In the previous lesson, we established how many real world phenomena are governed by differential equations rather than being modeled by known functions. In essence, sometimes we only know how things change, and not the values a quantity will have.

We also saw that some differential equations can be solved, but this is not always the case! In fact, most differential equations in the real world have no exact solutions. It may seem like they are useless in this case, but that is far from being true.

There are many ways to analyze and understand differential equations without solving them algebraically. One of the most powerful things we can do is visualize them. For now, we will only consider differential equations of the form

(AIHL students will learn a few more in the next lessons)

How do we interpret this equation? It says that at each point (x,y), we can calculate the slope using f(x,y). A very handy way to visualize these types of differential equations is to use a slope field. Essentially, we take a bunch of (x,y) points on a grid, and plot a small stick at each point whose slope is f(x,y). Here's an example of what that looks like: