Topic 7.3 key terms
- differential equation
- An equation for a derivative, such as ; it gives the slope a solution must have at each point.
- slope field
- A picture of a differential equation: at a finite set of points, a short segment with the slope the equation gives there. Equal lengths, no arrowheads.
- solution curve
- The graph of a function whose slope matches the differential equation at every point; on a slope field it follows the nearby segments smoothly.
- family of solutions
- A collection of functions that all satisfy the same differential equation, often infinitely many. For the smooth equations in this lesson, there is exactly one through each starting point.
- initial condition
- A specified value, such as (at the output is 1), that a solution must satisfy. For the smooth equations used in this lesson, an initial condition selects a unique solution.
- particular solution
- A specific solution that satisfies the given initial condition.
- equilibrium solution
- A constant solution , where the slope is zero at every point of the line; for the smooth equations used in this lesson, nearby solutions can approach it but never cross it.
Transcript
614 words · 2 parts
Five arches
It’s a calm night on this river, and up on the footbridge people are setting paper lanterns down on the water and simply letting them go. Nobody steers them, so each lantern goes wherever the water pushes it, and if you leave a camera’s shutter open, every one of them draws its own path in light.
Now look at the bridge. It has five arches, and wherever a lantern starts on that footbridge, it ends up floating under the same one. From up here you can see what’s going on. The paths swoop and curve in completely different ways, and then all of them close in on one perfectly straight lane.
The water at each spot only knows which way it’s pushing, a little direction and nothing more. Let’s keep just those directions and let the paths fade, because everything you’d need to draw them again is still right there.
By the end of this lesson, you’ll know exactly why every lantern picks that arch. You’ll also be able to look at the rule behind a river like this and draw any lantern’s path before you even let it go. Let’s start with just one lantern, freeze it right here, and ask what the water is telling it.
What the water knows
Let’s put a map on the water, with x measuring how far downstream we are and y measuring how far across, ten meters to each step. Our lantern is sitting at the point two, one, and right here the water pushes it along this little line, so at this exact spot its path has a slope.
This river follows one rule, the differential equation d y d x equals x minus y. It says the slope of a path at any point is that point’s x minus its y, so here that’s two minus one, a slope of one. It’s a made-up river that follows this rule exactly, so every lantern’s path has slope x minus y at every point it passes through.
If we pick some points and draw a little segment at each one with the slope the equation gives there, we get a picture called a slope field. Let’s start at the point zero, one, where x minus y is zero minus one, negative one, so we draw a little segment heading down with slope negative one.
Notice the point sits at a height of one, but its slope is negative one, and those two are really easy to mix up. At one, zero we get one minus zero, a slope of one, and at one, one we get one minus one, zero, so that segment lies perfectly flat. Let’s fill in the rest of these nine points the same way, and watch each little segment land on the graph as we go.
There’s a pattern hiding in this table. Every point where y equals x gives zero, and if you slide one step down from that diagonal, every slope is one. That means the slope only depends on how far above or below that diagonal you are, so we can fill in the whole field in a few seconds.
Every segment is the same length and none of them has an arrowhead, because a slope field only shows the tilt, so a slope of two means steeper, never longer. A slope field only ever shows a finite set of points, but that’s plenty, because the tilt changes smoothly from one point to the next.
Now let’s see you build a piece of this field yourself, right on the water. Turn each of these four dashes so its slope matches the rule, d y d x equals x minus y, at that point.