In the last subunit, I talked a bit about derivatives, maybe I talked a bit too much. But, the main point is that the derivative of a quantity is its instantaneous rate of change. It just tells you how fast a quantity is growing or shrinking at a particular moment in time. We are now ready to talk about differential equations. They're another type of dynamical system. In the next several lectures, I'll give you a number of different ways of thinking about differential equations, and how to solve them , and what those solutions might mean. To be honest, I'm not quite sure what order to do the next several lectures in, but I am confident that by the time we get to the end of them, you'll have a pretty good sense of what differential equations are, what their solutions mean, and how to think about them as dynamical systems. So, let's get started. I'll introduce differential equations by comparing them to iterated functions, the first type of dynamical system we studied. So, here's an iterated function. This notation makes it clear that we need an initial value and then we can always get the next value by applying the function to the current value. So the next value in the orbit or itinerary is a function of the current value. And this function will likely have different values depending on what the current value is. A differential equation is an equation involving the derivative of a function. So here the function is x of t. And this says that the derivative of x is a function of x. Alright, let's unpack this. The derivative of x that means the instantaneous rate of change of x so this tells us how x is changing how x is changing depends on x is a function of x If you tell me x, I can tell you how fast x is changing. Here, if you tell me xn, I can figure out the next value of xn. When one solves an iterated function, one needs a seed and from that, one can figure out the orbit. the first iterate, the second iterate, the third iterate, and so on. For the differential equation, you also need an initial condition, a starting point which you could write x 0 or you might write as x at time zero So you need an initial value for this variable whatever it is and then this function tells you how the function changes. It doesn't give you this information quite as directly as here where it just tells you the next value. Here, the differential equation is telling you how fast the function is changing at any given x. The solution to a differential equation is not exactly a discrete orbit, but it is a continuous function. So the solution would be a function x of t t there's x of t and who knows... again, I'm just making up an example. So, rather than a time series plot which jumps around is only discrete, here this is a smooth curve. It starts at whatever the initial condition is and then it grows or shrinks according to whatever the instructions are that it recieves from the differential equation. Here's a way that I like to think of differential equations. Differential equations are a dynamical system. A system that changes in time according to a well specified rule. And here's, roughly speaking, an example of a rule that can give you a way to think about this. So, here's a navigation device built into an i-phone. And, there are similar things built into many cars. And you tell it where you are starting. And then it gives you a set of directions and those set of directions get you to the destination that you entered. So here, I've entered how to get to Bangor, Maine from where I am on campus. And if I press start, it will give me, in an annoying voice, some directions like this: GPS Voice: "Starting route to Bangor, ME. Head SW on Sea Urchin Rd." So, it gives me my first set of directions. And I do that, and I'm in a different location because I've done what I'm told, I follow the directions. And then it gives me a different set of directions based on where I am currently. And so it's continually updating what it's telling me. What it tells me varies as I move around, as my x value changes in the equation, and it's always telling me what to do. So, a differential equation is kind of similar to this except, the directions it's giving me aren't direct position. Things go here, go here, go here. But instead, they are always telling me how fast to go. what my derivative should be what my speed should be and in what direction I should point. So, the differential equaiton is telling me continually what the derivative should be what the derivative should be At every point, it is continually talking to me in its annoying little voice. And in that way, it specifies a curve through space. Kind of like, very similar to how an iterated function that rule continually applied specifies an orbit. So, differential equations are a different type of dynamical system, but they are very similar to iterated functions. It's a rule that specifies a path through space, or through whatever, as long as you give it an initial condition. So let's go back to looking at the equation. A differential equation is an equation of this form. The derivative of x is a function of x. So, in words, this equation says the following: the rate of change of x, how fast x is increasing or decreasing that's the derivative is given by, that's the equal sign, a function of x. So as x changes, this rule, this function is always telling me not the next value of x, but how x continues to change. So, this determines the change in x and from the change in x, we can figure out x, the thing itself we are interested in. For example, if you know the speed and direction, we can figure out your position as a function of time. In the next lecture, I'll do a specific example of this, and you'll see how it works.