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Using Matlab obtain the approximate solution of the equation0.2 sin € 0.5 = 0via the fixed point iteration method with starting point To 0.5. Show that 9 (r) ...

Question

Using Matlab obtain the approximate solution of the equation0.2 sin € 0.5 = 0via the fixed point iteration method with starting point To 0.5. Show that 9 (r) |< 1. Also obtain your solution using 10 ~l3 Comments on YOU solutions.Solve the system of equations30 - 2 =3 21 + W +2 =1, 1-U+22 = -1,using the Gauss-Seidal method with starting point ro

Using Matlab obtain the approximate solution of the equation 0.2 sin € 0.5 = 0 via the fixed point iteration method with starting point To 0.5. Show that 9 (r) |< 1. Also obtain your solution using 10 ~l3 Comments on YOU solutions. Solve the system of equations 30 - 2 =3 21 + W +2 =1, 1-U+22 = -1, using the Gauss-Seidal method with starting point ro



Answers

Use the fourth-order Runge-Kutta subroutine with h = 0.1 to approximate the solution to
$$ y^{\prime}=3 \cos (y-5 x), \quad y(0)=0 $$
at the points x = 0, 0.1, 0.2, . . . , 4.0. Use your answers to make a rough sketch of the solution on [0, 4].

So for this problem, we know that we're going to be using Oilers method. Um and we're told that we want to have an is equal to five. And since our interval goes from 0 to 1, R h is going to be one minus zero, so that will be one divided by N, which is going to be divided by five. So we can write this as 0.2. And now that we have our end and H information, we can go ahead and start filling out our table so we can fill out our t's of ends right away with essentially no calculation. Um, since we know that our step size is going to be 0.2, which is our H, that means that each teeth, with an increasing end, is going to increase by 0.2. So that tells us our T seven values are gonna be point point two 0.4 point 6.8 and one. And so now we could go ahead and fill out the far right column of our table. So we know that wise of end is going to be given to us by this big formula here. So if we want to plug in? Um, wise of one. Our formula is going to become why sub one minus one, which will be wise zero plus h, which is 0.2 times f of t seven miles, one wise it and minus one. So that's going to be times two teeth. So times two times t's of zero, because that's n minus one again. So ah, why someone is going to be equal to twice of zero is zero plus 0.2 times two times t subzero, which is also zero. So that's just going to be zero. And we could do the same thing for finding why some too. This time it will be equal to wise of one plus 0.2 times, two times t someone which is going to give us, um, zero plus 00.2 times two times 0.2, which ends up being 0.8 And so we can go ahead and go through each of our other values and calculate, um, you why value for each, uh, wise of end. And when we do that, we're going to get our remaining values are 0.24 0.48 and 0.8, so from here. We've gone ahead and estimated using Oilers method that, um Why of one is going to be equal 2.8 and next. We're going to be able to solve directly for Weiss of for why of one So we can write our our differential equation as D Y t t is equal to two teeth. And so from here, if we multiply both sides of our equation by DT, we can see that we're going to be able to use our separation of variables. Do you integrate each side of our equation here? And when we do that, our equation is going to become Why is equal to t squared plus C, and so we're going to have to use our initial condition. 00 sulfur see. So when we plugged that in, we're going to get zero is equal to zero squared plus c so we can see that sea is just going to be equal to zero. So our differential equation is just going to become wise equal to you. T square. So if we want to know why of one or we have to do is plug in one for teeth, we find that why have one is equal to one


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