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Let's summarize and review the key results and ideas
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from unit 2, on differential equations.
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A differential equation is a type of dynamical system,
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and we've looked at differential equation of this form,
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again, the derivative of some variable,
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is just a function of that variable.
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Remember, a dynamical system is a system that changes in time,
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according to a well specified rule,
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and a differential equation is such a dynamical system.
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The differential equation specifies the derivative of X,
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as a function of X - that's the rule,
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and the derivative, which if you haven't had calculus,
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might be a new term,
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- is just the instantaneous rate of change of X.
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It's how fast X is changing, at a particular moment, or instant in time.
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There are three main classes, or types of solution methods for differential equations:
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first, is qualitative, or geometric techniques.
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- there, we might sketch the right hand side of the differential equation,
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and from that we can figure out fixed points, and their stability;
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we can draw a phase line, as I've done here,
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and we can sketch the general shape of solution,
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the directions in which the solutions are going.
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We can't however get an exact form for X of t,
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we don't necessarily know how fast the solutions go,
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but this method is good for giving an overall feel
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for the long-term behaviour of solutions,
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- where do orbits go? how many fixed points are there?
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and what are there stabilities?
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Another approach to solving a differential equation, is: computational.
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So, I presented Euler's method,
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and one could use Euler's method, or a fancier version - one of these Runge-Kutta methods,
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to figure out to X of t, and one does so, step-by-step.
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We start with an X value - we use that X value to figure out the derivative,
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the derivative tells us the X value a little while later,
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and we can go back here, plug X in, the new X,
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and figure out the new derivative,
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- a derivative we can use to figure out the X value a little later,
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and so on, and so we're constantly swapping back and forth
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between these two sides of the equation.
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I think Euler's method gets at the heart of what a differential equation is:
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a rule specifying how a quantity changes.
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Computational techniques like this are reliable
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and they work for all well posed equations.
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It does require the use of some computer software, or spreadsheet.
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These results are often called numerical results
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because the end result is a table of numbers, not a formula,
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but it's very easy to plot that table of numbers,
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and get a feel for what the solutions are doing.
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The last type of solution method is
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one that I actually have only hinted at so far,
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and that's a type of method called analytic.
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So, here, the task is to find a formula
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for the solution X of t, using calculus.
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So calculus - there's a well-developed machinery
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for finding derivatives, doing derivatives backwards, and so on.
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So, depending on your point of view this can be
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a lot of fun, or not so much fun at all.
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I myself had mixed feelings about it.
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My first differential equations class in college
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felt like all the worst parts of calculus 2 coming back to haunt me.
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Later on, in grad school, learning some other techniques
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for solving differential equations: power series, and Laplace transforms,
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- I enjoyed a lot, but anyway, it's a very different approach from the first two.
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It uses all the machinery of calculus.
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The bad news is, is that most nonlinear equations,
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and that's what we'll be studying in this course by an large,
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cannot be solved analytically.
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In some cases, one can even prove that there isn't an analytical solution,
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so, geometric, or numerical and computational solutions are necessary.
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Moreover, even for equations that can be solved analytically,
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doing so does not always lead to intuition, or understanding.
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You might do a bunch of calculus tricks, and get a weird-looking formula,
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and that's certainly a valuable thing,
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but it might not give you a feel, or intuition,
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for what the equations are doing.
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In my view, many certainly not all, but many differential equations text books
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place too much of an emphasis on analytic techniques.
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This course will focus on qualitative and computational solutions,
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- I think they're much better suited for dynamical systems and chaos,
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particularly for an introductory course like this.
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OK, let me review some of the key terminology from this unit,
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and it's actually almost the same as the terminology from the previous one.
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So, differential equations, just like iterated functions,
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have fixed points, and a fixed point is a point that doesn't change.
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In differential equations one often calls a fixed point an equilibrium point,
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but it means the same thing.
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A fixed point X is fixed, if it's derivative is 0.
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If your derivative is 0, you aren't changing,
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and if you aren't changing, you're a fixed point.
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Fixed points have stability as well - they can be stable, or unstable.
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A fixed point is stable if near by points
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move closer to the fixed point when iterated,
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or, perhaps I should say, if you have an initial condition
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near the fixed point, and you solve the differential equation
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it moves closer to that - to the fixed point.
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A stable fixed point is also called an attracting fixed point,
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or an attractor, or in differential equations, it is sometimes called a sink,
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because you can imagine a lot of solutions all head into this one point,
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so it looks like water going down a drain.
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A fixed point is unstable if near by points move further away from it,
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and an unstable fixed point is called a repelling fixed point, or a repellor,
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- it's also called a source: you can imagine a lot of solution lines
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or solution curves emanating from this repelling fixed point.
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So, source and sink - I don't think I'll use those terms much,
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- but they're pretty standard, so you might encounter them elsewhere.
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Just as we did for iterated functions,
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we can draw a phase line for differential equations,
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and it lets us see, all at once, the long-term behaviour for all initial conditions.
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In a phase line we lose time information, so for example here,
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I know that solutions move towards 9, but I don't know how fast.
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The phase line here is for a differential equation that has an attractor at 9.
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- things are getting closer to 9, and a repellor at 1,
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- things are getting pushed away from 1.
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So, differential equations are a type of dynamical system,
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and in dynamical systems, one of the goals of study,
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is to classify, and characterize, the sorts of behaviours that we see.
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So, for differential equations, what have we seen?
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- well we've seen: Fixed points (stable and unstable).
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- Orbits can approach a fixed point.
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- Orbits can tend towards infinity,
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or tend towards negative infinity - they can move off the ends of the phase line,
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and that's pretty much it,
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and additionally, an orbit cannot increase and then decrease,
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- it's rate of change is only a function of X - it's value,
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so that means that cycles or oscillations are not possible.
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In the next several units we'll see that
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differential equations are capable of doing some more interesting things
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but we'll have to go to higher dimensions
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in order to see this more interesting and exciting behaviour.