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In modelling the velocity chain slipping off horizontal platlor; the dillerential equation I0/y "/x is derived: Suppose the initial condition is Y(V) = Euler m...

Question

In modelling the velocity chain slipping off horizontal platlor; the dillerential equation I0/y "/x is derived: Suppose the initial condition is Y(V) = Euler method for solving y' f(x.y) Y(xo) given by Ya+hf (sn. Jn) where lixed stepsize_ and Yn y(Xn) Apply one step ol Euler' $ method t0 the initial value problem given above Apply one step the improved Euler method given byJn+1Jn + [J (xn. Vn) + f(n +hYn + hf(xn. Yn))] with h 0.1 to the initial value problem given above: For the i

In modelling the velocity chain slipping off horizontal platlor; the dillerential equation I0/y "/x is derived: Suppose the initial condition is Y(V) = Euler method for solving y' f(x.y) Y(xo) given by Ya+hf (sn. Jn) where lixed stepsize_ and Yn y(Xn) Apply one step ol Euler' $ method t0 the initial value problem given above Apply one step the improved Euler method given by Jn+1 Jn + [J (xn. Vn) + f(n +hYn + hf(xn. Yn))] with h 0.1 to the initial value problem given above: For the initial value problem given above, the (Wo methods given in part (a) and (b) as well &s the Runge-Kulta method ol order were run On computer with 0.05 and 0.025 to obtain approximations the solution a X The absolute errors 0l these approximations aF Shown in the following table. Note that the notation in the table such that an entry like 95166(-3) is shorthand for 95166 10-- Absolute error of approximations to >(2) Method 0.05 h = 0.025 Ratio Euler $ method 0.401786-4 0.195446-4 2.056 Improved Euler method 0.95166(-3) 0.26047(-3) 3.654 Runge-Kutta method 0.16374(-4) 0.94541(-6) 173319 The last column ol "Ratio" contains [or euch method the ratio Absolute error with 0.05 Absolute error With 0.025 For euch ol the three methods, IS known that the absolute error using stepsize 0l h is approximately Ch" for some positive constant € and integer Use the given ratios estimate the value of [or euch method_



Answers

(a) Use Euler's method with each of the following step sizes to estimate the value of $ y(0.4), $ where $ y $ is the solution of the initial-value problem $ y' = y, y(0) = 1. $
(i) $ h = 0.4 $ (ii) $ h = 0.2 $ (iii) $ h = 0.1 $
(b) We know that the exact solution of the initial-value problem in part (a) is $ y = e^x. $ Draw, as accurately as you can, the graph of $ y = e^x, 0 \le x \le 0.4, $ together with the Euler approximations using the step sizes in part (a). (Your sketches should resemble Figure 12, 13, and 14.) Use your sketches to decide whether your estimates in part (a) are underestimates or overestimates.
(c) The error in Euler's method is the difference between the exact value and the approximate value. Find the errors made in part (a) in using Euler's method to estimate the true value of $ y(0.4), $ namely $ e^{0.4}. $ What happens to the errors each time the steps size is halved?

Okay, so the Oilers method is a recursive method. So which has the formula? Why of then is equal to why? Of n minus one plus h times f of X and minus one comma y and minus one. Okay, so here f of X y is just equal to why, okay? And then we have why of zero is equal to zero and ex of zero is equal to sorry. Why of zeros, people to one, an ex of zero is equal to zero. Okay, so now, for one we have that h is equal to zero point for. So why one is going to be equal to y zero, which is able to one plus eight reaches 0.4 times and or why of 0.1, right? See why? Why zero? This is equal to one plus 0.4 times why of zero which is equal to this. So why, if one is equal to 1.4 and the next one is going to be just x zero plus each, which is equal to zero plus h, which is point for. All right, So now we have our estimate. Why one? Here is our estimate for one point, or is our estimate for, um, the function at 0.4. So this is our oilers method for that. Now, for two, we have a CI equals 20.2. So now we're gonna have a white one. So this legal to one plus 0.2 times why zero, which is equal to one. So this is equal to one plus point to which is equal to 1.2. Um, x one, it's going to be just zero plus 0.2. So we have 0.2 now. Why, too? This isn't going to be s. So why? And minus one. Here, this is 1.2 plus h your 0.2 times. Why one? Right? So he 1.2 plus 0.2 times wide. One. Why one is equal to 1.2. Hey, so we have 1.2 plus 0.2 times 1.2. So this is equal to 1.2 plus 0.24 This is equal to 1.44 Then we have that X two is gonna be 20.2 plus point to just point for. So we're done here. So are estimate for H equals your 0.2 is 1.44 now for three. You know, here we have that h is gonna be 0.1. So we're going to do this four times, right? So we first we have. Why one this is one of equal to again. So why zero plus, uh, h why each y zero? So one plus 0.1 times why zero is one. So 1.1. Why to it's going to be 1.1 plus 0.2 times. And then why one? Why one is times 1.1 here. So it's 1.1. Uh, Lenny shot here 1.1 plus 0.22 So this is going to be 1.32 Okay, then let's keep calling. Why? Three, This is able to 1.32 plus 0.2 times 1.32 We put that into a calculator, then we get one point, uh, 584 Then why four is going to be so 1.584 plus 0.2 times 1.584 lugging all that look healthier. Get we get what? Sorry, I just realized I have the wrong, um, each here, let me fix all of this. So this is incorrect here. Okay, So, starting back from here, uh, from wanted to. So 1.1 plus 0.1 times 1.1. All right, this is going to be equal to sew 1.1 plus, um, plus 0.11 This is equal to 1.21 So then 1.21 plus 0.1 times 1.21 So we have, um, point to one plus Oh, 0.1 to 1. No. So 1.21 plus 0.1 to 1. So this is able to one point three three one. And then lastly, we have so 1 23 31 plus 0.1. Oh, this should be a 00.1 here. 0.1. Your 0.1. Uh oh, no. Yeah, that's that's right. And then times 1.331 So our last and final answer is gonna be 1.46 for one. And this is gonna be our estimate for, uh, with a jingles 0.1. So that's for part a now part B. It tells us that Our exact answer is going to be equal to, um, you have that are exact. Enters the school to e to the X. So, um, now, now we need to accurately draw, um, our solutions. So is your 0.10 point 20.30 plate for here as X. And this is our Why here. And one's gonna start up one, uh, 01 Here. Let's see, um, our first answer. Give us 1.4. So I'm gonna pull it 1.4. Come up here. Let's see, um, our high standards should be about 1.46 so I'm gonna make that, uh, 1.5 up here. So our first solution had 1.46 That's like, you know, maybe we're sorry. 1.4. It's Let's say that's here. That's one. Just one here and then to okay, when h was even the point, too. We had are 1.2 and then 1.44 Okay, so 1.22 is maybe somewhere down here and then 1.44 is somewhere up here. So that one's gonna be, you know, to this one's one and two up here, then now for, uh, for three. Okay, we have 1.1, so we need to divide this up a little more. So 1234 Um, so maybe, actually, this should be Let's see here. 1.4 will be here with my 44 will be about here. 1.2 1.2 is Let's see 1.2 is here. 1.23 Um, so So 11.1 here and then 1.21 here. Oh, it's all right. Yeah, about here than 1.33 It's about here. And then 1.46 That's a is about here. So that's for three then lastly, um, e to the X right has a function. Uh, the value of E to the 0.4, uh is going to be equal to 1.49 So closer to 1.5. So about here. All right, so this is e to the X, like so. Okay. Okay. So here we can figure out that our estimates, um, underestimates because our graph is underneath the true curve. Now for part C, we need to find the error. So me true value of E to the 0.4 is approximately 1.4918 for 91 eat, um, 1.4918 So if we take the absolute ever So let's do for H equals 0.4, right? So we have 1.4 minus 1.4918 This is equal to are approximately equal to I see your point 90.9 18 Then if a chick was your 0.2, you have one point 44 1.44 minus 1.4918 Thank you. Huh? This is gonna be 0.518 And then, lastly, we have h equals zero point one. So we have one point 4641 minus 1.4918 This is equal to you're a point zero point. Row 278 or 0 to 7. Just 277 Okay, so now, as you can see from here, the ever is halved every time we, uh, half the step size error is how All right, then we're done.


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