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JENNIFER WEXLER: Last time we looked at differential equations
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from almost entirely an analytic perspective,
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and we found solutions to some differential equations,
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looked a little bit at their graphical solutions,
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but focused primarily on the analytic anti-differentiation pieces.
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This time we're going to look almost exclusively
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at a graphical analysis of differential equations using a tool
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called slope fields.
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So slope fields, as you might think from the name,
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are quite literally fields of slopes.
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So I'm going to put up my field- here is my field,
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the coordinate plane- and the little dots
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are going to help guide me as to where I'm
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going to draw the particular little slopes for the differential equation
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that I'm going to create a picture for.
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And the differential equation that we're going to start with
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is dy dx equals x plus 1.
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That's a differential equation we've seen before.
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In fact, it was one of the ones we solved analytically last time.
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This time I'm going to create a picture on this field of all these little slope
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segments, because I want to keep in mind that what dy dx is really telling
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me is a slope of a tangent line.
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And so I'm just going to pick a point, I think I'll start with 0,2,
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and I'm going to look at the differential equation
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and say what does that differential equation tell me
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about the slope at 0,2?
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Well, the slope is x plus 1.
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So that's 0 plus 1.
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So that's 1.
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So to help me visualize that, at the point 0,2
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I'm going to draw a little segment of slope approximately 1.
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That would be the slope of a line tangent
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to a solution curve right there.
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If I picked another point, say dy dx at 0,3,
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since this differential equation does not depend on y
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it doesn't matter that I changed the y to 3.
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The slope at that point would still be 1.
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And I could draw a little parallel segment at 0,3
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showing that a tangent line there would also have a slope of 1.
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And in fact anywhere I go on the y-axis I
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would see these little slope segments of slope 1.
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If I moved to another x-coordinate, say the coordinate negative 1, something.
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Again, it doesn't matter what the y-coordinate is,
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and so all I care is that the x is negative 1, negative 1 plus 1 is 0.
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So if I go to points where the x-coordinate is negative 1- that's
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along this column right here, this vertical line-
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I would have horizontal slope segments representing slopes of tangent lines
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at that particular x value.
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And I could continue on to fill in slope segments at all of these points,
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and I'm going to take a moment to do so and maybe give you
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a chance to do so as well.
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And so here we have a more complete slope
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field that shows not only the segments that we had already seen together,
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with slopes of one and slopes of zero, but I
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can see that as the x is increased- as x gets bigger here
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the slopes will get steeper.
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I see some increasingly steep slope segments.
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And as x becomes more and more negative, I
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see some increasingly steep negatively sloped segments.
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And as we had discussed before, along any vertical line,
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since the differential equation only depends on x, all of those little slope
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segments are parallel.
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And I could try to follow and find a specific solution
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curve- you may recall those solution curves were parabolas- I would start
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somewhere and I would follow that little slope until I got to the next slope
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segment, and continue to try and follow the trend.
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And I could see in that slope field, one of the parabolas
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that was a particular solution to our differential equation
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that we had seen before.
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Slope fields are a great way to visualize solutions,
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but it's a lot easier to see the solution
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when we use some computer generated slope fields, because we'll
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get to see more little slope segments coming together.
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And so we will now look at not only this particular slope
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field created by a computer, but a second slope field,
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and do a little bit of a comparison of the two.
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So here we have the slope field that we just created by hand:
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dy dx equals x plus 1.
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So that's the slope field associated with this differential equation,
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with the little parallel segments along vertical lines.
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Here we have a slope field that's showing a different trend.
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It's showing little parallel segments across horizontal lines.
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And so, just like these parallel segments
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told us that this differential equation depended only
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on x, these parallel segments tell me that the differential equation depends
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only on y.
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That I can change x as I move left and right along the slope field and nothing
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happens to my little slope segments.
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This differential equation for this slope field
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is also one we've seen before.
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dy dx equals y.
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We saw that last time.
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We know from last time that those solution curves
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are exponential functions, and if you follow
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the curves-- the slope segments to find a curve,
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you can see an exponential function, potentially in that slope field.
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Here's another one if I start here, there's another exponential function
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that I'm seeing in the slope field.
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And right along here, along the x-axis, we
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see these solutions that are all horizontal slope segments.
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All of these points have a y-coordinate of 0,
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and so the slopes all have to be 0.
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And we saw that solution last time as well, y
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equals 0 is a solution curve for this particular differential equation.
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So some differential equations depend only
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on x, some differential equations depend only on y.
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And we're going to now look at one that depends on both x and y
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So in this differential equation, we see that the derivative depends
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on both x and y, we see an x and a y on the right hand side of the equation.
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And so if I were to choose a point, for example the 0.22,
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then I can find the slope at that point.
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I chose a point where the x and the y-coordinates were equal,
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and I see that that gives me a horizontal slope for my tangent line,
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and that would be true wherever I have x equals y.
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And so along that diagonal I see a number of horizontal tangent lines.
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If I were to pick a point where the x was bigger than the y, say 3 comma 2,
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then 3 minus 2 is 1, and I would have 3,2 a little 45 degree angle
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tangent line.
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And everywhere in this area I would have positively sloped segments.
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If instead I were to pick a point that, where the x was less than the y,
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like 2 comma 3.
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Now I have a slope of negative 1.
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So if I go to the point 2 comma 3 my slope is negative 1.
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And everywhere up here, where the x is less than y,
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I would have negatively sloped segments.
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So if you want some practice you could take some time to try and create
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the rest of this slope field by hand.
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I'm going to move to a computer generated version.
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So here we have a computer generated version of the slope field
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that we were just working on by hand.
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The slope field for the differential equation dy dx equals x minus y.
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And I can see in this slope field the little horizontal slopes at 1,1,
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right all along the diagonal where y equals x.
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You'll notice in the computer generated slope field I'm not
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stuck with just these integer points.
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I'm allowed to see a little more detail at other points.
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But I do see those horizontal segments exactly where
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I would expect to see them.
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I do see those positively sloped segments
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exactly where I expect to see them.
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And I do see those negatively sloped segments
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exactly where expect to see them.
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And you can see there are no-- if I pick any vertical line,
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those slope segments are not all parallel.
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So I know that this has both an x and a y in it; same thing for a horizontal,
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right?
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There's no parallel slope segments along horizontal and vertical lines.
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And so I know that my differential equation
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must have a derivative that depends on both coordinates.
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So I'm now going to move to a dynamic graph
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so we can look at some of the particular solutions
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within this field of slopes that shows us the general solution.
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So now we have this graph right here, that's
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the same slope field that we were just looking at for dy dx equals x minus y.
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But in addition to the slope field, there's
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a particular solution sketched in.
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It happens to be the particular solution that
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contains the 0.01 that's a solution to this differential equation
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that we've seen previously.
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Which solution curve appears depends on what point I know the curve contained.
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So now I have a very different looking solution curve
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that goes to the 0.4,0.5 negative 2.4.
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If I were to change that point again, now that's
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the solution curve for the curve that contains negative 1.3, negative 2.6.
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And you may notice, as I drag this around,
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that the curves have many different shapes.
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There's one that's a straight line, that is a solution to this differential
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equation.
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They are all related, they are all part of the same family of solutions,
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and changing that known condition, that point on the curve,
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allows us to see the different variations in the family.
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So before moving on you'll have a chance to create a slope field by hand,
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and you'll also have a chance to look at some computer generated slope fields
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and interpret the information found in them.
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