Green's functions: the genius way to solve DEs

Green's functions is a very powerful and clever technique to solve many differential equations, and since differential equations are the language of lots of physics, including both classical mechanics, electrodynamics, and even quantum field theory, so it is important to know how it works. Of course, this includes some explanation and perhaps a pretty different motivation for Dirac delta function, which is pretty weird, but also not really when you think about it a different way.

Correction: in 19:11, the Green's function lacks a factor of 1/m.

This video simply aims to introduce the Green's functions, what it is supposed to do, how the motivation of it all comes to be, and why it works. If you do need a lot more than introductory knowledge on Green's functions, and you are comfortable in basic differential equation solving, here are some links:

https://brilliant.org/wiki/greens-functions-in-physics/ (not sponsored)
https://www.roe.ac.uk/japwww/teaching/fourier/fourier_lectures_part4.pdf
My further write-up on the examples of Green's functions (how to find): https://drive.google.com/file/d/1D6E857eTvqM1CQgS1vYwcLqhLeGFL-aV/view?usp=sharing

For those who want some answers for the exercise towards the end of Chapter 3, i.e. around 15:47:

Essentially, what I intended was that using that momentum change = integral of force over small period of time, you can obtain the first answer (by a similar definition of delta function in 1D), and I am expecting "point impulse / impulse" on Q2.

For Q3: It is supposed to be that "applied force can be thought of as a 'continuous sum' of point impulses".

For Q4: the Green's function describes the displacement of the oscillator after we apply an impulse. For this reason, Green's function is usually called the "impulse response".

For Q5: Exactly copying the "adding different charge distributions (implies) adding up the electric potential", so in this case, "adding different forces (implies) adding up the displacement"

For Q6: From the formula that x(t) = int G(t, tau)*F(tau) d(tau), we can interpret that the displacement is a continuous sum of the impulse responses.

I stopped saying anything more because (1) the video is already very long, (2) this video assumes only basic knowledge of calculus (it is actually better if you don't know too much of the rigour in real analysis, since this is really hand-wavy - and it has to be! Otherwise this would be a lecture in distribution theory, which I am not quite well-versed in), and (3) this really just aims to provide motivation for Green's functions and doing examples would make this more "textbook-y" than it already is.

Of course, the link to the Wikipedia table of Green's functions:
https://en.wikipedia.org/wiki/Green%27s_function#Table_of_Green's_functions

Note: they don't state the boundary / initial conditions explicitly, and they don't even use x and xi, or t and tau, usually just their difference. Usually it is that the Green's functions vanish when the position is far away from the origin, and for those involving time, 0 before time tau, assuming that tau is greater than 0 (the so-called "advanced" Green's function)

A little bit of remark after viewing the video once again: in some places, it is a little bit quick, so please treat yourself by pausing if necessary: YouTube allows you to do so! In my defense, different people require different time to pause, and also I don't want too much of dead air, so... that is probably also how a lot of other math videos on YouTube are doing right now.

Video chapters:
00:00 Introduction
01:01 Linear differential operators
03:54 Dirac delta "function"
09:56 Principle of Green's functions
15:50 Sadly, DE is not as easy

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Видео Green's functions: the genius way to solve DEs канала Mathemaniac