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So now we want to compare our three methods Euler, R.K., four and R.K. four or five.

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And since so far, we have only implemented this for solving first or the differential equations.

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We have to, of course, search for an example of where the differential equation was of First Order

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and some of the most difficult one that we have considered.

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So, for example, number two, two time amplified decay.

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And so I will copy these to cells here and let me make a new one and copy this.

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So here.

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And also this cell here.

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And then copy the whole thing and go to the bottom of the notebook.

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So here we have it.

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So when I run this, it gives me the solution for our oily methods.

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So what we will add next is, of course, we will add new solutions.

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And therefore, I would call this solution with the old methods solution.

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Then I have to, of course, also update this one here and here.

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And now I will calculate all sort of solutions with our other two solvers.

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And we have only we modified the general syntax of the Euler function.

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So the syntax here is really the same for the OK four in the arc, four or five.

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So we can just change the name here, and it should work.

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So this one will be our K four solution r k four.

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Then we have our K four five.

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And like this, and now we can just go ahead, copy this line of code and updates here, the R.K. four

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and R.K. four or five.

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And of course, also here for the Y component.

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And when I run this, we do not see anything really, so it seems as if all three solutions are pretty

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much the same and they are on top of each other.

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So this is already a good sign because there is no large deviation comparing the three methods.

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So now we go ahead and do something that we also did previously.

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When we compare the methods that were implemented in the solve, underscore IVP solver of the integrate

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module.

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We will compare these differences here in the way value.

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And so I will go ahead and copy this code here

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and I will write T and paragraph off Y and then here I will just subtract the test value, basically.

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So what we did previously is we find already here a test value test T and then test Y.

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And the only thing that we have to make sure I think this one is not correct because here we are missing

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a one because we only always start at two zero and then we go and next steps to h.

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So we have in total and mixed +1 values.

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And so now we also have here and makes +1 values.

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Since I updated this, I have to rerun the South and now I can just write here

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basically minus test y.

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And then we do this for all three lines.

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And now just to compare them, I will add colors.

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So you know, they right blue.

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Here I write red and green.

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And then since you see this line here so thick, I will also change the catapult to just a regular plot

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so that the points are connected by lines.

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And if you won this, then we see we have a blue line here, which so let me just tell you again what

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this shows here.

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So this is basically the error of our method compared to the analytical solution.

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And you see the blue curve, which is the only method has a very large error compared to the other two

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methods.

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So in fact, the green and the red one, you cannot even see anything different from zero here.

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So what I will do next is I will copy this paste and just comment or delete this line of code so that

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we do not show the blue one anymore and just concentrate on red and green.

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And you see this one.

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Or this time we have a 10 to the power of minus 10 arrows.

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So these values have to be multiplied with 10 to the power of minus 10.

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So this will be a very, very small value compared to this one, which is just 10 to the power of minus

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three.

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And you see, the red one has then an arrow, which is the Arc four method and the arc for five is an

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even more accurate.

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So we have really, really good solutions here and a very high accuracy.

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And this is really the take home message of the section we can, of course, implement just are straightforward,

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simple solutions like the older methods.

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But we always have to keep in mind that this brings about some error and that there are other methods

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that are much more accurate.

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We have seen that deriving these problem.

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The methods is, of course, very difficult.

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In fact, we didn't do it because it was too difficult here to time consuming.

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But if we just look at Wikipedia or other websites or books, you can find these tables here.

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And it didn't take us more than five minutes to implement these tables here.

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Of course, it's a bit tedious to write on all of these numbers, but you just have to do it once and

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then you have implemented.

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These are k four or five methods, and you see the error is really remarkably small.

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So here we have two times 10 to the power of minus three for the other method.

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We have 10 to the power of minus 10 for the quarter full methods and 400 cutoff for five methods.

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I don't even know.

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Let's see.

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We have, yeah, four times 10 to the power of minus 13.

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So really, really small error.

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The very high accuracy.

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I hope you see now that these methods to when we put our methods are very helpful and very useful.

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And of course, one thing that I have to mention, as well as in our following codes, we could of course,

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use this module here that we have implemented our K for five, which was a bit time consuming, but

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maybe just five to 10 minutes to write on all the coefficients.

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But of course, in the following mode we are going to use is we are going to use the solver of integrated

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solve, underscore IVP and then rewrite method, for example, R K for five.

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And then we have basically the same thing with even an adaptive step size, which is even better.

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So just let's get back to this example here, where we have really large step sides, step sizes at

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certain ranges.

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Where to code releases?

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OK, that function doesn't change that much here.

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It's OK to make a large step, and the arrow will not be too large.

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And so we can save a lot of time because here we only have to calculate a very few points compared to

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maybe 500 points if we would have chosen an equidistant list of points for this range here.