 In this video, though, then we will start looking at new medical methods, new medical. Up until now, we've really looked at analytical methods. In other words, we've built a few recipes by which we can solve differential equations or sets of differential equations. Now, that's usually not the norm. It's not easy to do. We do get our equation y prime, and it is set as some function of x and y. And the vast majority of those you can't really solve through the analytical methods that we've had before. And you can only solve them numerically. And we have to have ways of doing that. By numerically, we mean, which means we're just going to be able to plot points, plot points, plot points, and thereby build up our graph or our set of solutions. Now, most of these will have initial value problems. In other words, out of a whole family of equations, set of solutions, we'll just have one set by our initial value problem. So what I want you to think back at, so that's in essence what we're going to do. We're going to build up points, which is going to approximate a solution. And that's really what your computer does if it draws a graph of a differential equation that uses these numerical methods. Numerical meaning number. We're just going to get numbers, numbers, numbers, which is the x, y values that we plot and that will build up our graph and approximate the solution curve. So just think back, I had an equation y equals x squared. And that was really simple. It was this parabola right there. And I could get the slope at any point. I could get the slope at any point by just taking the derivative of that point. In other words, y prime was going to be 2x, which is a straight line. As you can imagine. And the slope of that line, the slope of that line changed everywhere. But we could work out an equation for that line. And we usually say mx plus c. Now this is not the y of that y. This is the y, yes. Maybe we should call this y sub 1. This was y sub 1. And let's call this y sub 2. And this y sub 2 is the equation of this line anyway, but it has a slope. And that slope is this first derivative. Remember that? The slope is just that first derivative. So what we actually have there is the slope. There's nothing other than that. We are given the slope. But in differential equations, is this the slope anyway? Any way in the x, y coordinate system. So there's a little point there and there's a little line element going through that point. And that is our slope exactly at that point. And that is why we could build up, we could build up these little slope elements everywhere. And if we combine them, we had, you know, that was one of the solutions. That was initial value problem. Usually it's not. It's this whole field of little line elements that make up all the sets of, all these solution, possible solution of any of solutions. Initial value problem. So we're just going for one. So we know what the slope is at every point. And remember this is a tangent line. But if I, if I had to extend things a little bit, let's make them a bit bigger. There was a point and there was a point. There's also this secant line. Okay, which just is a slope of this line connecting these two points. And it approximates, it's sort of the average of, of all the slopes here. And it was very easy to get this slope. It was just delta x and delta y. So the slope of this line that it's linked. We're not talking x squared plus y squared equals the hypotenuse squared. It's not Pythagoras. We just want the slope of this. So the slope of this secant line here was just going to be delta y divided by delta x. Delta y divided by delta x. So we could imagine that if we started here and we knew how far we were going to go across delta x. And we knew the slope. We knew that at this, so that's our value where we start, x of 1. This is our second value, x of 2. We know I'm going from x of 1 to x of 2. What the new, that could have been y sub 1 and y sub 2. Forgive me, that has nothing to do with that y sub 1 and y sub 2. But we know we could work out what this delta y is. How much we're going from now to the next spot. We could work that out if we knew the slope. Because I could just bring delta y on its own. And then this is going to be n times delta x. So if I knew the slope at that point, which I know, I know what it is. I'm giving the slope. And I know what this delta x value is. I can just work that out. Now remember this is, let's put it properly. It's y sub n plus 1 minus y sub n. So I'm going from 1 to 2 or 2 to 3 or 3 to 4. It's going to equal the slope, which is there. Which I'm giving there. I'm just writing it as that now. And x sub n plus 1 minus x sub n. So I'm just going from x sub 1 to x sub 2. So that's exactly what I have. Now I'm going to clean the board. And with this information, this is nothing new. This is really elementary school mathematics that we have here. I'm going to use this to develop our first numerical method, which is called Euler's method. After the famous mathematician Euler. Euler's method, which is very simple. I'll give you a little clue as to where we're going to go. We're going to say, well, we're going to move this across. We know what the slope is by using this equation. And we're just going to work out what the new point is. Now remember here, the slope is just going to be here. So we're actually going to end up there. So this is going to be an approximation. And you can well imagine the smaller we make this delta x, the smaller, the closer we will be able to stick to the curve. So what we're going to do is just have approximations. But you can very clearly see where this method is going to come from.