 Welcome to this quick recap of Section 7.5, Modeling with Differential Equations. Modeling is the process of creating a differential equation that represents the behavior of some real-world phenomenon. We can do this if we understand how a quantity changes, that is, if we know something about its derivative, but what we want to know is the actual value of the quantity. Since a differential equation relates the derivative of a quantity to its value, this is exactly the situation we can be in. To see what we mean by this, let's take a look at a few examples. Suppose we want to know information about the number of animals in a population, such as the number of wolves on Isle Royale, or the number of bacteria in a Petri dish. We don't necessarily know this information directly, but we often know about how the population's size changes. For example, we might know the rates of reproduction or death of some of these animals. If we know that, we can use it to construct a relationship between the number of animals and the rates of reproduction and death. We might be interested also in the volume of garbage in Lake Michigan. If we know about the rate at which water and garbage come into the lake and the rate at which water and garbage flow out of the lake, we can construct a relationship between these rates of change and the actual volume of garbage. Finally, we might be interested in the amount of money in a bank account. Money changes based on the interest rate, which is money increasing, or the frequency of deposits or withdrawals, which would be the money increasing or decreasing. Again, everything on the right here is related to the derivative of each item on the left. Modeling necessarily involves approximations and simplifications. You should be able to take a look at each of these situations and see how they leave out some details that might affect the quantity in the left-hand column. For example, for the volume of garbage in Lake Michigan, we probably would want to account for the garbage leaving by different methods, such as sinking to the bottom, washing up on shore, or evaporation of the water in the lake, changing the concentration of the garbage. By leaving some of these out, we make for a simpler model that might be easier to understand, but it also doesn't work quite as well as a more complicated model. This is a fundamental choice we have to make when constructing a model. Although we'll mainly work with examples in this section, there are some common steps that we take when creating a model. So here are some common steps to follow, and in each example that you see in the future videos and in the textbook, try to see where each of these steps is happening. The first and most important step is to identify the quantity in which we're interested and to be very precise about this. For example, we might name the volume of garbage in Lake Michigan in cubic meters as P of T, where P is measured in cubic meters, and T is the number of days since some starting point. By being precise, this helps us identify other related rates more carefully, and make sure that our units will match up when we start to put the differential equation together. Next, we'll identify factors that cause this quantity to change. As we've described before, this could be the rate of water and garbage coming into the lake and leaving it, but we could add other factors that are related to the change in garbage. Once we have all of this collected, we'll write a differential equation that relates the quantity and its rate of change. Our fundamental tool here is that the net rate of change, that is the derivative of our quantity, is equal to the rate of things coming in minus the rate of things coming out. This fundamental tool applies in many situations, not just garbage in Lake Michigan. It also applies to bank accounts, animals living and dying, and other things that we'll see in this section. Finally, once we have a differential equation, we can use all of our existing tools from Chapter 7 to understand the behavior of solutions. This includes slope fields, Euler's method, and solutions to separable differential equations. Anything we've done in previous sections could apply at this point. Now that we've seen this, let's take a look at some of these examples and see how they're constructed in practice.