 Once we have the direction field, we can start to draw some trajectories. What's a trajectory, you ask? Suppose we have the direction field for a differential equation. An initial value corresponds to a value of y of t zero, and that in turn corresponds to a point in the ty plane. Now remember, we're looking for a solution. The solution to the differential equation corresponds to the function y of t, which will tell us the y values for all possible t values. But that's a curve in the ty plane. And if we view t as time and the curve as the trace of a moving point, then the curve is the trajectory of the solution. A very useful idea when looking at direction fields are the ideas of isoclines and nullclines. We define them as follows. In a direction field, an isocline is a curve along which the derivative is a constant and a nullcline is an isocline along which the derivative is zero. And in fact, we've actually been graphing nullclines all along. We just haven't used the term yet. So for example, given the direction field shown, let's sketch several trajectories. And a good guideline is, as a general rule, take a starting point in each region defined by the nullclines and also a starting point on each of the nullclines. So our nullcline splits the plane into two regions, the top and the bottom. So let's pick a point in the top region, how about right here, which is conveniently enough where we've drawn a direction arrow. So there's two things to keep in mind. The important one is that the direction arrows are tangents to the trajectories. And that means that the curve goes roughly in the same direction as the direction arrow. And so what we might do is we might follow the arrow for a short distance. And that takes us near to another direction arrow right here. And again, the direction arrow should be tangent to the trajectories, so we'll follow this arrow for a short distance. It's also useful to draw the portion of the trajectory leading up to our chosen starting point. If we look at the direction arrows before our starting point, we might guess that the portion of the trajectory before the starting point looks something a little bit like this. So these are trajectories above the nullcline. Let's pick a starting point below the nullcline, how about here, and we'll follow the direction arrow for a short distance to get part of the trajectory. This takes us to another direction arrow, so we should follow that direction arrow. And we'll try to find the part of the trajectory leading up to our starting point. And finally, we should pick a point that actually starts on the nullcline as well. So how about here? The direction arrows tell us that we should go to the right. And this takes us to a direction arrow that says we should go to the right. And if we try to sketch the trajectory leading to this point, we might draw something like this. And so this gives us three distinct trajectories for this direction field. Now with a kind and gentle math teacher, you'd always be given the direction field. But you don't have that sort of math teacher, you have me. And that means you may have to sketch the direction field from the differential equation before trying to sketch some of the trajectories. So let's sketch the direction field for y dot equals 2y minus t. So remember y dot is another way of writing dy dt. This is a form that physicists like to use because of Newton. So again, the easiest way to proceed is we'll find where y dot is equal to 0. And so this will occur along the line y equals 1 half t. And remember, this is where our derivative is going to be 0. And while this is what we've been doing before, we do have a fancy new name for it. This is either the nullcline or the y dot equals 0 isocline. So we'll sketch the y dot equals 0 isocline. And remember that for any point on this isocline, y dot is equal to 0. And that means our direction arrows will be horizontal and pointing to the right. Now you'll notice that we've drawn the direction arrows through the nullcline. And that's because the direction arrows will actually be tangent to the trajectory of the curve. So as the curve passes through the nullcline, its tangent will be horizontal. And so our direction arrows show what that tangent will look like. So if we're above the nullcline, remember 2y minus t will be greater than 0. And since 2y minus t is equal to y dot, which means that y dot will be greater than 0, so the direction arrows will point upward and to the right. Moreover, the greater the value of y, the greater the value of y dot, so the direction arrows are steeper the higher we go. So let's sketch a few of those direction arrows. On the other hand, if we're below the nullcline, 2y minus t will be less than 0. So y dot will be less than 0. So the direction arrows will point downward and to the right. So let's draw some trajectories. Again, as a general rule, pick a starting point in each region defined by the nullclines and also a starting point on each nullcline. So if I start back here, my direction arrow points us to the right and upward, so we'll follow that for a short distance, which brings me to a place where my direction arrow tells me to go right and upward. And once again, we're at a place where our direction arrow says go right and upward. And we can also sketch the portion of the trajectory leading to our initial point, which might look something like this. Or we can pick a starting point below the nullcline, like here, and we follow our direction arrow. And we can draw the trajectory leading to that point. What if I start on the nullcline itself? Our direction arrow takes us to the right, so we can draw the trajectory. And now we're in the region that takes us right and down. So we'll follow a trajectory that looks something like this. And again, the trajectory that led up to this point might look something like this.