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We're almost at the end of this unit, and differential equations.
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Differential equations are a vast topic,
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- entire books have been written about the subject,
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- big ,large, thick books - so what we've done in this unit
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is just the beginning - it's just scratched the surface.
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Nevertheless, I hope we've covered enough about differential equations
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and how to think about them
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so that you can view them as dynamical systems
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and then we'll be able to study their properties
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in the next several units.
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I'll end this unit by first talking a little bit about
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some notation and terminology that I haven't gotten into yet,
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and then I'll summarize the main ideas
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about differential equations that are presented in this unit.
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After that, I'll say a little bit about Newton, and Laplace,
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and determinism, and this will set the stage for the next unit
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where will encounter chaos, and the butterfly effect.
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So far we've looked at equations of this form:
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dX/dT - some function of X,
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but there are other types of differential equations,
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and some vocabulary and terminology
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associated with these different types,
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that I think we should go over.
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So, if the right hand side of the equation does not depend on time
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- that's the case here - the derivative is a function just of X,
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- how fast the cold beer is warming up,
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is a function only of the temperature of the beer.
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For such a situation, we say that the equation is autonomous.
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In a sense it's just doing its own thing, it doesn't depend on time,
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but if there is time dependence here,
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then one would say
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the right hand side of the equation is non-autonomous.
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That might arise if, say, the room temperature in which the beer
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or hot coffee was, was varying what time
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- it warmed up in the day, and cooled off at night,
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then, in order to figure out how fast something's cooling off,
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we would need to know not only its current temperature,
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but also the time day.
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In any event, we will be studying only autonomous equations in this course.
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There are lots of fun and interesting non-autonomous equations,
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but we'll get plenty of mileage out of autonomous equations.
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Another key term about differential equations is
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the idea of the order of a differential equation.
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The order of a differential equation is just
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the highest derivative that appears in the equation.
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Here, this is just the first derivative of X,
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and so we would say that this is first order.
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If there were first and second derivatives around
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then we would say it's second order.
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The second derivative - it's not important for this course,
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but as you might guess - it's the derivative of the derivative,
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- the rate change of the rate of change.
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In this course we'll study only first-order equations.
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At first this might seem like a limitation,
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because there's some very important second order equations:
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Newton's law of motion - perhaps the most important for mechanics and chaos,
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is a second order equation,
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however, it's possible to convert a second order equation
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into a system of two first order equations.
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That's not immediately obvious,
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and we're not going to actually do that in this course,
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- we could talk about in the forums if people are interested,
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but the main point is that we'll only study first order equations
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but that's not limiting us in any way.
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OK, a little bit more terminology:
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a differential equation is called ordinary
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if it contains only ordinary i.e. full or total derivatives.
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So, ordinary differential equations - that's what we'll talk about here
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is often written ODE, and in contrast,
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a non ordinary differential equation is not a peculiar differential equation
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but it's called a partial differential equation,
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and it involves partial derivatives - things like: this:
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- this is a version of the wave equation.
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We won't cover partial differential equations in this course,
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we'll study only ordinary differential equations.
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Next, I want to say a little bit about
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the important ideas of existence and uniqueness.
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So, let me state this result,
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and then I'll talk about its implications.
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So, consider a differential equation of this kind,
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- that's what we've been studying in this unit,
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and the initial condition is given,
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so, we know the rule, we know the starting point.
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If this function of X is a nice function,
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- by that I mean: its continuous, it doesn't have any jumps,
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and it's smooth, it doesn't have any sudden kinks or bends in it,
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then, the solution to this equation exists and is unique.
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So, let's think about what this means, and why it matters.
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First of all, - these conditions:
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that this function is smooth and continuous,
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they're met in most physical applications and in most modelling applications.
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It's very rare that these conditions aren't met.
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So, what that means is, is that if I asked you a question like this:
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here's a differential equation, I tell you the starting point,
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- that there's one, and only one, solution,
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So, that if I find a solution, and you find a solution.
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they're the same solution - there's one, and only one out there,
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and I hope that this result isn't surprising.
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I've talked throughout this unit
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about how this differential equation is a rule,
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- it's a rule that tells how the rate of change of X,
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is related to X, and so what this result says is that
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the rule specified by this equation,
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and the starting point, is unambiguous - there is only one solution to it,
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there's only one way to follow the rule - it's an unambiguous rule.
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So, in brief, this result,
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which is proved in most differential equation's text books,
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says that: differential equations of this sort are well-behaved.
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If we have a reasonable right hand side, we're guaranteed to have one,
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and only one, solution - this rule is unambiguous.