 So one of the important tools in analyzing differential equations is known as a direction field. So remember that our goal is to be able to provide an analytic solution to every differential equation. Unfortunately, most differential equations are unsolvable. And what's important to remember is that in math and in life, your inability to solve a problem does not make the problem go away. And so the problem of solving the differential equation still exists. And so we might proceed as follows by trying to solve the problem. What can we say about the solutions to a differential equation if we can't actually find a solution? And this leads to the idea of a qualitative analysis of solutions. Since a differential equation involves a derivative or several, we might recall what the derivative itself says about a function. And so remember, if the derivative is positive, the function is increasing, and if the derivative is negative, the function is decreasing. One way to represent this information is to produce a direction field. And the idea is this. Suppose we have a first order differential equation. So remember that in principle, we can write this as a derivative equal to some function of the independent variable t and the solution to the differential equation y of t. And the idea is that our solution will be y of t and we can graph this in the ty plane. And our direction field shows how the dependent variable y changes as the independent variable t increases. Now it's important to recognize there's a bunch of things running around here. There's our independent variable t. There's our function y of t and if you keep all of these things, then the notation gets a little bit cluttered. And so we usually adopt the following convention. Since the independent variable is often understood, we usually exclude it from our function notation. And so while we are supposed to write first order differential equations this way, we often drop this dependent variable t when it's part of a function expression and write it this way. Well, let's see how that might work. So let's graph a direction field for the differential equation y prime of t equals 3t plus 12. So remember the important feature about the derivative is its sign. And that the sign of the derivative can change at a critical value, where the derivative is zero or undefined. And this means the first step is we want to find the critical values, the places where the derivative is zero or undefined. So again, since t is the understood independent variable, we can exclude it from our function notation. So instead of writing y prime of t, we can just write y prime and then it's equal to 3t plus 12. Where we have to keep this t because this is not part of our function notation. We want to know where the derivative is zero or undefined. So y prime is zero when t is equal to negative 4. And so our first step is in the ty plane, we'll graph t equals negative 4. And remember, if it's not written down, it didn't happen. We should label our t axis, our y axis, and our line t equals negative 4. And we want to find the sign of the derivative in every region. On the right side of t equals negative 4, we have t greater than minus 4. To find the sign of the derivative, we can pick a test point, say t equals, oh, I don't know, 1,000. And we find our derivative y prime will be positive. And so y is increasing. And so our direction field is going to show us what happens to our function. So we want to draw the direction field with arrows that are pointing to the right because the independent variable is increasing and upward because the dependent variable is also increasing. And so in this region, we might draw a representative arrow pointing to the upper right. And similarly on the left side of t equals negative 4, t is less than negative 4. So a test point might be t equals minus 1 million. And we find y prime is negative. So y is decreasing as t increases. So in our direction field, we'll draw arrows pointing to the right because our independent variable t is increasing, but down because y is decreasing. And so we'll draw an arrow pointing to the lower right. Now, because the direction field is a field, it can be helpful to fill out more direction arrows. And here's an important idea. Since we already know the sign of the slope in our regions, we know it's positive over here and negative over here, which means our arrows are pointing up on this side and pointing down on this side, we'll focus on the magnitude, the absolute value of t and y, and how it affects the magnitude of y prime. And the important idea here is that increasing the magnitude of t or y moves us away from the origin. On the other hand, increasing the magnitude of y prime, that's the slope, makes our direction arrows more vertical. So the first thing to notice here is that y prime does not actually depend on y. And what this means is that above and below any direction arrow is another arrow with a similar direction and magnitude. So we'll copy this one direction arrow that we graphed above and below. On the other hand, since y prime equals 3t plus 12, then increasing the magnitude of t will increase the magnitude of y prime. And so the direction arrows will be steeper the further we venture from the origin. So to the right, we'll draw some steeper direction arrows.