particle would move depending on the initial conditions’, but what does it mean for us to place a ‘particle’ somewhere and call it a initial condition? These are probably very stupid questions Also also, @ 2:01 you condense the graph of f(x) and f’(x) into just the x-axis. I’m having a hard time explaining this question, but in short, why are we allowed to do that, and how does that interact with us understanding the graph as a ‘particle in vector fields’? Also also also, you say that this technique gives us some understanding around first order differentials, but can we also do this to higher order differentials? Can we only do this with differentials ‘next to each other’? Or can you do this with f’(x) plotted against f’’’’’’(x)? And lastly (by the grace of the gods), what does it mean for the slope to be vertical? Thank you

[@virtually_passed]

Hi Tori, thanks for your questions. Let me start at the beginning. x_dot is just a mathematical notation and it's defined to be: x_dot = dx/dt (the derivative of position, x, with respect to time, t) So that means that the equation that I showed at 0:47 is: x_dot = 1/2 x^2 - 2 which is the same as dx/dt = 1/2 x^2 - 2 Notice that the Left Hand Side is dx/dt and on the Right Hand Side is just some function of x. In general, these types of equations can be written as: dx/dt = f(x) In English, this equation means "The velocity of a particle is determined purely by the position of that particle, x". So for example, if my particle was at a position x = 4 meters, then my particle will have a velocity dx/dt = f(4) = 1/2 (4)^2 - 2 = 6 meters per second to the right. You mentioned in your comment that speed is f'(x). This is not true. In this context, speed is just f(x) (because f(x) = dx/dt). -------------------- In the chart at 1:24 I DON'T plot f'(x) vs f(x)! Instead, on the y-axis I plot x_dot and on the x-axis I plot x. This means I'm essentially plotting speed versus position. This is why the equation shown looks like a parabola because: x_dot = 1/2 x^2 - 2 -------------------- I thought I'd add in a few extra details. I hope this doesn't confuse you. While it can be easily seen from the equation above that dx/dt is a function of x, a more subtle point is that x is also a function of time. x = x(t) So, if you like, another way of writing out the equation above is to say: dx/dt = f(x(t)) But this is a minor point that isn't worth too much attention in my opinion. -------------------- It's true that this technique can be extended to higher order differential equations. Equations like: d^2x/dt^2 + 3 * dx/dt + 4 * x = 0 And that's because equations like these can be rewritten in vector form as dX/dt = f(X) Where X is a vector = [x; dx/dt]. I talk about this more in my second episode here: kzbin.info/www/bejne/f2q4dnWIrsZnjpI --------------------- You asked, "what does it mean for a slope to be vertical"? I'm not sure where you are referring to this in the video, but here's the answer: To put it crudely, A horizontal slope is where the derivative = 0. And a vertical slope is where the derivative = +- infinity. For example, if I have some equation y = 1/x then the slope is dy/dx = - 1/x^2. Notice that dy/dx = +-infinity when x approaches 0. This means the slope is vertical when x = 0. ----------------------- I hope this makes sense. This can be a really challenging topic so if you want some more calculus background I'd highly recommend looking at some calculus videos by the Khan Academy. Best of luck! Let me know if you have any further questions :) :)

[@lemurpotatoes7988]

@@virtually_passed I'm brand new to your channel, here because someone in EleutherAI recommended your videos. It's really encouraging to see you take the time to answer questions like this, makes me look forward to getting to know your videos better.

[@virtually_passed]

@@lemurpotatoes7988 Hi thanks for the comment. Ah I'm embarrassed to say I've never heard of EleutherAI before but I'm glad someone is recommending my videos :)

[@mohinuddin8943]

Which programming language had been used to find the nice animation?

[@virtually_passed]

I used manim which is a library in python

[@hamzamohamed7935]

This is pure beauty

[@virtually_passed]

Thanks!

[@niccoboa]

Great! I loved it!

[@virtually_passed]

Glad you liked it!

[@gaganaut06]

How are these animations made?

[@virtually_passed]

Hello, I used python to code these animations. Specifically I used "manim" - a mathematical animation library. I did the final editing using filmora pro

[@gaganaut06]

@@virtually_passed looks great

[@algorithms_mit]

Could you please share the code of animation

[@Tabu11211]

Very cool

[@virtually_passed]

Thanks :)

[@Boeing773B]

I still don't get it...

[@willie333b]

#SoME3

[@haithamalbadi2192]

hello , i want to learn math from zero

[@fernandolk4536]

Jesus!