 If one differential equation is useful in modeling many problems in the sciences, then two or more differential equations are even more useful, except they do impose new challenges. In many situations two or more variables interact, so the military spending of a country is dependent on the military spending of its neighbors, or the number of wolves is a function of the number of rabbits, or the position of a planet depends on the gravitational attraction of all the other planets. We'll focus on systems where our independent variable is t, usually representing time, our dependent variables are x, y, and so on, and all of our derivatives are taken with respect to t. This produces a system of differential equations. How do we solve systems? I don't know. But remember, your inability to solve a problem does not make the problem go away. And so we might begin by considering the phase plane. Before we try to solve systems of differential equations, let's consider what we can say about the solutions. And so if we have a system of two equations, we can begin the analysis of our model using a phase plane. This time our null clients correspond to the places where the time derivatives are zero, and one important idea here is that the independent variable t does not appear in the phase plane. So for example, suppose I have a system of differential equations, dx dt equals something, dy dt equals something. So again, here we have our variables x and y, which are assumed dependent on t, and our derivatives are taken with respect to t. Let's sketch the phase plane and sum of the trajectories. So we'll find our null clients. Remember the null clients are where the derivative is equal to zero. Well, here there's actually two derivatives. So the null client where dx dt equals zero corresponds to the curve zero equals 3x plus 4y minus 24. Well, that's actually a straight line through the point zero six and eight zero. So for any point on this null client, dx dt equals zero. Since dy dt might not be equal to zero, this means that if a trajectory passes through a point on this null client, the trajectory will have some vertical component, but have no horizontal component. And so trajectories pass through this null client vertically. Now in previous problems, we'd drawn the direction arrows, but since we don't know the sign of dy dt, we don't know if the trajectories are passing through the null client going upward or downward. So rather than draw direction arrows now, we'll wait until we have more information. The null client dy dt equals zero, corresponds to the curve zero equals 2x plus 15y minus 30, which is a straight line through zero two and 15 zero. And again, for any point on this null client, dy dt is equal to zero. And again, since dx dt might not be zero, that means that if a trajectory passes through a point on this null client, the trajectory will have some horizontal component, but have no vertical component. And so trajectories pass through this null client horizontally. And again, since we don't know dx dt, we can't commit ourselves to the direction of the trajectories, so we'll wait until we have some more information. So note that the null clients divide the phase plane into four regions. We'll consider what happens in each of the four regions. So in this first region, the x dt is greater than zero, so x is increasing, dy dt is also greater than zero, so y is increasing. So if we started a point in this region, we begin moving right and upward. So we might draw a trajectory that starts at some point and moves right and upward. And it's important to note that since this is true for all points in the region, we further note that if we start near either null client, we move away from the null client. And so we won't cross the null client from this region. In this third region, dx dt is negative, so x is decreasing, and dy dt is also negative, so y is also decreasing. So if we started a point in this region, we begin moving left and downward. So we might draw a trajectory that looks like this. And as before, if we start near either null client, we move away from the null client. Now in the second region, dx dt is negative, so x is decreasing, and dy dt is positive, so y is increasing. So if we started a point in this region, we begin moving left and upward. Now if we start near the dx dt equals zero null client, while dx dt is going to be close to zero, dy dt is still positive, and so we are still moving upward. So we'll cross the null client going upward, and this takes us into the region that carries us right and upward, so our trajectory will continue that way. On the other hand, if we start near the dy dt equals zero null client, we're still moving leftward. So we'll cross the null client going leftward, and this takes us into the region that carries us left and downward, so our trajectory will continue that way. In the fourth region, dx dt is positive, so x is increasing, and dy dt is negative, so y is decreasing. So if we started a point in the region, we begin moving right and downward. And again, if we start near the dx dt equals zero null client, we're still moving downward, so we'll cross the null client going downward, and this takes us into the region with trajectories moving left and downward, so we'll continue that way. On the other hand, if we start near the dy dt equals zero null client, we're still moving rightward, so we'll cross the null client going rightward, and this takes us into the region with trajectories moving right and upward, so we'll continue that way. And one final note. Notice that at the intersection of the null client, dx dt is zero and dy dt is zero. So if you start here, you don't move horizontally, and you don't move vertically. And because of this, we say that this intersection point of the null client is a fixed point. There are, however, different flavors of fixed points, and here the important idea is that even if a trajectory starts near the fixed point, it will move away from the fixed point. Because of this, we say the fixed point is unstable.