 So many differential equations arise from problems in physics, and chemistry, and biology, and finance, but mostly physics. Okay, actually, differential equations pretty much show up any place you do any sort of modeling with mathematics. So here's the important question. Given some real-world situation, how can you produce a differential equation? In some cases, we're given the rate of change directly, and since the rate of change is a derivative, we can immediately write down a differential equation. For example, here's a differential equation that arises from physics. Let R be the distance of an object from the center of the Earth. The acceleration of the object is inversely proportional to the square of the distance. Write a differential equation to solve for the distance R at time t. So to analyze this, if R is the distance of an object from the center of the Earth, R prime of t will be the rate of change of the distance. This is what we would call the velocity of the object. R double prime of t will be the rate of change of velocity. That's the acceleration. And our assumption is that the acceleration is inversely proportional to the square of the distance. And so we have the rate of change of velocity. The acceleration is inversely proportional to the square of the distance. And so we can write that this way, where k is a constant of proportionality. Now if the universe were a kind and gentle place, you'd always be given rates of change as rates of change. Unfortunately, we don't live in that universe, and most differential equations don't mention anything about a rate of change. So how can we create a differential equation? A common situation is to try and model a quantity f as a function of time. And this leads to the following general approach. Let delta t be some small amount of time. Determine delta f, the amount by which f changes in that small amount of time. Recover an equation for delta f over delta t. And then we'll apply a little calculus. As delta t goes to zero, this ratio, delta f over delta t goes to the derivative of f with respect to t. For example, suppose a container holds v, liters of water, and suppose that salt is added at a rate of m kilograms per minute. If pure water is added to the rate of r liters per minute while the salt solution is removed at the same rate, let's write a differential equation giving the concentration of salt in kilograms per liter t minutes after the start of the experiment. We'll make the simplifying assumption that the concentration of salt in the container is uniform. Since the volume remains constant, we only need to know how many kilograms of salt is in the container at any time, t. Suppose there's currently a kilograms of salt. Then the concentration of salt is going to be a over v kilograms per liter. And now let's run things for a very short amount of time. So first we'll stop the clock. Now let's run things for a very short amount of time. So notice several things have happened during that short amount of time. First, some amount of salt solution has been removed from the container. In time delta t, we'd remove r delta t liters, and this would decrease the amount of salt by a over v r delta t kilograms. That's the volume of solution that we've removed times the concentration of the solution. At the same time, some pure water and salt have been added. And since we're adding salt at a rate of m kilograms per minute, we would have added m delta t kilograms of salt. And so we might summarize in time delta t, the amount of salt would change by the m delta t that we've added minus the amount a over v r delta t that we've removed. Now both of these terms have a factor of delta t, so let's factor that out. And we'll try and get an expression for delta a over delta t. And now we'll add a little bit of calculus. Let ting delta t go to zero makes delta a go to d a delta t go to d t. And so delta a over delta t goes to d a d t. And that gives us our differential equation. And once we find a, the concentration will be a over v. Oh, and