 Welcome to the GVSU Calculus Screencasts. In this edition, we'll talk about equilibrium solutions to autonomous differential equations. Recall that a differential equation is autonomous if it's of the form dy dt is f of y, where y is a function of y alone, not t at all. So in this screencast, we'll discuss what's meant by equilibrium solutions of autonomous differential equations. Let's consider an example. Let's take the differential equation dy dt is y times y minus 3. Now an important solution to this differential equation is one in which the solution is constant, when things don't change, or when we think of things as being in equilibrium. Now a function is constant when its derivative is 0, so the constant solutions to our differential equation will occur when dy dt is 0. Pause the video for a moment and determine those constant solutions to this differential equation. Resume when you're ready. On our example, we will have constant solutions when dy dt is 0. That's when y times y minus 3 is 0. And this occurs when y equals 0 or when y equals 3. So y equals 0 is a constant solution to this differential equation, and y equals 3 is a constant solution to this differential equation. And we can see these solutions pretty clearly, these constant solutions. When we look at our slope field, it looks pretty clear that there is a constant solution y equals 3 and a constant solution y equals 0 to this differential equation. Now if we examine that slope field a little bit more closely, we can see that there are really two different kinds of behaviors for solutions to this differential equation that depend on the initial condition. Notice that we have an initial condition where y is bigger than 3, say, oh, right here maybe. Then when y is bigger than 3, dy dt is positive, and that forces the solution to our differential equation to drift away from that equilibrium solution y equals 3. Equilibrium solutions that are like this are said to be unstable or repelling. The solutions are repelled away from that equilibrium solution. On the other hand, if we have an initial condition where y is less than 0, say, right here maybe, then dy dt is positive, and that forces our solution to move toward the equilibrium solution. Similarly, if we start with a point where y is between 0 and 3, say maybe right here, then dy dt is negative, and that forces our solution to be attracted toward that equilibrium solution y equals 0. And equilibrium solutions like this are said to be stable or attracting, because they attract nearby solutions. To summarize, an equilibrium solution to an autonomous differential equation is a solution where the derivative is 0. So equilibrium solutions are constant solutions. They have 0 derivative. And if we have a system that's modeled by our differential equation, that system is said to be in equilibrium because nothing changes. An equilibrium solution is stable if nearby solutions are attracted to it. And an equilibrium solution is unstable if nearby solutions are repelled away from it. That concludes our screencast on equilibrium solutions to autonomous differential equations. Please come back soon.