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So in the following, we want to, of course, solve more examples of differential equations and not

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just the specific ordinary differential equation that characterizes a radioactive decay.

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So what we're going to do is we are going to define a new function of herself, which basically mimics

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these two cells here.

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And in order to do this, let me just copy the code from these two cells.

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And also this one.

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And now, of course, we will change these cells and define a function.

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So to define the function, we write death.

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And then we give it a name.

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And I recall it.

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Euler Ozdemir, you can give it any name, basically, that is not given for other functions already.

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And now we have to specify here the arguments and here will be the code.

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And the code will, of course, be pretty similar to what we have written down here.

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So let me first write down what we need as arguments we need definitely f where f will be the function

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of the differential equation.

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So when I scroll up here, it will be this one here.

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So basically the first derivative of Y with respect to T, then we will need a starting time.

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This is not really necessary because most of the time will be we will be starting at zero but could

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be helpful to to find a starting time, then we will need a starting value.

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So this will be essentially this one here.

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So I will write y zero for the starting value of Y.

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Then we need the number of iterations

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which is and max, and then we need to step six.

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Which is H.

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And now I, of course, have to add all of these arguments to the function.

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So I write f comma, t zero comma, y zero comma and max comma.

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Page Sorry, it's too.

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It's like this.

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All right.

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So now we have our arguments and we have here.

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This commons define what these arguments mean.

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So this is a recommendation from me that you do this for the functions that you define yourself so that

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later on you understand what these arguments actually describe.

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So now we need to get going.

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So what we definitely are going to need is this one here.

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So the preparation of the arrays and then we will also need the loop, of course.

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But now we must change this maybe a bit because maybe it's not really correct anymore.

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For example, we provide here the function F, so we do not need this command here anymore.

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We already have our F.

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And so here we are.

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Right Y is equal to y plus f with the arguments t comma y and we multiply by h then what we also need

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since we do not.

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Yeah, since basically we have this time t zero now is we need to have time, an internal time running

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in the system.

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So of course, we could just add here a t0, but I think it's a bit more.

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It's it gives you a bit of a better overview.

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If you define here a variable called T, it's not really necessary.

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You could also solve this differently, but I will define your variable T.

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And so in each time step, we will increase T by H.

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Of course, here just starting value, we will now use T and not zero.

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And we will adhere to T values.

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So I write just values append T and I think now we are almost finished.

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We just need to load this y0 zero here.

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So we need to write y is equal to y zero.

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And I think this should already be sufficient.

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So you see, we haven't really changed much.

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Basically, we have just changed the function.

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We have changed the starting value and then we have done this t0, which was optional, however.

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So you see, it's really not difficult once you solve a specific example to define a function.

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And this is also my recommendation that when you are not quite sure how to define the function, just

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try to solve an example without a function and then use the solution to define the function later on.

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Because you see, it was pretty easy to do this.

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So the only thing that's missing here is the output, because all of this code is just internal in this

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function, it will give no output yet.

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So we need to write to return.

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So we could write print, then it would just print what we whatever we want as the output.

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But we want to actually use return so that we are then able to assign these values to variables.

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So we print an array of T values and Y values.

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So I have run this, of course, it gives me no output.

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And now we can test this at our example.

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This is always the best thing you should do first, when you define a function and you have already

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solved an example, try to solve the example now with the function and see if the solution is still

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correct.

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So what we need is we need to call, of course, this function, so we write oiler potty and then we

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must provide all of these arguments.

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So we provide a function f, which I will call F40.

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So this function we still have to define.

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So let me write this down here so that we do it later on, so define function.

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And here we call the other method.

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And then we need to provide the t0, which I will just use zero, as in our previous example, then

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for the Y0 we will use, yeah, we used one previously, but I will write here why zero zero is equal

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to one.

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We could also do it like this t0 and T zero is equal to one.

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Sorry, two zero.

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Then we have.

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And Max and Max was equal to 20.

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I think it was equal to 200.

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And age was equal to zero point zero one.

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OK, so now the only thing that's left to do is to define the function.

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And this is not so easy, maybe at if you have never done it before, because here we have provided

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this with the arguments T and why.

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So we must define a function that also has these arguments.

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So we cannot just right here minus y.

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So I write the I'm sorry, the fine f o d e.

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And then we provide the same arguments as above.

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T come away, of course.

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And then the only thing the function does is give the value of the functions, so return and the value

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of the function is minus Y.

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And I think now it should work.

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So I run this and it gives me a new solution, which is a list of X. Well, I sort of T values and the

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list of Y values.

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So now we can store any solutions in the variables, so I will call this solution.

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And now we can plot this so we can just take the plot from here.

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And changed the T values to solution the first element, so index zero and then the Y values is the

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other element.

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And you see, it still gives the correct result, even though now the solution has been determined by

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calling our all the method function that we have just programmed in this lecture.