 Welcome to this quick recap of section 7.2, Qualitative Behavior of Solutions to Differential Equations. This is also about slope fields. As a reminder, a differential equation, which will abbreviate DE, is an equation involving an unknown function and its derivative. We can write this dy dt is equal to some function involving both t and y. A DE describes the slope of a solution, y of t, at the point ty. We can see this by reading the differential equation written above by reading it as the slope of y is given by some function involving both t's and y's. We can calculate the slope of a solution at a point by substituting values for t and y into the DE's rule. Then we can draw a small part of the tangent line on a graph at the point ty. By drawing many such lines, we get what's called a slope field. Here's an example in which each of these green line segments is a small part of a tangent line to one of the solutions to this differential equation. Remember that the derivative of a function gives you the slope of the tangent line, and so this shows many different tangent lines to many different functions, all of which solve this differential equation. We can use a slope field to explore solutions to a differential equation. In this example, we're interested in the initial condition y of 0 equals 1. In other words, we want to follow a solution that goes through the point 01, which is plotted on the graph. We'll follow from the point following along the green line segments to the right. We can see that our solution starts out by going upwards, continues upwards, but at some point it starts to turn as we follow the green line segments. It goes downwards, and then drops off steeply as the slopes become more and more negative. This is one solution, the one that goes through the point 01, but we can also do this with many other solutions going through other nearby points. Here's several of them. Remember that a slope field is showing us all possible solutions to a differential equation in a sort of big picture way, so by looking at this we can see overall what solutions to this differential equation looks like. Remember, every differential equation has many different solutions. A particular kind of differential equation that we're interested in is called an autonomous differential equation. This is one in which the independent variable t does not appear in the DE's rule or formula. We can write this as dy dt is equal to some function involving only y. For an autonomous DE, the slopes depend only on the y value. In this slope field for an autonomous DE, by looking along a horizontal line we can see that the slopes don't change because t does not come into the formula that tells us the slope. For an autonomous differential equation, a special type of solution called an equilibrium solution is a solution y such that dy dt is equal to zero. If we write dy dt is a function of y, then an equilibrium solution called y makes f of y equal zero. Remember, y is a function, and so what we're saying is that we have a function y whose slopes are always zero. That means that y is a constant function, and its graph, as shown here, will always be horizontal. The red line on this slope field is an equilibrium solution, and we can see that it never changes as it goes across the graph. And so an equilibrium solution is a constant solution, a horizontal line. If we plot several other solutions to this differential equations that are not equilibrium solutions, we can see that they're pulled towards the equilibrium solution. We call this a stable equilibrium in which solutions are attracted towards the equilibrium. If we imagine that this differential equation models the number of bacteria in a colony, we could say that the equilibrium solution is some key number of bacteria and that the number of bacteria always either grows or dies off until it reaches that value at which they live happily at some constant number of bacteria. Here's a different differential equation slope field. First, make sure that you can identify where the equilibrium solution is in this slope field. Here the red line shows this equilibrium solution, again horizontal with all slopes equal to zero. But if we plot a number of different solutions, we can see that, as we read left to right, these solutions are repelled away from the equilibrium. We call this an unstable equilibrium, and the solutions are pushed away or repelled from that equilibrium. This is something like walking on a tightrope. If you're perfectly on the tightrope, you stay constantly on it, and if you tip ever so slightly to one side or the other, you'll be repelled away from it. Now that we've seen these, let's take a look at them in action.