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- [Voiceover] So we have the differential equation, the derivative of y with respect to x is equal to y over six times four minus y. And what we have plotted right over here is the slope field or a slope field for this differential equation and we can verify that this indeed is a slope field for this differential equation, let's draw a little table here, so let's just verify a few points, so let's say x, y, and dy/dx. So let's say we start with, I don't know, let's start with this point right over here, one comma one, when x is one, and y is one, well when I look at the differential equation, 1/6 times four minus one, so it's 1/6 times three, which is 3/6, which is 1/2, and we see indeed on the slope field, they depicted the slope there if a solution goes to that point, right at that point, its slope would be 1/2. And as you see, it's actually only dependent on the y-value, it doesn't matter what x is as long as y is one, dy/dx is going to be 1/2, and you see that's why when x is 1 1/2, and y is one, you still have a slope of 1/2 and as long as y is one, all of these sampled points right over here, all have a slope of 1/2, so that, just looking at that, that makes us feel that this slope field is consistent with this differential equation, but let's try a few other points just to feel a little bit better about it, and then we will use a slope field to actually visualize some solutions. So let's say, let's do an interesting point, let's say we have this point, actually no, that's at a half point, let's say we have this, let's see, let's say we do this point right over here, so that's x is equal to one, and y is equal to six, and we see the way the differential equation is defined, it doesn't matter what our x is, it's really dependent on the y that's going to drive the slope, but we have 6/6, which is one, times four minus six, which is negative two. So it's negative two, so we should have a slope of negative two and it looks like that's what they depicted, so as long as y is six, we should have a slope of negative two. Have a slope of negative two, and you see that in the slope field. So hopefully you feel pretty good that this is the slope field for this differential equation, if you don't, I encourage you to keep verifying these points here, but now let's actually use the slope field, let's actually use this, to visualize solutions to this differential equation based on points that the solution might go through. So let's say that we have a solution that goes through this point right over here. So what is that solution likely to look like? And once again this is going to be a rough approximation, well right at that point it's going to have a slope just as the slope field shows, and as our y increases, it looks like our slope, it looks like our slope... so at this point I should be, actually let me undo that, so if I keep going up at this point when y is equal to two I should be parallel to all of these segments on the slope field that y is equal to two, and then it looks like the slope starts to decrease as we approach y is equal to four, and so if I had a solution that went through this point, my guess is that would look something, and then now the slope decreases again as we approach y is equal to zero. And of course we see that, because if when y equals zero this whole thing is zero so our derivative is going to be zero. So, a reasonable solution might look something like this, so this gives us a clue of, look, if a solution goes through this point, this right over here might be what it looks like. But what if it goes through, I don't know, what if it goes through this point right over here? Well then, it might look like... It might look like this by the same exact logic. So it might look like this, so just like that, we're trying to get a sense, we don't know the actual solution for this differential equation, but we're starting to get a sense of what type of functions or the class of functions, that might satisfy the differential equation. But, what's interesting about this slope field, is it looks like there's some interesting stuff if our solution includes points between where the y value is between zero and four, it looks like we're going to have solutions like this, but what if we had y-values that were larger than that or that were less than that, or exactly zero or four? So for example, what if we had a solution that went through this point right over here? Well, that point right over here, the slope field tells us that our slope is zero. So our y-value is not going to change, and as long as our y-value doesn't change, our y-value is going to stay at four, so our slope is going to stay at zero, so we actually already found this is actually a solution to the differential equation, y is equal to four is a solution to this differential equation. So, y is equal to four, and you can verify that that is a solution, when y is equal to four, this right-hand side is going to be zero, and the derivative is zero for y is equal to four. So that is a solution to the differential equation. And the same thing for y is equal to zero. That is also a solution to the differential equation. Now what if we included points, what if we included this point up here, and actually, let me do it in a different color, so that you could see it, let's say our solution included that point. Well then it might look something like this, and once again, I'm just using the slope field as a guide to give me an idea of what the slope might be as my curve progresses, as my solution progresses. So solution that includes the point zero five, might look something like this, and once again, it's just another clue. A solution that includes the point zero negative 1 1/2, might look something like... might look something like this. So anyway, hopefully this gives you a better appreciation for why slope fields are interesting. If you have a differential equation that just involves the first derivative, and some x's and y's, this one only involves the first derivative and y's. We can plot a slope field like this, not too much trouble if we essentially just keep solving for the slopes, and then we can use that slope field to get a conceptual or visual understanding of what the solutions might look like given points that the solutions might actually contain.