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Lastly, we can use these pictures to sketch
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the general form of solutions to this differential equation.
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So here's a different equation: Newton's Law of Cooling,
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- that specifies the derivative - how the temperature changes,
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as a function of temperature, and here's a plot of the right hand side,
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the same thing I should before, that shows how the rate of change
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in Celsius per minute - degrees Celsius per minute,
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00:00:27,828 --> 00:00:32,173
depends on the temperature, and here's the phase line
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- we have a stable, attracting fixed by at 20,
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and we can use this to sketch solutions T(t).
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So this phase line is similar to the phase line for iterated functions,
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now I'll sketch some solutions that are similar to the time series plots for iterated functions.
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Let me draw some axes... So here are my axes,
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and now on the horizontal axis is time - t in minutes,
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and then here's temperature - degrees Celsius,
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So I know that, - let's say, do this point in purple,
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if my starting value is 5 - so I'm going to start somewhere here,
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I know I'll increase until I get 20 - I know I'm going to start at a rapid rate of increase
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because this function has a large value - the rate of cooling is large,
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and the rate of cooling gets smaller and smaller, as I approach 20.
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So, I don't know the exact details but I know the curve has to look something like this:
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I'm going to approach the stable fixed point at 20,
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and initially, I'm warming up very quickly - this function is large
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and then the rate at which I'm warming up decreases as I approach 20.
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I could do another solution - suppose we started with a different beverage
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at, say, 45 degrees - then I would cool off very rapidly
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- this is very large and negative so I'm losing temperature quickly,
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and this might look something like that.
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So, I can't from this qualitative picture figure out the exact functional form,
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or the exact timing of this - I'll show you how to do that in the next lecture,
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but we can get an awful lot of information.
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So, this is called a qualitative analysis of a differential equation,
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- we sketch the right hand side, and we see where is the function increasing,
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and where is it decreasing - it's increasing whenever the derivative is positive,
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it's decreasing whenever the derivative is negative.
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From there, we can immediately draw a phase line,
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and from that phase line we can sketch the general shape of solutions.
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So, I think this is a lot of information just from a little bit of geometry and common sense
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and very often this will be enough to analyse a differential equation.