Quoting Steven Strogatz, “Since Newton,

mankind has come to realize that the laws of physics are always expressed in the language

of differential equations.” Of course, this language is spoken well beyond the boundaries

of physics as well, and being able to speak it and read it adds a new color to how you

view the world around you. In the next few videos, I want to give a sort

of tour of this topic. To aim is to give a big picture view of what this part of math

is all about, while at the same time being happy to dig into the details of specific

examples as they come along. I’ll be assuming you know the basics of

calculus, like what derivatives and integrals are, and in later videos we’ll need some

basic linear algebra, but not much beyond that. Differential equations arise whenever it’s

easier to describe change than absolute amounts. It’s easier to say why population sizes

grow or shrink than it is to describe why the have the particular values they do at

some point in time; It may be easier to describe why your love for someone is changing than

why it happens to be where it is now. In physics, more specifically Newtonian mechanics, motion

is often described in terms of force. Force determines acceleration, which is a statement

about change. These equations come in two flavors; Ordinary

differential equations, or ODEs, involving functions with a single input, often thought

of as time, and Partial differential equations, or PDEs, dealing with functions that have

multiple inputs. Partial derivatives are something we’ll look at more closely in the next video;

you often think of them involving a whole continuum of values changing with time, like

the temperature of every point in a solid body, or the velocity of a fluid at every

point in space. Ordinary differential equations, our focus for now, involve only a finite collection

of values changing with time. It doesn’t have to be time, per se, your

one independent variable could be something else, but things changing with time are the

prototypical and most common examples of differential equations.

Physics (simple) Physics offers a nice playground for us here,

with simple examples to start with, and no shortage of intricacy and nuance as we delve

deeper. As a nice warmup, consider the trajectory

of something you throw in the air. The force of gravity near the surface of the earth causes

things to accelerate downward at 9.8 m/s per second. Now unpack what that really means:

If you look at some object free from other forces, and record its velocity every second,

these vectors will accrue an additional downward component of 9.8 m/s every second. We call

this constant 9.8 “g”. This gives an example of a differential equation,

albeit a relatively simple one. Focus on the y-coordinate, as a function of time. It’s

derivative gives the vertical component of velocity, whose derivative in turn gives the

vertical component of acceleration. For compactness, let’s write this first derivative as y-dot,

and the second derivative as y-double-dot. Our equation is simply y-double-dot=-g.

This is one where you can solve by integrating, which is essentially working backwards. First,

what is velocity, what function has -g as a derivative? Well, -g*t. Or rather, -g*t

+ (the initial velocity). Notice that you have this degree of freedom which is determined

by an initial condition. Now what function has this as a derivative? -(½)g*t^2 + v_0

* t. Or, rather, add in a constant based on whatever the initial position is. Things get more interesting when the forces

acting on a body depend on where that body is. For example, studying the motion of planets,

stars and moons, gravity can no longer be considered a constant. Given two bodies, the

pull on one is in the direction of the other, with a strength inversely proportional to

the square of the distance between them. As always, the rate of change of position

is velocity, but now the rate of change of velocity is some function of position. The

dance between these mutually-interacting variables is mirrored in the dance between the mutually-interacting

bodies which they describe. So often in differential equations, the puzzles

you face involve finding a function whose derivative and/or higher order derivatives

are defined in terms of itself. In physics, it’s most common to work with

second order differential equations, which means the highest derivative you find in the

expression here is a second derivative. Higher order differential equations would be ones

with third derivatives, fourth derivatives and so on; puzzles with more intricate clues. The sensation here is one of solving an infinite

continuous jigsaw puzzle. In a sense you have to find infinitely many numbers, one for each

point in time, constrained by a very specific way that these values intertwine with their

own rate of change, and the rate of change of that rate of change. I want you to take some time digging in to

a deceptively simple example: A pendulum. How does this angle theta that it makes with

the vertical change as a function of time. This is often given as an example in introductory

physics classes of harmonic motion, meaning it oscillates like a sine wave. More specifically,

one with a period of 2pi * L/g, where L is the length of the pendulum, and g is gravity. However, these formulas are actually lies.

Or, rather, approximations which only work in the realm of small angles. If you measured

an actual pendulum, you’d find that when you pull it out farther, the period is longer

than what that high-school physics formulas would suggest. And when you pull it really

far out, the value of theta vs. time doesn’t even look like a sine wave anymore. First thing’s first, let’s set up the

differential equation. We’ll measure its position as a distance x along this arc. If

the angle theta we care about is measured in radians, we can write x and L*theta, where

L is the length of the pendulum. As usual, gravity pulls down with acceleration

g, but because the pendulum constrains the motion of this mass, we have to look at the

component of this acceleration in the direction of motion. A little geometry exercise for

you is to show that this little angle here is the same as our theta. So the component

of gravity in the direction of motion, opposite this angle, will be -g*sin(theta). Here we’re considering theta to be positive

when the pendulum is swung to the right, and negative when it’s swung to the left, and

this negative sign in the acceleration indicates that it’s always pointed in the opposite

direction from displacement. So the second derivative of x, the acceleration, is -g*sin(theta).

Since x is L*theta, that means the second derivative of theta is -(g/L) * sin(theta).

To be somewhat more realistic, let’s add in a term to account for air resistance, which

perhaps we model as being proportional to the velocity. We write this as -mu * theta-dot,

where -mu is some constant determining how quickly the pendulum loses energy. This is a particularly juicy differential

equation. Not easy to solve, but not so hard that we can’t reasonably get some meaningful

understanding of it. At first you might think that this sine function

relates to the sine wave pattern for the pendulum. Ironically, though, what you’ll eventually

find is that the opposite is true. The presence of the sine in this equation is precisely

why the real pendulum doesn’t oscillate with the sine wave pattern. If that sounds odd, consider the fact that

here, the sine function takes theta as an input, but the approximate solution has the

value theta itself oscillating as a sine wave. Clearly something fishy is afoot. One thing I like about this example is that

even though it’s comparatively simple, it exposes an important truth about differential

equations that you need to be grapple with: They’re really freaking hard to solve. In this case, if we remove the damping term,

we can just barely write down an analytic solution, but it’s hilariously complicated,

involving all these functions you’re probably never heard of written in terms of integrals

and weird inverse integral problems. Presumably, the reason for finding a solution

is to then be able to make computations, and to build an understanding for whatever dynamics

your studying. In a case like this, those questions have just been punted off to figuring

out how to compute and understand these new functions. And more often, like if we add back this dampening

term, there is not a known way to write down an exact solution analytically. Well, for

any hard problem you could just define a new function to be the answer to that problem.

Heck, even name it after yourself if you want. But again, that’s pointless unless it leads

you to being able to compute and understand the answer. So instead, in studying differential equations,

we often do a sort of short-circuit and skip the actual solution part, and go straight

to building understanding and making computations from the equations alone. Let me walk through

what that might look like with the Pendulum. Phase space

What do you hold in your head, or what visualization could you get some software to pull up for

you, to understand the many possible ways a pendulum governed by these laws might evolve

depending on its starting conditions? You might be tempted to try imagining the

graph of theta(t), and somehow interpreting how its position, slope, and curvature all

inter-relate. However, what will turn out to be both easier and more general is to start

by visualizing all possible states of the system in a 2d plane. The state of the pendulum can be fully described

by two numbers, the angle, and the angular velocity. You can freely change these two

values without necessarily changing the other, but the acceleration is purely a function

of these two values. So each point of this 2d plane fully describes the pendulum at a

given moment. You might think of these as all possible initial conditions of the pendulum.

If you know this initial angle and angular velocity, that’s enough to predict how the

system will evolve as time moves forward. If you haven’t worked with them, these sorts

of diagrams can take a little getting used to. What you’re looking at now, this inward

spiral, is a fairly typical trajectory for our pendulum, so take a moment to think carefully

about what’s being represented. Notice how at the start, as theta decreases, theta-dot

gets more negative, which makes sense because the pendulum moves faster in the leftward

direction as it approaches the bottom. Keep in mind, even though the velocity vector on

this pendulum is pointed to the left, the value of that velocity is being represented

by the vertical component of our space. It’s important to remind yourself that this state

space is abstract, and distinct from the physical space where the pendulum lives and moves. Since we’re modeling it as losing some energy

to air resistance, this trajectory spirals inward, meaning the peak velocity and displacement

each go down by a bit with each swing. Our point is, in a sense, attracted to the origin

where theta and theta-dot both equal 0. With this space, we can visualize a differential

equation as a vector field. Here, let me show you what I mean. The pendulum state is this vector, [theta,

theta-dot]. Maybe you think of it as an arrow, maybe as a point; what matters is that it

has two coordinates, each a function of time. Taking the derivative of that vector gives

you its rate of change; the direction and speed that it will tend to move in this diagram.

That derivative is a new vector, [theta-dot, theta-double-dot], which we visualize as being

attached to the relevant point in this space. Take a moment to interpret what this is saying. The first component for this rate-of-change

vector is theta-dot, so the higher up we are on the digram, the more the point tends to

move to the right, and the lower we are, the more it tends to move to the left. The vertical

component is theta-double-dot, which our differential equation lets us rewrite entirely in terms

of theta and theta-dot. In other words, the first derivative of our state vector is some

function of that vector itself. Doing the same at all points of this space

will show how the state tends to change from any position, artificially scaling down the

vectors when we draw them to prevent clutter, but using color to loosely indicate magnitude. Notice that we’ve effectively broken up

a single second order equation into a system of two first order equations. You might even

give theta-dot a different name to emphasize that we’re thinking of two separate values,

intertwined via this mutual effect they have on one and other’s rate of change. This

is a common trick in the study of differential equations, instead of thinking about higher

order changes of a single value, we often prefer to think of the first derivative of

vector values. In this form, we have a nice visual way to

think about what solving our equation means: As our system evolves from some initial state,

our point in this space will move along some trajectory in such a way that at every moment,

the velocity of that point matches the vector from this vector field. Keep in mind, this

velocity is not the same thing as the physical velocity of our pendulum. It’s a more abstract

rate of change encoding the changes in both theta and theta-dot. You might find it fun to pause for a moment

and think through what exactly some of these trajectory lines say about possible ways the

pendulum evolves for different starting conditions. For example, in regions where theta-dot is

quite high, the vectors guide the point to travel to the right quite a ways before settling

down into an inward spiral. This corresponds to a pendulum with a high initial velocity,

fully rotating around several times before settling down into a decaying back and forth. Having a little more fun, when I tweak this

air resistance term mu, say increasing it, you can immediately see how this will result

in trajectories that spiral inward faster, which is to say the pendulum slows down faster.

Imagine you saw the equations out of context, not knowing they described a pendulum; it’s

not obvious just-looking at them that increasing the value of mu means the system tends towards

some attracting state faster, so getting some software to draw these vector fields for you

can be a great way to gain an intuition for how they behave. What’s wonderful is that any system of ordinary

differential equations can be described by a vector field like this, so it’s a very

general way to get a feel for them. Usually, though, they have many more dimensions.

For example, consider the famous three-body problem, which is to predict how three masses

in 3d space will evolve if they act on each other with gravity, and you know their initial

positions and velocities. Each mass has three coordinates describing

its position and three more describing its momentum, so the system has 18 degrees of

freedom, and hence an 18-dimensional space of possible states. It’s a bizarre thought,

isn’t it? A single point meandering through and 18-dimensional space we cannot visualize,

obediently taking steps through time based on whatever vector it happens to be sitting

on from moment to moment, completely encoding the positions and momenta of 3 masses in ordinary,

physical, 3d space. (In practice, by the way, you can reduce this

number of dimension by taking advantage of the symmetries in your setup, but the point

of more degrees of freedom resulting in a higher-dimensional state space remains the

same). In math, we often call a space like this a

“phase space”. You’ll hear me use the term broadly for spaces encoding all kinds

of states for changing systems, but you should know that in the context of physics, especially

Hamiltonian mechanics, the term is often reserved for a special case. Namely, a space whose

axes represent position and momentum. So a physicist would agree that the 18-dimension

space describing the 3-body problem is a phase space, but they might ask that we make a couple

of modifications to our pendulum set up for it to properly deserve the term. For those

of you who watched the block collision videos, the planes we worked with there would happily

be called phase spaces by math folk, though a physicist might prefer other terminology.

Just know that the specific meaning may depend on your context. It may seem like a simple idea, depending

on how well indoctrinated you are to modern ways of thinking about math, but it’s worth

keeping in mind that it took humanity quite a while to really embrace thinking of dynamics

spatially like this, especially when the dimensions get very large. In his book Chaos, James Gleick

describes phase space as “one of the most powerful inventions of modern science.” One reason it’s powerful is that you can

ask questions not just about a single initial state, but a whole spectrum of initial states.

The collection of all possible trajectories is reminiscent of a moving fluid, so we call

it phase flow. To take one example of why phase flow is a

fruitful formulation, the origin of our space corresponds to the pendulum standing still;

and so does this point over here, representing when the pendulum is balanced upright. These

are called fixed points of the system, and one natural question to ask is whether they

are stable. That is, will tiny nudges to the system result in a state that tends back towards

the stable point or away from it. Physical intuition for the pendulum makes the answer

here obvious, but how would you think about stability just by looking at the equations,

say if they arose from some completely different and less intuitive context? We’ll go over how to compute the answer

to a question like this in following videos, and the intuition for the relevant computations

are guided heavily by the thought of looking at a small region in this space around the

fixed point and asking about whether the flow contracts or expands its points. Speaking of attraction and stability, let’s

take a brief sidestep to talk about love. The Strogatz quote I referenced earlier comes

from a whimsical column in the New York Times on mathematical models of love, an example

well worth pilfering to illustrate that we’re not just talking about physics. Imagine you’ve been flirting with someone,

but there’s been some frustrating inconsistency to how mutual the affections seem. And perhaps

during a moment when you turn your attention towards physics to keep your mind off this

romantic turmoil, mulling over your broken up pendulum equations, you suddenly understand

the on-again-off-again dynamics of your flirtation. You’ve noticed that your own affections

tend to increase when your companion seems interested in you, but decrease when they

seem colder. That is, the rate of change for your love is proportional to their feelings

for you. But this sweetheart of yours is precisely

the opposite: Strangely attracted to you when you seem uninterested, but turned off once

you seem too keen. The phase space for these equations looks

very similar to the center part of your pendulum diagram. The two of you will go back and forth

between affection and repulsion in an endless cycle. A metaphor of pendulum swings in your

feelings would not just be apt, but mathematically verified. In fact, if your partner’s feelings

were further slowed when they feel themselves too in love, let’s say out of a fear of

being made vulnerable, we’d have a term matching the friction of your pendulum, and

you two would be destined to an inward spiral towards mutual ambivalence. I hear wedding

bells already. The point is that two very different-seeming

laws of dynamics, one from physics initially involving a single variable, and another from…er…chemistry

with two variables, actually have a very similar structure, easier to recognize when looking

at their phase spaces. Most notably, even though the equations are different, for example

there’s no sine in your companion’s equation, the phase space exposes an underlying similarity

nevertheless. In other words, you’re not just studying

a pendulum right now, the tactics you develop to study one case have a tendency to transfer

to many others. Okay, so phase diagrams are a nice way to

build understanding, but what about actually computing the answer to our equation? Well,

one way to do this is to essentially simulate what the world will do, but using finite time

steps instead of the infinitesimals and limits defining calculus. The basic idea is that if you’re at some

point on this phase diagram, take a step based on whatever vector your sitting on for some

small time step, delta-t. Specifically, take a step of delta-T times that vector. Remember,

in drawing this vector field, the magnitude of each vector has been artificially scaled

down to prevent clutter. Do this repeatedly, and your final location will be an approximation

of theta(t), where t is the sum of all your time steps. If you think about what’s being shown right

now, and what that would imply for the pendulum’s movement, you’d probably agree it’s grossly

inaccurate. But that’s just because the timestep delta-t of 0.5 is way too big. If

we turn it down, say to 0.01, you can get a much more accurate approximation, it just

takes many more repeated steps is all. In this case, computing theta(10) requires a

thousand little steps. Luckily, we live in a world with computers, so repeating a simple

task 1,000 times is as simple as articulating that task with a programming language. In fact, let’s write a little python program

that computes theta(t) for us. It will make use of the differential equation, which returns

the second derivative of theta as a function of theta and theta-dot. You start by defining

two variables, theta and theta-dot, in terms of some initial values. In this case I’ll

choose pi / 3, which is 60-degrees, and 0 for the angular velocity. Next, write a loop which corresponds to many

little time steps between 0 and 10, each of size delta-t, which I’m setting to be 0.01

here. In each step of the loop, increase theta by theta-dot times delta-t, and increase theta-dot

by theta-double-dot times delta-t, where theta-double-dot can be computed based on the differential

equation. After all these little steps, simple return the value of theta. This is called solving the differential equation

numerically. Numerical methods can get way more sophisticated and intricate to better

balance the tradeoff between accuracy and efficiency, but this loop gives the basic

idea. So even though it sucks that we can’t always

find exact solutions, there are still meaningful ways to study differential equations in the

face of this inability. In the following videos, we will look at several

methods for finding exact solutions when it’s possible. But one theme I’d like to focus

is on is how these exact solutions can also help us study the more general unsolvable

cases. But it gets worse. Just as there is a limit

to how far exact analytic solutions can get us, one of the great fields to have emerged

in the last century, chaos theory, has exposed that there are further limits on how well

we can use these systems for prediction, with or without exact solutions. Specifically,

we know that for some systems, small variations to the initial conditions, say the kind due

to necessarily imperfect measurements, result in wildly different trajectories. We’ve

even built some good understanding for why this happens. The three body problem, for

example, is known to have seeds of chaos within it. So looking back at that quote from earlier,

it seems almost cruel of the universe to fill its language with riddles that we either can’t

solve, or where we know that any solution would be useless for long-term prediction

anyway. It is cruel, but then again, that should be reassuring. It gives some hope that

the complexity we see in the world can be studied somewhere in the math, and that it’s

not hidden away in some mismatch between model and reality.

At 3.00 how the derivative of a sine wave like function gives out a straight line .

The slope is changing constantly so it must be the function of cos right ?

A straight line derivative is for a function which has uniform slope of the original function

Or am I missing something ?

I think this is my favourite video, ever.

There should be a Nobel prize for YouTube videos and this channel should be the first one to win it!

anus

Give a medal to this channel..

Hands down the best video on youtube!

"THEY'RE REALLY FREAKING HARD TO SOLVE" – in the thickest Irish accent :')

which tool you use for making these videos.

21:38 I didn't expect to realize something about my romantic life while watching videos on DEs… but thanks!

24:11 I just realized how amazing that application is.

Why I can't run it on Python?

Now m feeling the class I 'd attended for DE is something else..😂😂 best explanation and much more stuck with the originality of how things happen

All I know is my affection to loving your channel is exponentially accelerating upward.

I just see the relation between channel's name and the logo. Umm nice.

just watched the full add so u get paid.if u start a pantheon i will totally back u up

you are so so so stunning!!

I'm really confused… Isn't that representation of vectors: untrue?

This is just brilliant.

10:56

Man, you just violated PEP 8

Yo I don't understand shit why am I here

Not sure why I don't connect with this style…try as I may. More about performance…less about instruction. Too slick for me.

GREAT VIDEO

I no longer watch your videos because I'm force-fed the automatic translation of video description.

your code might have some error as if i add print(theta(10)) it just prints the value of THETA_0 and not the actual value……

# Physical constants

g = 9.8

L = 2

mu = 0.1

THETA_0 = np.pi

THETA_DOT_0 = 0

#defining the OED

def get_theta_double_dot(theta, theta_dot):

return -mu * theta_dot – (g/L) * np.sin(theta)

#Solution to the differential equation

def theta(x):

theta = THETA_0

theta_dot = THETA_DOT_0

delta_t = 0.01

for time in np.arange(0, x, delta_t):

theta_double_dot = get_theta_double_dot(

theta, theta_dot

)

theta += theta_dot * delta_t

theta_dot += theta_double_dot * delta_t

return theta

print(theta(10))

output: 3.14.592653589793

Wonderful, wonderful.

Charts and examples reinforce intuition.

It turns out that a lot of time and effort has been made.

Thanks very much and thanks. It was great, very much

I'm a 17 year old who just finished a high school calculus course. I did physics last year. I comprehended about half of this video, so I was wondering what resources I can use and what I need to study to be able to fully understand the concepts presented in this video

your an artist of a kind..

Jeeezus Christ, my head just imploded. I went to engineering school, undergrad and masters, but differential equations was such a mystery to me because I'm such a visual learner because I'm human, not a dog or a bat. I paid 10s of thousands of dollars to my unviersity to teach me these things but it never really made sense, so many why's? Years of just staring and pondering, then this guy just made my whole world click in one video, albeit in a very intense way.

Great job man, if I had the money to spare I'd give you a lot, but for now, I'll just support you on Patreon. Guys like you who can make such 'traditionally complex' knowledge accessible to the masses will be the real heroes of the future and you all need to be rewarded amply. From my heart, thank you sooo much, there is nooo way to quantify the impact these videos to aspiring engineers and scientists, actually perhaps you can.

Could you do some finite element method? Maybe as a continuation of the PDE episode. These are so good

thanks a lot

"This equation from (pause), chemistry…"

Lmao

Really enjoyed the tangent on code. Is there a good method / software to plot solutions to differential equations? I am starting uni this october, so it might help me out to be able to plot diff equations.

When you say X=Ltheta, I am confused. This equation merely says the “ Length “ of the curve, rather the “ Position “ of the curve in 2-d space whose 2nd derivative is the acceleration. How can you substitute a 2-d vector with a real number.

Dude, I took calculus and passed but this still confused the shit out of me

can you make videos on cauchy residue theorem (complex analysis).it would be a nice topic to truly visualise andd understand instead of mugging up the formulas

May I add something to your point at the end? Another important thing to learn from chaos theory is that there are times when we KNOW we can’t know something, and so we should not place our faith in those predictions. For example, in facing climate change, because we know our predictions about climate behavior is so chaotic, but we can be reasonably sure it’s not a good idea not to emit so much carbon, a solution that we can change as the system changes, like atmospheric carbon extraction machines, are MUCH better than geoengineering solutions like seeding the upper atmosphere with sulfur to lower temperatures. Understanding the extent of our ignorance is a very important part of understanding what to do. “I don’t know” does not need to be an excuse for inaction, sometimes it’s just part of describing the problem.

this is mind blowing god , i have a braingasm

this is impressive

For that unsolved DE involving air resistance, could we write a series solution for it?

can't wait for your explanation for wave equation and Schrodinger equation! Thanks so much for the great series!

amazing

At 8:30 how second derivative of x is replaced by L times second derivative of theta?

What kind of software do you use? Python?

Genius, as usual

Spanish?

Thank you for always reminding why I love math!

good

The internet has raised the bar for good education a lot, I hope teachers in schools and Unis will follow

Amazing!!! thanks a lot!

Moon roationat

instructions unclear, ignored my partner for a few days meanwhile watched your videos, she started ode with another guy making d(her)/dt independent of me. Please solve. Btw amazinggg video, and lots of loveeee

beautifully explained..great work!! 🙂

You didn't think we'd notice the Taus on the theta axis of the phase space but I did, you dirty tauist

You're videos are awesome and I enjoy them very much. Nevertheless I have to point out that air resistance is calculated using the square of v (among other factors). I do understand that you simplified it for easier understanding, but it probably would be better to refer to it as simply friction since viewers could get a wrong impression about how air resistance works.

I couldnt understand State space model at university class.

Finally understand what the model says.

Thank you

8:50 forgot friction of the rotating pole to the connector pieces on the mast eh? dw

8:50 forgot friction of the rotating pole to the connector pieces on the mast eh? dw

My 8th grade math teacher insisted that math was beautiful. I didn't understand what she meant until I started watching your videos.

Beautiful ,simple, elegant…thank you!

Those who possess the knowledge are not good at expressing it, and those who can express do not have enough knowledge about it.

And here you are, someone who got the knowledge and the ability to express it. Thank you for that.

Shit, now there's even equation for love!

I feel if you are a professor in courses of Science/Engineering, you will make a lot more Einsteins. Trust me, I never understood, how to make use of all the Differential Equations and Physics together even though it has been taught to me in the classes of few best institutes. You made me connect the dots! Thank you @GrantSanderson @3Blue1Brown

I love this channel. Is there a stationary point in the universe?

if only such good videos have existed 10 years ago 😀

you're really cool, not just an awesome teacher!

Haha "…and another from, uhh, 'chemistry'…"

in the beginning example, with Vo and V1 changing, shall not the shape of parabola also change?

This is one of the BEST videos I've ever had the pleasure of watching on YouTube, and an AMAZING explanation of DE's. I love the presentation, the artwork, the "simplicity" (not of the subject, but of the conveyance), and it should pretty much be shown at the beginning of any course in DE's or Numerical Methods.

This is an awesome video

sir what yr name? and what softwere u use for videos?

how i can understand videos of this person.i am unable to understand.

from where should i start?

so we are using matrix vectors to compactly encapsulate information about these rates of change of different orders for easy analysis

You youngsters have it damned easy these days. When I learned these concepts,, I did literally have to "do the math", as my only assistant was an 8 dollar scientific calculator from radio shack. It had no clue what a derivative was…. I think you folks don't really get a true grasp of the concepts when a computer does your work for you.

Hi Grant, I hope you read my comment. Can you please make a video of maxwell's equations?

and still, some people disliked the video… WTH

TBH most engineers don't solve diff equations analystically but numerically, hopefully 3b1b makes a numerical analysis video soon

What is the name of the video with rabbits and foxes?!

Does anyone know where I can learn about differential equations with an understanding in Algebra 2?

Oh my god, 24:20, that’s the problem, that is the problem with most simulations, that needs to go, like, completely.

amazing

Saw this video once cause I was curious, then my math for physics teacher got all crazy and jumped from derivatives to this and well, here we are again.

(I'm in secondary school)

Amazing video, thanks Grant ! For those who might be interested, I managed to reproduced the phase space for the same differential equation with a small python script : https://gist.github.com/OmarAflak/c08e98a6d32c12231899f7ffd8c89b40

Thank you for making these educational videos; they are mesmerizing! I appreciate your explanations, analogies, animations, and background music. You are making math accessible to more people! These videos are calming yet thrilling. I look forward to learning from you!

Some notes on the intended use of this series. I was deliberate in using the phrase "tour of differential equations", as opposed to "introduction to" or "essence of". I think of the relationship between watching this series and taking a course as being analogous to the relationship between touring a city vs. living in it. You'll certainly see a lot less with the tour since you're spending less time overall, but the goal will be to walk around some of the most noteworthy monuments and town centers with helpful context given to you by a guide. And just as someone who lives in a city may very well have never gone to visit some of the historical sites of their town, despite living there for years, many differential equations students may not always get the chance to zoom out and appreciate the central cornerstones of the subject amidst all the computations they are learning.

I hope you enjoy the tour, but at the same time know that it is, by design, very different from taking courses on the subject.

16:54 – Actually, it really is obvious to me (I guess I have enough experience to know that).

There are currently 185 people who wasted their time and money on women's studies, fine art, communications, or psychology weighing in on this video's value. It's not too late, the STEM fields are open to everyone.

Please sub spanish

The world runs on for loops

@7:33, move the yellow gravity and pink vector such that the gravity vector aligns itself with the perpendicular line, you will see alternate angles forming.. and alternate angles are equal..

You Aura is increasing video by video. Hats off to you!

Sin subs en español :/

Hello Sir plz explain the concept # Equation of Planes

I realize inertial/non-inertial points of reference are outside the scope of this discussion –but gravity isn't a "force." Maybe you could go into a greater detail later (or reference an external link).

I think I love you.

Thank you

Funniest math video ever

Almost as depressing as primes, however extremely beautiful.

Is there a serious for specific kind of diff. eq like legendre,Bessel etc.? I would be glad if you answer

Thank you so much sir for this amazing video and it's really help me ……. thanks a lot……