Sketching slope fields
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Worked example: equation from slope field
- [Voiceover] We've already seen scenarios where we start with a differential equation and then we generate a slope field that describes the solutions to the differential equation and then we use that to visualize those solutions. What I want to do in this video is do an exercise that takes us the other way, start with a slope field and figure out which differential equation is the slope field describing the solutions for. And so I encourage you to look at each of these options and think about which of these differential equations is being described by this slope field. I encourage you to pause the video right now and try it on your own. So I'm assuming you have had a go at it. So let's work through each of them. And the way I'm going to do it is I'm just going to find some points that seem to be easy to do arithmetic with, and we'll see if the slope described by the differential equation at that point is consistent with the slope depicted in the slope field. And, I don't know, just for simplicity, maybe I'll do x equals one and y equals one for all of these. So, when x equals one and y is equal to one. So, this first differential equation right over here, if x is one and y is one, then dy/dx would be negative one over one or negative one. Dy/dx would be negative one. Now, is that depicted here? When x is equal to one and y is equal to one, our slope isn't negative one. Our slope here looks positive. So we can rule this one out. Now, let's try the next one. So, if x is equal to one and y is equal to one, well then dy/dx would be equal to one minus one or zero. And, once again, I just picked x equals one and y equals one for convenience. I could have picked any other. I could have picked negative five and negative seven. This just makes the arithmetic a little easier. Once again, when you look at that point that we've already looked at, our slope is clearly not zero. We have a positive slope here, so we can rule that out. Once again, for this magenta differential equation, if x and y are both equal to one, then one minus one is once again going to be equal to zero. And we've already seen this slope is not zero here, so rule that one out. And now here we have x plus y, so when x is one and y is one, our derivate of y with respect to x is going to be one plus one, which is equal to two. Now, this looks interesting. It looks like this slope right over here could be two. This looks like one. This looks like two. I would want to validate some other points, but this looks like a really, really good candidate. And you can also see what is happening here. When dy/dx is equal to x plus y, you would expect that as x increases for a given y your slope would increase and as y increases for a given x your slope increases. And we see that. If we were to just hold y constant at one but increase x along this line, we see that the slope is increasing. It is getting steeper. And if we were to keep x constant and increase y across this line, we see that the slope increases. And, in general, we see that the slope increases as we go to the top right. And we see that it decreases as we go to the bottom left and both x and y become much, much more negative. So, I'm feeling pretty good about this, especially if we can knock this one out here, if we can knock that one out. So, dy/dx is equal to x over y. Well, then when x equals one and y equals one, dy/dx would be equal to one, and this slope looks larger than one. It looks like two, but since we are really just eyeballing it, let's see if we can find something where this more clearly falls apart. So, let's look at the situation when they both equal negative one. So, x equals negative one and y is equal to negative one. Well, in that case, dy/dx should still be equal to one because you have negative one over one. Do we see that over here? So, when x is equal to negative one, y is equal to negative one. Our derivative here looks negative. It looks like negative two, which is consistent with this yellow differential equation. The slope here is definitely not a positive one, so we could rule this one out as well. And so we should feel pretty confident that this is the differential equation being described. And now that we've done it, we can actually think about well, okay, what are the solutions for this differential equation going to look like. Well, it depends where they start or what points they contain. If you have a solution that contains that point, it looks like it might do something like this. If you had a solution that contained this point, it might do something like that. And, of course, it keeps going. It looks like it would asymptote towards y is equal to negative x, this downward sloping. This essentially is the line y is equal to negative x. Actually, no that is not the line y equals negative x. This is the line y is equal to negative x minus one, so that's this line right over here. And it looks like if the solution contained, say, this point right over here, that would actually be a solution to the differential equation y is equal to negative x minus one and you can verify that. If y is equal to negative x minus one, then the x and negative x cancel out and you are just left with dy/dx is equal to negative one, which is exactly what is being described by this slope field. Anyway, hopefully you found that interesting.