4.2.3

[STARTS AT 0:59]

Instructor: We're going to apply these to consider this initial value problem-- y prime equals 3x squared minus y squared plus 1. Where we have y at 0, the initial value is 2. Now use Euler's method with a step size of 0.1-- OK, that is going to be our h, that is going to be our h-- to generate a table of values for the solution for values of x between 0 and 1. 

We actually did this, in a way, with our direction fields, so keep that in mind. These two tie together very well. And I'm not going to go from 0 to 1 all the way, because going by 0.1's, that's going to take a while. But I am going to do enough steps here so that hopefully you can see how you could extend this and give you some information to check with. 

So here is what our table is going to consist of. We have x n minus 1, y n minus 1. That is the point we are given. And this is actually the information that we are after. We're after a set of points that represent our solution. Our first point is going to be 0, 2, because that's what's given to us. 

Next, we want to find f of x n minus 1, y n minus 1. That'll be finding the slope at that point. We'll go ahead and do that here. Going to keep all of that visible if I can for the time being. 

So I'm going to evaluate my slope function, 3x squared minus y squared plus 1 at the point, 0, 2, to give me how my graph and my solution is changing at that point. So that is 3, 0 squared, minus 2 squared plus 1. That tells me my slope at that point is negative 3. 

OK. Now, actually, let's give a little more space because the next one may need more. Next, we're going to find the next y value. OK, we already know the next x value because it's going to be 0.1, and then 0.2, and then 0.3, and then 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1. We don't need to worry about the x values. They increase by 0.1. 

However, we need to use these formulas to find the new y value. So this says our new y value, y sub n, is equal to y sub n minus 1, plus h, which will be 0.1 at every stage, times f of x n minus 1, y n minus 1. So for this particular one, this is going to tell us the y, y 1-- OK, that's our next step-- is going to be equal to our y n minus 1, which is 2, plus 0.1 times negative 3, the value we just found in the previous column. 

Well, that tells us that our y value is going to be, what is that? Negative 0.3, so 1.7. That is going to be built into our next step. That value is going to come down here. 

Now we'll begin the process again. 3 times 0.1 squared, minus 1.7 squared, plus 1, which is negative 1.86. Now we'll use that y value, negative 1.86. Actually, hold on. We want the last y value, so the 1.7. That negative 1.86 is our slope. Plus 0.1 times negative 1.86. And of course, I am rounding here. That is going to be 1.514. And again, that will become our new y value for our point. Now again, those points, what we're after, those represent our solution. 

OK. Now at this point, I want to write what I'm doing and then extrapolate for you. 1.514 squared plus 1. That is negative 1.172-ish. OK? And then we would take our values of 1.514 plus 0.1 times negative 1.172196 is what that is. And that is a value of 1.3968. 1.3968. 

And we would continue this process. And I'm going to fill in the blanks as to what the final values are so that you can compare as you do your own calculations. That'd be 1.3287. 0.5 is 1.3001. 

Then we have 1.3061, 1.3435. At 0.8, 1.4100. At 0.9, 1.5032. And 1.6202. Those would be the values we would calculate using Euler's method.