Like

Report

(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?

a)

(i) $y(0.4)=1.4$

(ii) $y(0.4)=1.44$

(iii) $y(0.4)=1.4641$

b) We see that the estimates are underestimates since they are all below the graph of $y=e^{x}$

c) Each time the step size is halved, the error estimate also appears to be halved (approximately).

Differential Equations

You must be signed in to discuss.

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

in part A. You're giving the queen initial value and you're asked to use the process value of function y 14 His wife. Why initial value is why zero people want part one report A. They're asked Approximate why of 14 even step size inch 14 So using oil admitted you have why, but one people to why zero? Which is the same thing y zero y zero or one What the stepped sides fines function y the value of 01 warm one plus 14 or 1.4. Why? To equal to why one Which four plus our step side for fine Why have one the Orient approximation y 0.4, do you 11 14 report one part two you're asked crossing using a step size but line point to And you see that again Why would one want people to Why hero or one set size? Want to fines our function? Why you're one one 1.2 from zero for y people to 111 what? Except fine our function wide one one 1.2 1.2 Well, 1.4. It's excise. While I went for y two one warm Brian's report to part three. You ask me sex one. That would be four. Why one people to why you is one 11 for one. You're one one 11 or 1.1. That that Why? Why one one? Lots of tips. One your function. Why one one 1.1. Simply 1.1 is equal to one 1.1. People want one. What? Three y two or 1.1. But that size the value of a function Wine one. Want to want to 21 people. 1.1. Finally. Well, last year we have Why or why three or 1.31? Well, I want one highs Valuable country. Why? 131 one, 1.4 or one. So we have Why? For approximately y for one point force or one part we're giving the exact solution. Are you sure you problem your ass? Do you draw the graph of using our boiler cock solution? You were asked to evaluate. They're not prosecutions under overestimate. So we're giving that the solution value problem. Why? People's were asked to draw this from 0.4 on the X axis. Now you have. This is all going to do first. This is details either way. Very useful. For what? Here. I want one y direction. You have but this draw this table of here. Pickle don't for you. 0.4. I want to be need to the point for crossing one point or 91 This is very, very close to five. What is 1.5 max? One point for you, you know, Grab our solution here. Black. We know since that he didn't hear with one 11 He else had a point for just under five. But news is exponential. Cross somewhat once, right? White one. It needed a 1111 of five without 111 0.2. Happen. You want to for one point. You want so a little over. I think he's one. So with this, it looks almost like a straight line. Like you're different set sizes. Our nation on one. So we have from this one point this prosecution coursing 01 another point approximation. You have that X one y one people 2.4, 1.4. He knows that the west here. This is consistently on your the real value function. Grab function. What to going to? Really? So we have that want why you want to? 01 We have one x one. Why one you want to one week that closer? And then why x two y two people. 14 I want one for four. It wise. Pretty hard to see, but all right, but the better President report one finally graph from three. I got it in blue Euro one wanted on. We also have X one. Why? One 111 1.1. Okay. Yeah. Uh huh. One. So you see it? They hold endurance to me. But the recipe What I see you have to find error where there's method. You're given that Aaron holders Has this formula hair you Exactly. My this proximate. So this era Pretty positive. One. Sybil Roosevelt, Whether we're overrun Drastic meeting and see that part one really said step size h was going forward. Error is Ongondo Approximate You know that one for one point or nine. Want me minus proximate? You, Wyatt don't is equal to get out point 091 part two step side Want to in our air 1.49 a minus that absolution, Wife for se. Why is for for it's legal to you hear these errors. Action value you four 00 and finally part three days except Sunday. One. Our Internet oxen one point or non one. What is 1.46 is equal to one euro, he said, Since you have these three. After that, she's here. You're nine. And by 20 27 set, it would take ratio of his errors over points. This is approximately one point and want five to 77. So you see that as a step size each is error? No. You see that not only the Kruse's I without so