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11: Problem 2Previous ProblemProblem ListNext Problempoint)Let y(t) be the solution to dt 8te satisfying y(0)(a) Use Euler's Method with time step h 0.2 to app...

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11: Problem 2Previous ProblemProblem ListNext Problempoint)Let y(t) be the solution to dt 8te satisfying y(0)(a) Use Euler's Method with time step h 0.2 to approximate y(0.2) , y(0.4), . ..,y(1.0) .2 0.43 0.64,0.85/1.0(b) Use separation of variables to find y(t) exactly: y(t)Compute the error in the approximations to y(0.2) , y(0.6) , and y(1). ly(o.2) y1 |ly(0.6) Y3 |ly(1) Ys |Note: You can earn partial credit on this problem:

11: Problem 2 Previous Problem Problem List Next Problem point) Let y(t) be the solution to dt 8te satisfying y(0) (a) Use Euler's Method with time step h 0.2 to approximate y(0.2) , y(0.4), . ..,y(1.0) . 2 0.4 3 0.6 4,0.8 5/1.0 (b) Use separation of variables to find y(t) exactly: y(t) Compute the error in the approximations to y(0.2) , y(0.6) , and y(1). ly(o.2) y1 | ly(0.6) Y3 | ly(1) Ys | Note: You can earn partial credit on this problem:



Answers

Let $y(t)$ be the solution to $\frac{d y}{d t}=t e^{-y}$ satisfying $y(0)=0$. (a) Use Euler's Method with time step $h=0.1$ to approximate $y(0.1), y(0.2), \ldots, y(0.5)$ (b) Use Separation of Variables to find $y(t)$ exactly. (c) Compute the errors in the approximations to $y(0.1)$ and $y(0.5)$.

Okay, so we're given why? Teeny is a solution of why I thought physical T. He figured. Why satisfying? Why you got to go report a for asked to use Oilers message to with its step size agency, which is open one to approximate. Why joke? One to why, so a point. But all right, let's start with why you appoint one what X is equal to. Why not? Actually, let me do something else. So we know. Why not? Is he cool too? Why? No. Is he good to go now? Why one is you could wind no 10.1. You do? Why not? Which is so plus our step size 0.1 times our function at mm. Peanuts. And why not? That's just this thing over here. With Marty equal to zero and I wiII equal to Joe. Here we go. So all times each and the result that used me. So So why one is so that Why do you look so why so a point too. Which is equal to why one of his own steps eyes open One tie t 10.1 times e to our Why would you think himself? Thank you me 0.1 times. Quite one. That gives me 0.1 Okay, for y three have y 0.3 is equal to why two witches opens. Don't one of us are step size 0.1 Times are tee too, which is 0.2. I'm e to the group, so point so one. Which is why D'oh! Well, this is equal to you. Write it. Found here that's equal to so 0.29 Eat Joe's only 997 Okay, moving on. What's wife? Or that he puts you? Why so important for which is equal to why three witches. So 0.2980097 plus our step size with no 0.1 times g to the three, which is 0.3 times e to the negative value over here. Which is why three appoint so 2980097 Okay, buddy, this we get wife or is equal to so 0.5 feet 9 to 015 seven. Okay. Now, for WiFi, that's equal to by 0.5. The last one, actually, which is equal to wife. What? Which is zero point. So I teach 9 to 157 plus our step size Europe on one times in Are, or Tea boy, which is open for E t o. Making himself until 89 to 157 butt, Plug this into a chance. Later you get so when So. 96631 for 39 Okay, now that we've found why you 2.5? We move on to party, which asks us to the separation to find Mikey. Exactly. Okay, you know that why I got is equal to t. You couldn't What? That's the wine you got to d G got to G C o. Why Moving this over here. We need you over here. We get, do you want? Mmm E Y times. The Y is equal to G g intolerable sides. We get the Y is equal to G squared over to see taking Elena. Both sides get Why is he quitted? Ellen squared over to see you know that wise so is equal to go. So Ellen no over to seek. Is he good to go? Well, that's this Ellen of seeing not taking the year. Both sides get Eve's. Oh, you got to see which is equal to one. All right, I'm putting this back into our formal. We get why d is equal to Owen. Um, t squared over two. Plus. What? Okay, you notice now for part two. You were asked to find the errors in our approximation. Okay, so that's this. Why, uh, t minus y of approximate. Yeah. Okay, so we're gonna do this for cheesy could 0.10 point 2013 and opened for two. Starting to do appoint one are Y 0.1. We put that into here. What is? Why a box? I'm so 0.1. Well, that was this value over here. But you're so if you plug those two value, then just formal here, you get that This is equal to the whole point. So so Or 987542 I can't do the same thing for you. Get the air is zero points 00980 to 6 to seven. So why is it free? Minus y hawks? Don't you agree? You get this is equal to zero points. Don't want 4 to 1. It's it's And why? For minus y a pox of floor. One point for we get so point Still. 180488 four. And lastly for wise opened five minutes. Why approx? I'm 0.5 goodness into your calculator. Your zero point so to 1151 597

So in this problem we need toe find out the two steps for a for the other method. So recall that if the difference in the question why brain? Because to f off whitey f off TV So then from the island methods U N is ru n plus one. Yes. U N plus 1/4 d n u N Delta t Yeah, you zero is white. Zero de en is t n minus one lasted 30 and till 30 it's the step side. So using the starter, we need to find out the values off you want, friend, you. So let us simplify this. So I feel simplify. Awaiting u n plus one is U N plus and we have in this problem were given with So in this problem, why zero? Why Jiro is minus work a four d y is minus way t 00 That's what it is here. So u n plus the filter tea and ilter TV is finest minus Delta Is your point too? So, um, plus so if Philip lightest Alba getting u n minus U N times zero point what this implies u n one minus zero point. Oh, that's 0.8 U. N So this is the real lesson, that anxiety. So you one plus one is 0.8 times. Okay, so let's find out two steps. So step one. So you one will be 0.8. You zero. This is your 0.8 and you zero is why zero? That's minus one. So this is minus zero point it therefore have you any questions? Minus zero point it a step too. You to e questo 0.8 you won, which is a question. 0.8 times minus Serves reappointed. That is minus of 0.64 Therefore, you tools minus of 0.6 ft. So these are forced to stay for the idea, right? That.

This problem. We need toe solve our differences equation numerically. So basically we will apply the island's method on it. Asked you toe, find out the first two steeps off that. So let's recall that. So if the difference in equation why brain it goes to f off de vie, Then I learned my third states that so I learned states that why end plus one, We teach the end plus one. It approximates, um, is why and the n it approximates. Um, so this is nothing but buy off d n plus steep size delta t f four de en. Why, yeah, where this d n is given way. Okay. Previous escape plus no stoop sites. So in this problem we have steep is 0.5 t 00 Why did she do? Which is why zero is, too, And if off devise for what? Okay, so it's a player. Step one. Yeah. So Step one, the Steven will be t zero still 30 which is half, and hence the Y one will be while zero plus delta t 1/4 d zero. What is you know, which is? So why zero is too. Plus delta T is half claims f R P zero was still on 50 This is nothing but two plus half times Air Force two ways. This is two times two. So two plus half times four, which is nothing but two plus two, which is four. So why one is for so what I'm getting? Why one is for this completes the first day. Let's do this. Take to right. Uh huh. So t two is now even plus identity. The gun was half. Hostility was half. That's one. Therefore, why two will be why one plus until 30 times fr Stephen. And why one This is nothing. But so why one was the previous thing that we got was four president at his heart. Fourth divan is half and why one was for and therefore this is four plus half times. Two times. Why? So it's this. So this is nothing but four plus half times eight. So it's it and therefore what God is. Why do you question it? And this is a state too. So we got y one is four. And why? To his okay, it and this complete stir proof on this complete stuff Sunshine

Yeah, So in this problem, we need toe salt defensively. Question by I live method and we need to compute the first two steps for the island method. So we call that I learned my third is if the difference in the question why praying is there for TV then I lived with that states that the u N plus one is u N plus 1/4 t N u n Until dirty. Yeah, this u zero is zero. That's initial condition. De en is t n minus one plus tell dirty on this day already. Is this step size? Uh huh. So in this problem, turkey velvet for D y is tu minus fire. Why, Jiro, that is also you. Zero is one until 30 is zero point. What? So if you'll Psaltis then she went Plus one will be you will plus f off this with two minus u n times one. So this is your recurrence relation. Okay, so let's find out. Step one step one. So now you one will be you zero plus two minus you zero I am 0.1, but you zero is one tu minus one is one time 0.1. That is 1.1. So what I'm getting is humans. 1.1. Yes. Now step two. Calculate you too. Okay. So you're too will be you want. Plus tu minus you. One tires 0.1. So you won was 1.1 plus tu minus 1.1. Thanks. You 0.1. And this is 1.1. Plus, this is 0.9 times 0.1, and that is 1.1 plus 0.9 which is 1.1 night and therefore you group, is it just a one point once? Hence I have that to Steve's father. I learned that.


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