 Let's take a look at a few more examples of direction fields. So again, as a general strategy, given a first-order differential equation, we want to analyze the behavior of its solutions qualitatively, and we'll start by drawing a direction field. We'll find the critical lines and draw direction arrows in different parts of each region. So let's consider the differential equation y' equals ty. So first off, we want to find the places where the derivative is zero. So if y' is equal to zero, our differential equation becomes zero equals ty, and the solutions to this are t equals zero and y equals zero. So we'll graph these critical lines. Now we note that the critical lines divide the plane into four regions. In the first region, t is greater than zero and y is greater than zero, so our derivative is going to be greater than zero. So our direction arrow will point towards the upper right. In the second region, t is less than zero and y is greater than zero, so our derivative is going to be less than zero, and so our direction arrows will point to the lower right. In the third region, t and y are both negative, and so y' will be positive, and so our direction arrow will point to the upper right, and finally in the fourth region, t is positive and y is negative, so our derivative will be less than zero, and our direction arrows will point to the lower right. Again, it's helpful if we draw a few more direction arrows, so remember, once you've determined the sign, focus on the magnitude of the derivative, and the greater the magnitude of the slope, the steeper the line. Since y' equals ty, then as t increases in magnitude, y' also increases in magnitude, and this means that the direction arrows get steeper as we move horizontally away from the origin. So we can draw a few more direction arrows. The further out we are, the steeper the direction arrow should be. Again, since y' equals ty, then as y increases in magnitude, y' also increases in magnitude, so the direction arrows get steeper as we move vertically away from the origin. And again, we draw a few more direction arrows. The further away from the origin we are, the steeper we should draw those arrows. How about this differential equation? So we want to find where y' is equal to zero, so solving for y' setting it equal to zero, and solving gives us the critical lines t equals zero and t equals y. We graph the critical lines, and note that these divide the plane into four regions. So now we want to find the sign of the derivative in each of the four regions. So if we're in the first region, t is greater than zero, and y is greater than t. Since we have this factored form of the derivative, let's make use of it to find the sign of the derivative in that first region. So our derivative is going to be negative, and so our direction arrow points down and to the right. In the second region, t is less than zero, and y is still greater than t, and so our derivative is going to be greater than zero, and a direction arrow will point up to the right. In the third region, t is negative, and y is less than t, and so y' will be negative, and a direction arrow will point down to the right. In the fourth region, t is greater than zero, and y is less than t, and so that means our derivative will be positive, and our direction arrows will point up to the right. So again, to flesh out our direction field, we want to add a few more arrows. So again, we've already determined the sign of the derivative, and that translates into whether our direction arrows are pointing up or down, but the other important thing is the magnitude. And since we only care about the magnitude, we can consider the absolute value of y' Well, that's just the absolute value of the product t times t minus y, and the absolute value of a product is the product of the absolute values. And looking at this expression, we might note the following. As the magnitude of t gets larger, the magnitude of y' will also get larger. And so the direction arrows get steeper the farther we move from the origin to the left or right. And so we might draw a couple of steeper arrows to the further left or further right of the arrows we've already drawn. Likewise, as y gets farther from t, the magnitude of the absolute value of t minus y gets larger, and so the magnitude of y' itself gets larger. And that means the direction arrows get steeper the farther we move away from the line y equals t. And again, we might draw some steeper arrows further away from the line y equals t. Alternatively, since we're running out of room, this also means the closer we are to the line, the shallower our arrows are going to be. So we might draw these additional direction arrows as and this gives us a sketch of what our direction field looks like. Or consider another differential equation. We find where y' is equal to 0. Because this is a rational expression, it's possible that it might be undefined and that will occur if our denominator is 0, but we see that can't happen. So the only critical line is y equals 1. So we'll graph that critical line. If we're above the critical line, y is greater than 1, and a derivative will be negative. So our direction arrows point down and to the right. If we're below the critical line, y is less than 1, and so our derivative will be positive. So our direction arrows point up and to the right. And again, we would like to fill out the direction field. So let's see what happens. The farther we are from y equals 1, the larger the magnitude of 1 minus y. And so the magnitude of y' will also be larger. And so as we move away from the line y equals 1, the direction arrows get steeper. So let's draw a few steeper arrows further from y equals 1. On the other hand, the larger the magnitude of t, the smaller the magnitude of y' and remember, since we're talking slope, the smaller the magnitude of the slope, the more horizontal the line. And so far away from the origin in the horizontal direction, the direction arrows are going to be shallower. So that's going to be true whether we go right or left.