Euler's method | Differential equations| AP Calculus BC | Khan Academy

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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.

stanrocks

Euler is a math genius while Sal is a genius educator😍😍

Chairman

im an internarional student and Khan Academy is super helpful Thank You!

userpmzjah

You are a saint, even in college you pull through, god bless.

gigatoaster

Ur explanation of euler's method was very intuitive good job

chibuezendiokwelu

The cogs just clicked into place for me, thank you again Khan academy!!!

Ewetubedean

Sal, I love your videos and wanted to let you know how much I appreciate them. Keep up the good work.

sparrow

The Khan Academy helps me a lot of times. That was another one. My appreciation, and please, keep doing that you are really CAN!

romanglushenok

Thanks for all that you do on here this is great stuff !! Very Helpful =)

BBert

i still dont get how you went from 1.5 to 2.25 in the second table in the y column

LizBeReal

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.

speaketh

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?

Hunar

He just repeated it's going to look like three times lol 7:54 thought my computer was frozen

jacksewe

Thank you Sal. Your altruism is commendable. Thanks for the beautiful explanation.

sudandevkota

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.

fractalfan