Euler's method | Differential equations| AP Calculus BC | Khan Academy
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- Опубликовано: 12 сен 2024
- Euler's method is a numerical tool for approximating values for solutions of differential equations. See how (and why) it works.
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I think for people who are having trouble understanding this (me included) intuitively. Think about this, in the formula for eulers method y_old + Δx (dy/dx). This part of Δx (dy/dx) is just giving us the value of Δy which is then added to the y_old to give us our new y value. It can be derived from Δy/Δx ≈ dy/dx, multiplying by Δx to both sides. But the reason we multiply by Δx is because think about rise over run. For every change in Δx, Δy changes by a specific amount relative to the slope of the line. In the senario when Δx = 0.5 and dy/dx = 1, we are think about how much does "y" change when x changes by 0.5 when the slope is one, giving us Δy = 0.5, this can be applied when dy/dx is equal to different values.
Thank you much, I really had so much trouble thinking abt this one.
Thanks a lot stanley, really was scratching me head on this one
You explained this so succinctly! Thank you so much, I think I understand it better now :)
Your explanation was lucid and clever! And I have no idea what lucid even means!
Your Explanition was very good i am subcribing you
0:53 thanks for the motivation
Euler is a math genius while Sal is a genius educator😍😍
@@preenan5123 no this is sal in the video
im an internarional student and Khan Academy is super helpful Thank You!
Sal, I love your videos and wanted to let you know how much I appreciate them. Keep up the good work.
jack sparrow? is that you?
@@conflictsofthe20thcenturyc91 captain *
The Khan Academy helps me a lot of times. That was another one. My appreciation, and please, keep doing that you are really CAN!
Thanks for all that you do on here this is great stuff !! Very Helpful =)
Ur explanation of euler's method was very intuitive good job
Amazing I finally got it.
by the way, what is the name of that board-like program you use to explain?
You are a saint, even in college you pull through, god bless.
The cogs just clicked into place for me, thank you again Khan academy!!!
Thanks for interpreting the concept!!
I used the four x,y points that you found in the ∆x=1 .. and put them in a table and found ∆y, ∆^2y, ∆^3y and used the taylor expansion and got (1+x+x^2/2!+x^3/3!) which is exactly e^x .. so for that big ∆x how did i reach that accuracy?
If for some real number x and this ODE for y=e^x, we apply the method n times with the increment x/n for any natural number n, starting at zero, we will get (1+x/n)^n, which converges to e^x as n goes to infinity.
To people who got confused by the colouring: To match the table, the line segment from x = 0 to x = 1 should be purple (since it has a slope of 1), the next one green (slope 2), and the third one pink (slope 4). He hasn't really got to the orange line segment yet, which goes from x = 3 to x = 4.
It was nice for getting a better intuitive understanding of what is done.
Thank you Sal. Your altruism is commendable. Thanks for the beautiful explanation.
i still dont get how you went from 1.5 to 2.25 in the second table in the y column
it's because the slope at x=0.5 is 1.5 (dx/dy), so remember that he is stepping by 0.5 and not by 1, then "y" evaluated should be equal to "y" before plus 0.5 * "slope", which is 1.5, so... y = 1.5 + 0.5 * "1.5" = 2.25.
the formula is u take the 1.5 + (1.5)(0.5) = 2.25 .. If I'm not mistaken .. hehe
Delta X is 0.5, so after 0.5 "steps" in x axis he goes up in y axis half of the previous slope and then the new slope is equal to that y value. y values were 1, 1.5 and 2.25 because the slopes were 1 and 1.5 .
basically
y_new = ((dx/dy)_old * delta_x) + y_old
which works ... but i don't understand it's meaning
LizzieAthey
from the slope formula
slope=y2-y1/(x2-x1). you slope is equal 1.5 and (x2-x1)=0.5 and y1=1.5 so that will be equal: 0.5*1.5=y2-1.5.......and y2=2.25.
Pretty good explanation thanks man
Is there a value of delta x such that the approximated curve diverges from the actual function? If so, how would you find it?
Trial and error.
what we more or less get from this is, a ) a good local approximation and b) a terrible global approximation. global one is bad because
1.) often times, solutions to differential equations might not be unique and if we consider lim to infty or smth they often diverge differently (hence things like weather forevast etc are terrible for more than like a week)
2.) the error term if were lipschitz and the solution is thus unique contains an exp(x-x_0) times a constant depending on our problem functions max curvature and the delta_x we chose. so even if we dont curve a lot, we get a lot if error if x is far away from x_0...
this is also the reason the local approximation is pretty good, because we get an exp term of pasically one and the other term is pretty small if were feedinf the problem to a computer which calculates it which a small delta_x, keeping the error minimal.
tl:dr for unique solitions local approximation is good using small delta_x but the error grows exponentially so after some time the quality drops massively
اخيرا فيديو انجليزي مترجم عربي👍👍😍
thank you so much how nicely explained this euler method ♥♥
He just repeated it's going to look like three times lol 7:54 thought my computer was frozen
All hail finite element analysis!!
I love you Sal, you're doing God's work
that's so great and helpful. thank you
Good Video. Best platform for learning
thx for making me understand this
I just dont understand one thing. Why, in the first example, you straight assume that you multiply "y" by 2?
Gleb Khachatryan look at his initial conditions for x and y. for every time x increases by a particular factor, i.e delta x, y would increase by that factor plus 1
this was inspiring, thanks 💖
at 6:07 wouldn't y be 12, since our slope is 4 and y=mx so 4*3 would give us 12
That's only the case when the line goes through the origin. In our case, we're currently at (2, 4), and since the slope is now 4, y increases by 4 when x increases from 2 to 3, making the new y value 8.
thanks this helped me a lot
I'm almost understanding 'e' !
Isn't the last y value in the first chart supposed to be 7 since dy/dx is 4 and x is 3? 4+3=7?
+Mehecanogeesir The slope is now equal to 4. +1 for the x-value corresponds to +4 for the y-value. Since the previous y-value was 4, we now get 8.
Yes it should be 7.
It was very exciting, honest
Thank you sir.
6:01 you increase by 4, that means 4+3=7... not eight. Good explanation so far ^^, really apreciate it
No, 8 is correct. We're at (2, 4), and since the rise dy/dx = 4, y will increase from 4 to 8 when x goes from 2 to 3.
@@speakethUPS, my bad 😅
@@speakethUPS, my bad 😅
Clear explanation
Is it just me, or is euler's method basically just linear approximation plus a starting value.
Why do we increase y by the previous dy/dx? ?
+Swati Chow dy/dx denotes the slope. Given that y=dy/dx, and y precedes n increases in x as well as dy/dx. We are not always increasing by the slope (dy/dx).
why wasn't the expression just given to find y(n+1)=y(n) + h*y'(x(n),y(n)), where h is the step or delta x. This obviously caused some confusion.
The only problem with this is the exercises that come after. On the quiz for this topic, it solely relies on the formula for Euler’s method, which is not mentioned a single time in the video, so having the quiz be about the formula makes absolutely no sense and should be changed instantly
How do u get y(x)=e^x ?
I am sort of new at this stuff. It would have been nice if he used the formula for Euler's method to do the calculations along with the graph. Using the formula I got the following results: Can anybody tell me where I went wrong?Xo=0 Yo=1X1=1, Y1= 1+1(0+1)=2X2=2, Y2= 2+1(1+2)=5X3=3, Y3= 5+1(2+5)=12X4=4, Y5= 12+1(3+12)=27 and so on.
I figured out what I was doing wrong.
What were you doing wrong?
@@Anna-tl6oz well he did it last year lmfao
rockin it sal!
thank you for making it sound more complicated than needed
If you think that of this dead simple explanation, you're not gonna make it.
thanks for the eulers pronunciation. I thought it was you-ler
Thanks for translation Arabia
How’s y equal to 2 when x=1? When x=1, y should be e
You are supposed to use the derivative of the function , not the actual function
What software is he using to draw?
It just clicked, thanks
thx
0.51 You khan do it guys!!!
nice
Makes sense
The Bass in his Voice😶🌫🤩
Nice Video 🤙
THANKYOU, I had no idea what my text book was talking about!
This guy reminds me of Bane from Batman
What happens if you get to dy/dx is 0? It is a problem I am having!
Simo2009BORO simple enough, y(x) = constant
That's the solution, dy/dx is the rate of change of y with respect to x, if the rate of change is zero, the function must be a constant..
then find another means to make it in life please
OILER'S METHOD?
What is "analytic" and "numeric" methods? Can anyone please explain...? Thanks a lot in advance :)
Analytic is basically when you can use formulas to get a generalized answer. From there you can input any value and get an accurate answer. However, since that is not always possible you can use may numbers and find out the answer (in this case the shape the graph) numerically.
can not understand how u got 2.25 in the second table
Absolutely lost me @5:11...
dy/dx is the change in the y direction when x changes. so if we increase x by 1, "dy/dx = 2" dictates that we should increase y by 2. the statement dy/dx=2 reads as "per change in x, there is a change of 2 in y".
why did he add .5 by half of 1.5??? wasn't the pattern to add x to the slope?
The formula to get the next y value is this:
y + (dy/dx * DX)
y is the previous y value
dy/dx is the previous dy/dx value
DX is how much you’re changing by x
I dont get it.
does this guy still post ?
Wilfred yes
Lmao at people disliking this video because he doesn’t plug-and-chug using a preexisting formula. The derivation for that formula is incredibly simple, especially with this video as a reference. Think harder!
I love you.
Hi isn't it e^2=7.389 & e^3=20.086??
Good question. I think he's doing it completely wrong. If you plug 1 into x, y is e, and since dy/dx = y, dy/dx is e as well. He isn't even approximating the solution correctly.
@@Raptured_and_back tks....😊
No he isn't. He just picked big values of Δx so that the example is comprehensive, avoiding unnecessary calculations. If you pick Δx=1 you can see even from the first step that it is a bad approx since it gives e≈2. If you want better approximations you pick smaller Δx as he mentions in the video. His purpose is to explain, not to approximate.
This is so ridiculous
complete swag
Schulist Brook
This method has limitations, like the sin1/x, it fluctuate rapidly between -0.25 and 0.25
Bruce Xu smaller change in x then
who's here because of Hidden Figures? :P
Pineapple29 me
Yep, and briefly mentioned in the imitation game, so I thought it might be worth a look!
Hidden answers in exam
CHOOSE THE CORRECT OPTION:
The first approximate value of dy/dx =1-y ,y(0)=0 find value of y at x=0.1 using Euler's method is
A)0.9
B)1.2
C)1.0
D)0.1
Please reply quickly because tomorrow is my exam
In hindi
thanks for barely explaining it. Thanks.
Thanks.
Thanks
Thanks
Thanks.
Thanks.
I give up guys🎉
Greek
Hidden Figures brought me here
Very bad. At least show some equations. Error calculation.
What is there to show?
Euler's method can only solve first order equations.. Actually I have derived a technique that can solve any differential equation numerically, no matter what order it is..
Is there any known method that does so, or is it something new I have invented ???
maybe something new you invented
The only way to know is to try publishing it in a journal and have it peer reviewed. Also, if you can find a college professor, try asking them to check your mathematical proof.
Bkc
bu ne yaa sinirim kalktı
NO!!!!!! LMAO
thanks for nothing totally useless
forget the way this guy talks???
I have watched this video for more than 10 times, but I didn't get anything
I think this is wrong
Sayon Pak , don't think. First confirm and Ink it.
Why does he sound like a big joke
because he's laughing at your gpa as he is explaining