 Welcome to this quick recap of section 7.1, an introduction to differential equations. A differential equation is an equation involving an unknown function and one or more of its derivatives. Differential equations can model many physical phenomena, which is why they often show up in the sciences and engineering. For example, differential equations can model the growth of a population, for example of bacteria or rabbits or any living thing. Differential equations can model the decay of a radioactive substance, the amount of interest earned on a bank account, and many other things. The thing that all of these have in common is that they change over time, and differential equations, since they involve derivatives, fundamentally work with the idea of changing objects. A few examples. We can write differential equations using Leibniz notation, such as this first one, dy dt equals t sin of t. Or it can be used in Newton's notation, y' equals y. In the second one, notice that only the function appears. We want to find a function whose derivative is equal to itself. We're solving not just for a variable, but for an entire function. The variable in these differential equations is often t standing for time, since many of the phenomena described above change over time. A solution to a differential equation is a function that satisfies the given description or equation. For this example of a differential equation, you can check that y of t equals t squared is a solution. If you're given a solution like this, you can verify it by substituting it into the given a description or equation, and seeing that it matches the rule. It, however, is much more difficult to find a solution to a differential equation. We will learn a few techniques for doing that later in this chapter, but for now we're going to study the properties of a differential equation and what it means without trying to solve it. Differential equations typically have infinitely many solutions because many different functions can have the same derivative. In this example, there's several different curves drawn, all of which have the same derivative, dy dt equals 2t. This is related to the fact that there are infinitely many antiderivatives for any given function. However, if we specify what's called an initial condition such as y of 0 equals 1, this specifies one particular function which is a solution to the differential equation and goes through the specified point. When we have this extra initial condition, we call this an initial value problem, or IVP, which is a differential equation together with an initial value for the solution. Now that we've seen these definitions, let's take a look at how to work with differential equations.