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