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For the following problems, use Euler's Method with $n=5$ steps over the interval $t=[0,1]$ . Then solve the initial-value problem exactly. How close is your Euler's Method estimate?$$y^{\prime}=-4 y x, y(0)=1$$

$y=e^{-2 x^{2}}$$y_{\text { Ealer }}(1)=0.10692864, y_{\text { Exact }} \approx 0.13534$

Calculus 2 / BC

Chapter 4

Introduction to Differential Equations

Section 5

First-order Linear Equations

Differential Equations

Harvey Mudd College

University of Nottingham

Idaho State University

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to set up our Oilers methods. You know that we want five sub intervals, so that tells us that our end is going to be equal to five. And to calculate our h, we want to find the domain of our problems, which goes from 0 to 1 so that the length of one, um, divided by five because that's our end value. So our h will be 0.2, and from here we can go ahead in calculate each of our exit and values. Essentially, this h gives us our step size. So that means that each a subsequent except Ben will increase by one times R h. So that means that except one will be point to accept. It will be point for, um, and so on as 0.6 point eight and one. And from here, we're going to be able to calculate our wise of ends. So we know that first, want to calculate why someone and it's going to be equal to wise of n minus one, which is why subzero plus H, which is 0.2 times f of x zero. Why subzero? And so when we plug all this information, we know that y zero is equal to negative one, and this is going to be equal to or plus added to 0.2 times. Um, just why subzero, which is negative one. And this is going to end up giving us a solution of negative 1.2. I'm so little bit and put that in your table right here and next. We can calculate wise up to this is going to be equal to wise of one plus 0.2 times f of ex of one wise of one. I'm so we know that wise of one was negative. 1.2 and this is going to be added 2.2 times are negative 1.2, which will give us an answer of negative 1.44 So again, I'll fill that into our table right here, and all we have to do to complete our approximation is fill out the rest of our table in the same way. So our remaining values are for a wife are going to be negative. 1.7 to 8 negative. 2.736 and finally negative. 2.48832 So we see that using Oilers method. Um, this very bottom right number is going to be our approximation for why, um and for the last part of the problem, we want to compare our approximation to the exact solution. So we're giving the formula for our exact solution. And all we have to do to find of the exact solution at X E X is equal to one is plug in one for X into this equation. So that's gonna give us a negative e to the first power, which is actually equal to have negative 2.718 so we can see that this is fairly close, um, to our solution that we calculated it's off by about 0.23 which is a fairly small margin of

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