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Problem 24 Hard Difficulty

a) Use Euler's method with step size 0.2 to estimate $ y(1.4), $ where $ y(x) $ is the solution of the initial-value problem $ y' = x - xy, y(1) = 0. $
(b) Repeat part (a) with step size 0.1.

Answer

a) $y(0.6) \approx 0.525801$
b) $y(0.6) \approx 0.503424$

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Video Transcript

were given an initial value problem, were asked to use Oilers method with a certain steps size fine approximation of the value of the solution at some X. So in part A, we have the differential equation. Wide crime was co sign and X plus y. With initial value, Teoh equals zero. If you want to find approximation for why a 0.6 in part? A. We use step size each equals point to from our initial value, we have that X zero is equal to zero. And why zero is why FX zero, which is equal to zero as well. X one. This is X zero, plus her step size. So this is a point to why one is equal to y 00 plus our step size times function Co sign of X Plus y evaluated at X zero. Wiser. Just simply co sign of zero. Just one says his equal 2.2 x two is equal to x one plus or step size 0.4. And why to is equal to why one plus her step size times Our Function co sign of X Plus y evaluated that queen to 0.2. This is co sign of 0.4, although we could just keep it in this form, I'm going to approximate this. This is about equal to this is about equal to point 38 4 2122 x three This is going to be X two plus or step size is equal to 0.6, and we have that. Why have 0.6 is approximately equal to Y? Said three, which is equal to wise up to. I'm just going to write his wife's of to closer step size point. Two times a function evaluated that 0.4. Why, too? So this is co sign of 0.4 plus. Why to and this is about equal to point 5 to 5 811 He used the store function. You might get a more accurate value, but this is fairly accurate now in part B, whereas to you different steps size this time, step size of 0.1. So now we expect to have six steps instead of three. Again, we have that X zero is equal to zero and why zero is equal to zero. So x one is X zero plus or step size, which is 0.1, and why one is equal to y 00 plus or step size times Our function evaluated at x zero N. Y zero simply co sign of zero, which is equal to 12 This is 120.1 next to is equal to point to and why to is equal to why one plus step size times our function evaluated it x one plus why one this is going to be point to. And again, we could just leave this in this form, but or simplicity. This is approximately equal to point 19 eight 0067 Our third step X three is 0.3 and why three is equal to y two. Plus our step size times function evaluated that wide to at x two y two. This is going to be point to less. Why, too, this is approximately 0.290 19 zero to Step four. We have X four is equal 2.4 and why four is equal to y three, plus our steps sized 0.1 times are function of I redid it x three and why three so co sign of 0.3 plus why three, which is equal to approximately point 373 27 37 Step x five is equal 2.5 one More step after this one in WiFi is equal to why four plus our step size times Our function evaluated at X for Y for this is point for plus Why, for this is why side was y four plus or step size Times Co sign of 40.4 plus Why for which is approximately equal to 0.444 83 65 And finally, Step six. We have thanks six is equal to 0.6. We have that. Why 0.6? It's approximately equal to why six, which is equal to why five plus or step size times function. If I read it x five, Why five sue 0.5 plus why five. This is approximately equal to point by 03 for to for so we see that these answers out to the first digit at least are the same. We would expect that our second answer is going to be closer to the true value in our first answer