 And welcome to this quick recap of section 7.3 on Euler's method. Euler's method is an algorithm for finding approximate solutions to an initial value problem. An algorithm is a repeatable set of steps that can be used to get to an answer. In the previous section, 7.2, we found that we could estimate solutions to a differential equation by following a slope field. If we had an initial condition, we could follow the rough trend of the slopes to get an idea of what a solution would look like. Euler's method is a more precise way of doing this. Euler's method is named after a famous mathematician named Leonard Euler, and his name is pronounced Euler. Right now you should pause the video and say Euler out loud five times until you start to get used to it. Alright, now that we're back, let's take a look at how Euler's method works graphically, and then we'll take a look at the formulas that make this work. If we're given a differential equation with an initial condition, then we can start at the point represented by that initial condition. We want to follow the slope on the slope field for a little distance. To do that, we'll actually choose a fixed horizontal distance called delta t, and we'll follow the slope at our starting point for that distance. We'll also follow the slope up or down by however much it indicates we should. In this picture, the horizontal axis is the t-axis, so delta t represents a short horizontal distance. After we've arrived at that point, there's another slope, and so we'll follow that slope horizontally for a short distance delta t, and we'll follow up or down however much the slope indicates that we should. When we arrive at that point, there's another slope, and so on. At each point, we follow the slope for a short horizontal distance using the slope to determine how far up or down we should go along that time. Remember that even though we haven't drawn a bunch of slopes here, in fact at this point it looks like there's no slope, we've only drawn a selection of these, and we can use the given formula for the differential equation to calculate the slope at any point. That means that anywhere we end up, we can always find a slope to follow for this short horizontal distance. Here's a picture of a more exact solution given by a better delta t value, a much smaller delta t value. The size of delta t can improve the Euler's method approximation, but it comes at the cost of having to do many other calculations. You can see that the one in red that we calculated first looks reasonable and does follow the same general trend. Now let's take a look at how Euler's method works using some formulas that allow us to repeat these calculations precisely. In this method, we're given a differential equation, an initial condition, and a step size called delta t. This is the short horizontal distance that we'll be following. We'll start at our given point, t0, y0. In order to follow the slope at that point for a short distance, we're going to calculate the t and y values that we arrive at after following the slope. To get the t value is relatively easy. We take our old t value and add delta t to it. Take a look at where this is represented on the image to the right. This is a short horizontal distance. To get our new y value is a little more complicated. We take our old y value, where we began, and we want to add the change in y represented by the slope. To do that, we need to actually calculate this using the items that are highlighted on the screen right now. To do that, we'll use the slope given to us by the differential equation. Remember that we have a formula that tells us the slope at any point. So we'll substitute into that formula our original point, t0, y0. That gives us this slope, which we can interpret as rise over run, and we'll multiply it by the amount of run or the amount of horizontal distance. Together, this gives us a change in y distance or delta y. We'll take that and add it on to our original y value to get our new y value, which we call y1. So now we've arrived at a new point, t1, y1. You should take a look at the picture on the right to make sure that you understand how each of these is related to the previous one. Now that we've arrived at this new point, t1, y1, we can repeat the process to get to yet another point, t2, y2. Again, we add delta t on to our current t value, and we add this formula, the slope at our current point, times our delta t or step size. We add that on to our current y value. So at every step we take our existing t and y value, and we add on to them a formula based on delta t and on the slope at our current point. We can keep repeating this at any point. So if we're at a point tk, yk, where k represents how many steps we've taken, we can calculate our next point called tk plus 1, yk plus 1, by using these formulas. Once we do that, we'll arrive at a new point. We could use Euler's method to calculate the next step beyond that and so on. The formulas on screen right now are the fundamental formulas that allow us to use Euler's method to calculate an estimate for the solution to a differential equation. The picture here represents the general form of an equation, and we can think of this final y value as an estimate for the value of that solution at the given t value, tk plus 1. This method is called an iterative method, meaning that we iterate or repeat the same steps every time using the previous point in these formulas to get a new point. We can keep doing this for as long as we would like. If we'd like to simplify this a little bit, we can see that we're given this differential equation where the function f represents the slope. So instead of writing dy dt at this point, we can write f of tk, yk, which represents exactly the same thing. f and dy dt are both ways of writing the slope at a certain point. We can keep doing this until we've reached the desired t value, at which point we should stop, and we can look at that y value as an estimate for the value of our solution. So how good is Euler's method? Well, suppose that we follow Euler's method with a step size of delta t. Then the error in our estimate of any y value is proportional to the step size delta t. In words, that means that the exact value of the solution, if we knew how to calculate it, minus the estimate given by Euler's method is approximately equal to c times delta t, where c is a constant that depends on the differential equation. The way to understand this is to think that if I made delta t half as large, then the error, that is the difference between the exact value and the Euler's method estimate, would also be half as large. So by making delta t small enough, we can get an answer as precise as we would like. Now that we've seen the idea behind Euler's method, let's take a look at how to use it in practice.