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The function graphed here cannot possibly be a solution to this type of differential equation.
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Here's why: This differential equation says that the derivative,
3
00:00:12,612 --> 00:00:20,743
- the rate of change of X, is only a function of X, and that's not the case for this function.
4
00:00:20,743 --> 00:00:29,127
Here's a way to see that: So when X equals 5, here the function is increasing - that's fine,
5
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but here there's a contradiction - the function is decreasing.
6
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If the rate of change depends only on X,
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it can't have two different values at the same value X equals 5.
8
00:00:43,880 --> 00:00:54,736
This is an important point, so let me write this out.
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00:00:54,736 --> 00:00:58,859
Again, the rate of change of X, depends only on X,
10
00:00:58,859 --> 00:01:04,048
- that's what this equation says: the rate of change of X, is only a function of X.
11
00:01:04,048 --> 00:01:09,375
Another way to say that is: the same X value - if I put the same X value in here,
12
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say 5, it has to give the same rate of change,
13
00:01:12,576 --> 00:01:17,976
but, it clearly doesn't here - I put in 5, and I get a positive rate of change,
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I put in 5, I get a negative rate of change, here I get another positive one, and so on.
15
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So, this wriggling function cannot possibly be a solution to this equation
16
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and I can state that with certainty, even though I don't know what this f(X) might be,
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and this is a general result: differential equations of this form can never have oscillatory solutions,
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so, although these are important in engineering and science,
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they're a little bit boring from a dynamical systems point of view,
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- a particular X curve can only increase, or only decrease, or be at a fixed point. It can never turn around.