WEBVTT
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Today, once again,
a day of solving no
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differential equations
whatsoever.
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The topic is a special kind of
differential equation,
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which occurs a lot.
It's one in which the
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right-hand side doesn't have any
independent variable in it.
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Now, since I'm going to use as
the independent variable,
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t for time, maybe it would be
better to write the left-hand
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side to let you know,
since you won't be able to
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figure out any other way what it
is, dy dt.
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We will write it this time.
dy dt is equal to,
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and the point is that there is
no t on the right hand side.
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So, there's no t.
There's a name for such an
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equation.
Now, some people call it time
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independent.
The only problem with that is
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that sometimes the independent
variable is a time.
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It's something else.
We need a generic word for
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there being no independent
variable on the right-hand side.
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So, the word that's used for
that is autonomous.
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So, that means no independent
variable on the right-hand side.
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It's a function of y alone,
the dependent variable.
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Now, your first reaction should
be, oh, well,
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big deal.
Big deal.
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If there's no t on the right
hand side, then we can solve
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this by separating variables.
So, why has he been talking
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about it in the first place?
So, I admit that.
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We can separate variables,
and what I'm going to talk
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about today is how to get useful
information out of the equation
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about how its solutions look
without solving the equation.
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The reason for wanting to do
that is, A, it's fast.
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It gives you a lot of insight,
and the actual solution,
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I'll illustrate one,
in the first place,
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take you quite a while.
You may not be able to actually
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do the integrations,
the required and separation of
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variables to get an explicit
solution, or it might simply not
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be worth the effort of doing if
you only want certain kinds of
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separations about the solution.
So, the thing is,
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the problem is,
therefore, to get qualitative
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information about the solutions
without actually solving --
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Without actually having to
solve the equation.
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Now, to do that,
let's take a quick look at how
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the direction fields of such an
equation, after all,
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it's the direction field is our
principal tool for getting
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qualitative information about
solutions without actually
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solving.
So, how does the direction
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field look?
Well, think about it for just a
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second, and you will see that
every horizontal line is an
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isocline.
So, the horizontal lines,
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what are their equations?
This is the t axis.
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And, here's the y axis.
The horizontal lines have the
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formula y equals a constant.
Let's make it y equals a y zero
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for different values of the
constant y zero.
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Those are the horizontal lines.
And, the point is they are
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isoclines.
Why?
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Well, because along any one of
these horizontal lines,
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I'll draw one in,
what are the slopes of the line
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elements?
The slopes are dy / dt is equal
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to f of y zero,
but that's a constant because
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there's no t to change as you
move in the horizontal
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direction.
The slope is a constant.
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So, if I draw in that isocline,
I guess I've forgotten,
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as our convention,
isoclines are in dashed lines,
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but if you have color,
you are allowed to put them in
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living yellow.
Well, I guess I could make them
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solid, in that case.
I don't have to make a dash.
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Then, all the line elements,
you put them in at will because
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they will all have,
they are all the same,
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and they have slope,
that, f of y0.
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And, similarly down here,
they'll have some other slope.
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This one will have some other
slope.
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Whatever, this is the y zero,
the value of it,
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and whatever that happens to
be.
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I'll put it one more.
That's the x-axis.
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I can use the x-axis.
That's an isocline,
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too.
Now, what do you deduce about
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how the solutions must look?
Well, let's draw one solution.
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Suppose one solution looks like
this.
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Well, that's an integral curve,
in other words.
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Its graph is a solution.
Now, as I slide along,
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these slope elements stay
exactly the same,
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I can slide this curb along
horizontally,
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and it will still be an
integral curve everywhere.
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So, in other words,
they integral curves are
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invariant under translation for
an equation of this type.
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They all look exactly the same,
and you get them all by taking
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one, and just pushing it along.
Well, that's so simple it's
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almost uninteresting,
except in that these equations
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occur a lot in practice.
They are often hard to
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integrate directly.
And, therefore,
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it's important to be able to
get information about them.
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Now, how does one do that?
There's one critical idea,
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and that is the notion of a
critical point.
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These equations have what are
called critical points.
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And, what it is is very simple.
There are three ways of looking
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at it: critical point,
y zero;
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what does it mean for y0 to be
a critical point?
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It means, another way of saying
it is that it should be a zero
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of the right-hand side.
So, if I ask you to find the
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critical points for the
equation, what you will do is
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solve the equation f of y equals
zero.
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Now, what's interesting about
them?
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Well, for a critical point,
what would be the slope of the
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line element along,
if this is at a critical level,
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if that's a critical point?
Look at that isocline.
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What's the slope of the line
elements along it?
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It is zero.
And therefore,
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for these guys,
these are, in other words,
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our solution curves.
But let's prove it formally.
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So, there are three ways of
saying it.
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y zero is a critical point.
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It's a zero of the right-hand
side, or, y equals y0 is a
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solution to the equation.
Now, that's perfectly easy to
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verify.
If y zero makes this
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right-hand side zero,
it's certainly also y equals y0
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makes the left-hand
side zero because you're
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differentiating a constant.
So, the reasoning,
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if you want reasoning,
is proof.
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Maybe we can make one line out
of a proof.
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To say that it's a solution,
what does it mean to say that
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it's a solution?
It means to say that when you
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plug it in, plug in this
constant function,
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y0, the dy0 dt is equal to f of
y0.
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Is that true?
Yeah.
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Both sides are zero.
It's true.
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Now, y0 is not a number.
Well, it is.
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It's a number on this side,
but on this side,
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what I mean is a constant
function whose constant value is
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y zero, this function,
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and its derivatives are zero
because it has slope zero
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everywhere.
So, this guy is a constant
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function, has slope zero.
This is a number which makes
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the right-hand side zero.
Well, that's nice.
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So, in other words,
what we found are,
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by finding these critical
points, solving that equation,
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we found all the horizontal
solutions.
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But, what's so good about
those?
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Surely, they must be the most
interesting solutions there are.
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Well, think of how the picture
goes.
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Let's draw in one of those
horizontal solutions.
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So, here's a horizontal
solution.
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That's a solution.
So, this is my y0.
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That's the height at which it
is.
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And, I'm assuming that f of y0
equals zero.
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So, that's a solution.
Now, the significance of that
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is, because it's a solution,
in other words,
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it's an integral curve,
remember what's true about
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integral curves.
Other curves are not allowed to
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cross them.
And therefore,
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these things are the absolute
barriers.
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So, for example,
suppose I have two of them is
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y0, and let's say here's another
one, another constant solution.
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I want to know what the curves
in between those can do.
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Well, I do know that whatever
those red curves do,
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the other integral curves,
they cannot cross this,
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and they cannot cross that.
And, you must be able to
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translate them along each other
without ever having two of them
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intersect.
Now, that really limits their
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behavior, but I'm going to nail
it down even more.
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So, other curves can't cross
these.
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Other integral curves can't
cross these yellow curves,
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these yellow lines,
these horizontal lines.
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But, I'm going to show you
more, and namely,
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so what I'm after is deciding,
without solving the equation,
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what those curves must look
like in between.
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Now, the way to do that is you
draw, so if we want to make
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steps, everybody likes steps,
okay, so step one is going to
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be, find these.
Find the critical points.
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And, you're going to do that by
solving this equation,
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finding out where it's zero.
Once you have done that,
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you are going to draw the graph
of f of y.
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And, the interest is going to
be, where is it positive?
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Where is it negative?
You've already found where it's
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zero.
Everywhere else,
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therefore, it must be either
positive or negative.
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Now, once you have found that
out, why am I interested in
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that?
Well, because dy / dt is equal
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to f of y, right?
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That's what the differential
equation says.
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Therefore, if this,
for example,
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is positive,
that means this must be
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positive.
It means that y must be
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increasing.
It means the solution must be
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increasing.
Where it's negative,
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the solution will be
decreasing.
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And, that tells me how it's
behaving in between these yellow
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lines, or on top of them,
or on the bottom.
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Now, at this point,
I'm going to stop,
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or not stop,
I mean, I'm going to stop
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talking generally.
And everything in the rest of
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the period will be done by
examples which will get
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increasingly complicated,
not terribly complicated by the
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end.
But, let's do one that's super
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simple to begin with.
Sorry, I shouldn't say that
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because some of you may be
baffled even by here because
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after all I'm going to be doing
the analysis not in the usual
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way, but by using new ideas.
That's the way you make
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progress.
All right, so,
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let's do our bank account.
So, y is money in the bank
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account.
r is the interest rate.
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Let's assume it's a continuous
interest rate.
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All banks nowadays pay interest
continuously,
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the continuous interest rate.
So, if that's all there is,
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and money is growing,
you know the differential
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equation says that the rate at
which it grows is equal to r,
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the interest rate times a
principle, the amount that's in
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the bank at that time.
So, that's the differential
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equation that governs that.
Now, that's,
00:14:41.000 --> 00:14:47.000
of course, the solution is
simply an exponential curve.
00:14:46.000 --> 00:14:52.000
There's nothing more to say
about it.
00:14:49.000 --> 00:14:55.000
Now, let's make it more
interesting.
00:14:52.000 --> 00:14:58.000
Let's suppose there is a shifty
teller at the bank,
00:14:56.000 --> 00:15:02.000
and your money is being
embezzled from your account at a
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constant rate.
So, let's let w equal,
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or maybe e, but e has so many
other uses in mathematics,
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w is relatively unused,
w is the rate of embezzlement,
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thought of as continuous.
So, every day a little bit of
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money is sneaked out of your
account because you are not
00:15:26.000 --> 00:15:32.000
paying any attention to it.
You're off skiing somewhere,
00:15:31.000 --> 00:15:37.000
and not noticing what's
happening to your bank account.
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So, since it's the rate,
the time rate of embezzlement,
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I simply subtract it from this.
It's not w times y because the
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embezzler isn't stealing a
certain fraction of your
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account.
It's simply stealing a certain
00:15:54.000 --> 00:16:00.000
number of dollars every day,
the same number of dollars
00:15:59.000 --> 00:16:05.000
being withdrawn for the count.
Okay, now, of course,
00:16:03.000 --> 00:16:09.000
you could solve this.
This separates variables
00:16:07.000 --> 00:16:13.000
immediately.
You get the answer,
00:16:10.000 --> 00:16:16.000
and there's no problem with
that.
00:16:12.000 --> 00:16:18.000
Let's analyze the behavior of
the solutions without solving
00:16:17.000 --> 00:16:23.000
the equation by using these two
points.
00:16:20.000 --> 00:16:26.000
So, I want to analyze this
equation using the method of
00:16:24.000 --> 00:16:30.000
critical points.
So, the first thing I should do
00:16:28.000 --> 00:16:34.000
is, so, here's our equation,
is find the critical points.
00:16:34.000 --> 00:16:40.000
Notice it's an autonomous
equation all right,
00:16:38.000 --> 00:16:44.000
because there's no t on the
right-hand side.
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Okay, so, the critical points,
well, that's where ry minus w
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equals zero.
In other words,
00:16:51.000 --> 00:16:57.000
there's only one critical
point, and that occurs when y is
00:16:56.000 --> 00:17:02.000
equal to w over r.
So, that's the only critical
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point.
Now, I want to know what's
00:17:05.000 --> 00:17:11.000
happening to the solution.
So, in other words,
00:17:09.000 --> 00:17:15.000
if I plot, I can write away,
of course, negative values
00:17:14.000 --> 00:17:20.000
aren't of particularly
interesting here,
00:17:17.000 --> 00:17:23.000
there is definitely a
horizontal solution,
00:17:21.000 --> 00:17:27.000
and it has the value,
it's at the height,
00:17:25.000 --> 00:17:31.000
w over r.
That's a solution.
00:17:30.000 --> 00:17:36.000
The question is,
what do the other solutions
00:17:33.000 --> 00:17:39.000
look like?
Now, watch how I make the
00:17:35.000 --> 00:17:41.000
analysis because I'm going to
use two now.
00:17:39.000 --> 00:17:45.000
So, this is step one,
then step two.
00:17:41.000 --> 00:17:47.000
What do I do? Well,
I'm going to graph f of y.
00:17:45.000 --> 00:17:51.000
Well, f of y is ry minus w.
00:17:47.000 --> 00:17:53.000
What does that look like?
00:17:50.000 --> 00:17:56.000
Okay, so, here is the y-axis.
Notice the y-axis is going
00:17:55.000 --> 00:18:01.000
horizontally because what I'm
interested in is the graph of
00:17:59.000 --> 00:18:05.000
this function.
What do I call the other axis?
00:18:04.000 --> 00:18:10.000
I'm going to use the same
terminology that is used on the
00:18:08.000 --> 00:18:14.000
little visual that describes
this.
00:18:11.000 --> 00:18:17.000
And, that's dy.
You could call this other axis
00:18:14.000 --> 00:18:20.000
the f of y axis.
That's not a good name for it.
00:18:18.000 --> 00:18:24.000
You could call it the dy / dt
axis because it's,
00:18:22.000 --> 00:18:28.000
so to speak,
the other variable.
00:18:24.000 --> 00:18:30.000
That's not great either.
But, worst of all would be
00:18:28.000 --> 00:18:34.000
introducing yet another letter
for which we would have no use
00:18:32.000 --> 00:18:38.000
whatsoever.
So, let's think of it.
00:18:37.000 --> 00:18:43.000
We are plotting,
now, the graph of f of y.
00:18:40.000 --> 00:18:46.000
f of y is this function,
00:18:43.000 --> 00:18:49.000
ry minus w.
Well, that's a line.
00:18:48.000 --> 00:18:54.000
Its intercept is down here at
w, and so the graph looks
00:18:53.000 --> 00:18:59.000
something like this.
It's a line.
00:18:55.000 --> 00:19:01.000
This is the line,
ry minus w.
00:18:59.000 --> 00:19:05.000
It has slope r.
Well, what am I going to get
00:19:05.000 --> 00:19:11.000
out of that line?
Just exactly this.
00:19:08.000 --> 00:19:14.000
What am I interested in about
that line?
00:19:13.000 --> 00:19:19.000
Nothing other than where is it
above the axis,
00:19:18.000 --> 00:19:24.000
and where is it below?
This function is positive over
00:19:23.000 --> 00:19:29.000
here, and therefore,
I'm going to indicate that
00:19:28.000 --> 00:19:34.000
symbolically,
this, by putting an arrow here.
00:19:35.000 --> 00:19:41.000
The meeting of this arrow is
that y of t is
00:19:39.000 --> 00:19:45.000
increasing.
See where it's the right-hand
00:19:43.000 --> 00:19:49.000
side of that last board?
y of t is increasing when f of
00:19:48.000 --> 00:19:54.000
y is positive.
f of y is positive here,
00:19:52.000 --> 00:19:58.000
and therefore,
to the right of this point,
00:19:55.000 --> 00:20:01.000
it's increasing.
Here to the left of it,
00:19:59.000 --> 00:20:05.000
f of y is negative,
and therefore over here it's
00:20:03.000 --> 00:20:09.000
going to be decreasing.
What point is this,
00:20:08.000 --> 00:20:14.000
in fact?
Well, that's where it crosses
00:20:11.000 --> 00:20:17.000
the axis.
That's exactly the critical
00:20:15.000 --> 00:20:21.000
point, w over r.
Therefore, what this says is
00:20:19.000 --> 00:20:25.000
that a solution,
once it's bigger than y over r,
00:20:24.000 --> 00:20:30.000
it increases,
and it increases faster and
00:20:28.000 --> 00:20:34.000
faster because this function is
higher and higher.
00:20:34.000 --> 00:20:40.000
And, that represents the rate
of change.
00:20:37.000 --> 00:20:43.000
So, in other words,
once the solution,
00:20:40.000 --> 00:20:46.000
let's say a solution starts
over here at time zero.
00:20:44.000 --> 00:20:50.000
So, this is the t axis.
And, here is the y axis.
00:20:48.000 --> 00:20:54.000
So, now, I'm plotting
solutions.
00:20:51.000 --> 00:20:57.000
If it starts at t equals zero,
above this line,
00:20:55.000 --> 00:21:01.000
that is, starts with the value
w over r,
00:20:59.000 --> 00:21:05.000
which is bigger than zero,
a value bigger than w over r,
00:21:04.000 --> 00:21:10.000
then it increases,
and increases faster and
00:21:07.000 --> 00:21:13.000
faster.
If it starts below that,
00:21:12.000 --> 00:21:18.000
it decreases and decreases
faster and faster.
00:21:16.000 --> 00:21:22.000
Now, in fact,
I only have to draw two of
00:21:20.000 --> 00:21:26.000
those because what do all the
others look like?
00:21:24.000 --> 00:21:30.000
They are translations.
All the other curves look
00:21:29.000 --> 00:21:35.000
exactly like those.
They are just translations of
00:21:35.000 --> 00:21:41.000
them.
This guy, if I start closer,
00:21:39.000 --> 00:21:45.000
it's still going to decrease.
Well, that's supposed to be a
00:21:47.000 --> 00:21:53.000
translation.
Maybe it is.
00:21:50.000 --> 00:21:56.000
So, these guys look like that.
Let's do just a tiny bit more
00:21:57.000 --> 00:22:03.000
interpretation of that.
Well, I think I better leave it
00:22:06.000 --> 00:22:12.000
there because we've got harder
things to do,
00:22:13.000 --> 00:22:19.000
and I want to make sure we've
got time for it.
00:22:19.000 --> 00:22:25.000
Sorry.
Okay, next example,
00:22:23.000 --> 00:22:29.000
a logistic equation.
Some of you have already solved
00:22:31.000 --> 00:22:37.000
this in recitation,
and some of you haven't.
00:22:39.000 --> 00:22:45.000
This is a population equation.
This is the one that section
00:22:44.000 --> 00:22:50.000
7.1 and section 1.7 is most
heavily concerned with,
00:22:48.000 --> 00:22:54.000
this particular equation.
The derivation of it is a
00:22:52.000 --> 00:22:58.000
little vague.
It's an equation which
00:22:55.000 --> 00:23:01.000
describes how population
increases.
00:22:58.000 --> 00:23:04.000
And one minute,
the population behavior of some
00:23:02.000 --> 00:23:08.000
population, --
-- let's call it,
00:23:06.000 --> 00:23:12.000
y is the only thing I know to
call anything today,
00:23:10.000 --> 00:23:16.000
but of course your book uses
capital P for population,
00:23:15.000 --> 00:23:21.000
to get you used to different
variables.
00:23:18.000 --> 00:23:24.000
Now, the basic population
equation runs dy / dt.
00:23:22.000 --> 00:23:28.000
There's a certain growth rate.
Let's call it k y.
00:23:26.000 --> 00:23:32.000
So, k is what's called the
growth rate.
00:23:29.000 --> 00:23:35.000
It's actually,
sometimes it's talked about in
00:23:33.000 --> 00:23:39.000
terms of birthrate.
But, it's the net birth rate.
00:23:38.000 --> 00:23:44.000
It's the rate at which people,
or bacteria,
00:23:42.000 --> 00:23:48.000
or whatever are being born
minus the rate at which they are
00:23:47.000 --> 00:23:53.000
dying.
So, it's a net birthrate.
00:23:49.000 --> 00:23:55.000
But, let's just call it the
growth rate.
00:23:53.000 --> 00:23:59.000
Now, if this is the equation,
we can think of this,
00:23:57.000 --> 00:24:03.000
if k is constant,
that's what's called simple
00:24:01.000 --> 00:24:07.000
population growth.
And you are all familiar with
00:24:05.000 --> 00:24:11.000
that.
Logistical growth allows for
00:24:08.000 --> 00:24:14.000
slightly more complex
situations.
00:24:13.000 --> 00:24:19.000
Logistic growth says that
calling k a constant is
00:24:18.000 --> 00:24:24.000
unrealistic because the Earth is
not filled entirely with people.
00:24:26.000 --> 00:24:32.000
What stops it from having
unlimited growth?
00:24:32.000 --> 00:24:38.000
Well, the fact that the
resources, the food,
00:24:36.000 --> 00:24:42.000
the organism has to live on
gets depleted.
00:24:41.000 --> 00:24:47.000
And, in other words,
the growth rate declines as y
00:24:46.000 --> 00:24:52.000
increases.
As the population increases,
00:24:50.000 --> 00:24:56.000
one expects the growth rate to
decline because resources are
00:24:56.000 --> 00:25:02.000
being used up,
and they are not indefinitely
00:25:01.000 --> 00:25:07.000
available.
Well, in other words,
00:25:05.000 --> 00:25:11.000
we should replace k by a
function with this behavior.
00:25:09.000 --> 00:25:15.000
What's the simplest function
that declines as y increases?
00:25:14.000 --> 00:25:20.000
The simplest choice,
and if you are ignorant about
00:25:17.000 --> 00:25:23.000
what else to do,
stick with the simplest,
00:25:20.000 --> 00:25:26.000
at least you won't work any
harder than you have to,
00:25:24.000 --> 00:25:30.000
would be to take k equal to the
simplest declining function of y
00:25:29.000 --> 00:25:35.000
there is, which is simply a
linear function,
00:25:32.000 --> 00:25:38.000
A minus BY.
So, if I use that as the choice
00:25:37.000 --> 00:25:43.000
of the declining growth rate,
the new equation is dy / dt
00:25:41.000 --> 00:25:47.000
equals, here's my new k.
The y stays the same,
00:25:45.000 --> 00:25:51.000
so the equation becomes a minus
by, the quantity times y,
00:25:50.000 --> 00:25:56.000
or in other words,
00:25:53.000 --> 00:25:59.000
ay minus b y squared.
00:25:55.000 --> 00:26:01.000
This equation is what's called
the logistic equation.
00:26:01.000 --> 00:26:07.000
It has many applications,
not just to population growth.
00:26:05.000 --> 00:26:11.000
It's applied to the spread of
disease, the spread of a rumor,
00:26:11.000 --> 00:26:17.000
the spread of many things.
Yeah, a couple pieces of chalk
00:26:16.000 --> 00:26:22.000
here.
00:26:28.000 --> 00:26:34.000
Okay, now, those of you who
have solved it know that the
00:26:34.000 --> 00:26:40.000
explicit solution involves,
well, you separate variables,
00:26:40.000 --> 00:26:46.000
but you will have to use
partial fractions,
00:26:44.000 --> 00:26:50.000
ugh, I hope you love partial
fractions.
00:26:48.000 --> 00:26:54.000
You're going to need them later
in the term.
00:26:53.000 --> 00:26:59.000
But, I could avoid them now by
not solving the equation
00:26:59.000 --> 00:27:05.000
explicitly.
But anyway, you get a solution,
00:27:03.000 --> 00:27:09.000
which I was going to write on
the board for you,
00:27:06.000 --> 00:27:12.000
but you could look it up in
your book.
00:27:09.000 --> 00:27:15.000
It's unpleasant enough looking
to make you feel that there must
00:27:13.000 --> 00:27:19.000
be an easier way at least to get
the basic information out.
00:27:16.000 --> 00:27:22.000
Okay, let's see if we can get
the basic information out.
00:27:20.000 --> 00:27:26.000
What are the critical points?
Well, this is pretty easy.
00:27:23.000 --> 00:27:29.000
A, I want to set the right-hand
side equal to zero.
00:27:26.000 --> 00:27:32.000
So, I'm going to solve the
equation.
00:27:30.000 --> 00:27:36.000
I can factor out a y.
It's going to be y times a
00:27:35.000 --> 00:27:41.000
minus by equals zero.
00:27:39.000 --> 00:27:45.000
And therefore,
the critical points are where y
00:27:43.000 --> 00:27:49.000
equals zero. That's one.
00:27:47.000 --> 00:27:53.000
And, the other factor is when
this factor is zero,
00:27:52.000 --> 00:27:58.000
and that happens when y is
equal to a over b.
00:27:59.000 --> 00:28:05.000
So, there are my two critical
points.
00:28:03.000 --> 00:28:09.000
Okay, what does,
let's start drawing pictures of
00:28:08.000 --> 00:28:14.000
solutions.
Let's put it in those right
00:28:12.000 --> 00:28:18.000
away.
Okay, the critical point,
00:28:15.000 --> 00:28:21.000
zero, gives me a solution that
looks like this.
00:28:18.000 --> 00:28:24.000
And, the critical point,
a over b,
00:28:21.000 --> 00:28:27.000
those are positive numbers.
So, that's somewhere up here.
00:28:25.000 --> 00:28:31.000
So, those are two solutions,
constant solutions.
00:28:29.000 --> 00:28:35.000
In other words,
if the population by dumb luck
00:28:32.000 --> 00:28:38.000
started at zero,
it would stay at zero for all
00:28:35.000 --> 00:28:41.000
time.
That's not terribly surprising.
00:28:39.000 --> 00:28:45.000
But, it's a little less obvious
that if it starts at that magic
00:28:43.000 --> 00:28:49.000
number, a over b,
it will also stay at that magic
00:28:47.000 --> 00:28:53.000
number for all time without
moving up or down or away from
00:28:50.000 --> 00:28:56.000
it.
Now, the question is,
00:28:52.000 --> 00:28:58.000
therefore, what happens in
between?
00:28:54.000 --> 00:29:00.000
So, for the in between,
I'm going to make that same
00:28:58.000 --> 00:29:04.000
analysis that I made before.
And, it's really not very hard.
00:29:03.000 --> 00:29:09.000
Look, so here's my dy/dt-axis.
I'll call that y prime,
00:29:09.000 --> 00:29:15.000
okay?
And, here's the y-axis.
00:29:12.000 --> 00:29:18.000
So, I'm now doing step two.
This was step one.
00:29:16.000 --> 00:29:22.000
Okay, the function that I want
to graph is this one,
00:29:21.000 --> 00:29:27.000
ay minus b y squared,
or in factor form,
00:29:26.000 --> 00:29:32.000
y times a minus by.
00:29:29.000 --> 00:29:35.000
Now, this function,
we know, has a zero.
00:29:32.000 --> 00:29:38.000
It has a zero here,
and it has a zero at the point
00:29:37.000 --> 00:29:43.000
a over b.
At these two critical points,
00:29:43.000 --> 00:29:49.000
it has a zero.
What is it doing in between?
00:29:46.000 --> 00:29:52.000
Well, in between,
it's a parabola.
00:29:49.000 --> 00:29:55.000
It's a quadratic function.
It's a parabola.
00:29:52.000 --> 00:29:58.000
Does it go up or does it go
down?
00:29:55.000 --> 00:30:01.000
Well, when y is very large,
it's very negative.
00:29:59.000 --> 00:30:05.000
That means it must be a
downward-opening parabola.
00:30:04.000 --> 00:30:10.000
And therefore,
this curve looks like this.
00:30:07.000 --> 00:30:13.000
So, I'm interested in knowing,
where is it positive,
00:30:12.000 --> 00:30:18.000
and where is it negative?
Well, it's positive,
00:30:16.000 --> 00:30:22.000
here, for this range of values
of y.
00:30:19.000 --> 00:30:25.000
Since it's positive there,
it will be increasing there.
00:30:24.000 --> 00:30:30.000
Here, it's negative,
and therefore it will be
00:30:28.000 --> 00:30:34.000
decreasing.
Here, it's negative,
00:30:32.000 --> 00:30:38.000
and therefore,
dy / dt will be negative also,
00:30:36.000 --> 00:30:42.000
and therefore the function,
y, will be decreasing here.
00:30:41.000 --> 00:30:47.000
So, how do these other
solutions look?
00:30:44.000 --> 00:30:50.000
Well, we can put them in.
I'll put them in in white,
00:30:49.000 --> 00:30:55.000
okay, because this has got to
last until the end of the term.
00:30:54.000 --> 00:31:00.000
So, how are they doing?
They are increasing between the
00:30:59.000 --> 00:31:05.000
two curves.
They are not allowed to cross
00:31:04.000 --> 00:31:10.000
either of these yellow curves.
But, they are always
00:31:08.000 --> 00:31:14.000
increasing.
Well, if they're always
00:31:11.000 --> 00:31:17.000
increasing, they must start here
and increase,
00:31:15.000 --> 00:31:21.000
and not allowed to cross.
It must do something like that.
00:31:19.000 --> 00:31:25.000
This must be a translation of
it.
00:31:22.000 --> 00:31:28.000
In other words,
the curves must look like that.
00:31:26.000 --> 00:31:32.000
Those are supposed to be
translations of each other.
00:31:32.000 --> 00:31:38.000
I know they aren't,
but use your imaginations.
00:31:35.000 --> 00:31:41.000
But what's happening above?
So in other words,
00:31:38.000 --> 00:31:44.000
if I start with a population
anywhere bigger than zero but
00:31:42.000 --> 00:31:48.000
less than a over b,
it increases asymptotically to
00:31:46.000 --> 00:31:52.000
the level a over b.
What happens if I start above
00:31:50.000 --> 00:31:56.000
that?
Well, then it decreases to it
00:31:52.000 --> 00:31:58.000
because, this way,
for the values of y bigger than
00:31:56.000 --> 00:32:02.000
a over b,
it decreases as time increases.
00:32:00.000 --> 00:32:06.000
So, these guys up here are
doing this.
00:32:04.000 --> 00:32:10.000
And, how about the ones below
the axis?
00:32:06.000 --> 00:32:12.000
Well, they have no physical
significance.
00:32:09.000 --> 00:32:15.000
But let's put them in anyway.
Whether they doing?
00:32:12.000 --> 00:32:18.000
They are decreasing away from
zero.
00:32:15.000 --> 00:32:21.000
So, these guys don't mean
anything physically,
00:32:18.000 --> 00:32:24.000
but mathematically they exist.
Their solutions,
00:32:21.000 --> 00:32:27.000
they're going down like that.
Now, you notice from this
00:32:25.000 --> 00:32:31.000
picture that there are,
even though both of these are
00:32:29.000 --> 00:32:35.000
constant solutions,
they have dramatically
00:32:32.000 --> 00:32:38.000
different behavior.
This one, this solution,
00:32:37.000 --> 00:32:43.000
is the one that all other
solutions try to approach as
00:32:41.000 --> 00:32:47.000
time goes to infinity.
This one, the solution zero,
00:32:45.000 --> 00:32:51.000
is repulsive,
as it were.
00:32:48.000 --> 00:32:54.000
Any solution that starts near
zero, if it starts at zero,
00:32:52.000 --> 00:32:58.000
of course, it stays there for
all time, but if it starts just
00:32:58.000 --> 00:33:04.000
a little bit above zero,
it increases to a over b.
00:33:02.000 --> 00:33:08.000
This is called a stable
00:33:05.000 --> 00:33:11.000
solution because everybody tries
to get closer and closer to it.
00:33:11.000 --> 00:33:17.000
This is called,
zero is also a constant
00:33:14.000 --> 00:33:20.000
solution, but this is an
unstable solution.
00:33:17.000 --> 00:33:23.000
And now, usually,
solution is too general a word.
00:33:21.000 --> 00:33:27.000
I think it's better to call it
a stable critical point,
00:33:26.000 --> 00:33:32.000
and an unstable critical point.
But, of course,
00:33:30.000 --> 00:33:36.000
it also corresponds to a
solution.
00:33:34.000 --> 00:33:40.000
So, critical points are not all
the same.
00:33:37.000 --> 00:33:43.000
Some are stable,
and some are unstable.
00:33:40.000 --> 00:33:46.000
And, you can see which is which
just by looking at this picture.
00:33:46.000 --> 00:33:52.000
If the arrows point towards
them, you've got a stable
00:33:51.000 --> 00:33:57.000
critical point.
If it arrows point away from
00:33:55.000 --> 00:34:01.000
them, you've got an unstable
critical point.
00:33:58.000 --> 00:34:04.000
Now, there is a third
possibility.
00:34:03.000 --> 00:34:09.000
Okay, I think we'd better
address it because otherwise
00:34:09.000 --> 00:34:15.000
you're going to sit there
wondering, hey,
00:34:13.000 --> 00:34:19.000
what did he do?
Suppose it looks like this.
00:34:18.000 --> 00:34:24.000
Suppose it were just tangent.
Well, this is the picture of
00:34:24.000 --> 00:34:30.000
that curve, the pink curve.
What would the arrows look like
00:34:31.000 --> 00:34:37.000
then?
What would the arrows look like
00:34:35.000 --> 00:34:41.000
then?
Well, since they are positive,
00:34:38.000 --> 00:34:44.000
it's always positive,
the arrow goes like this.
00:34:41.000 --> 00:34:47.000
And then on the side,
it also goes in the same
00:34:45.000 --> 00:34:51.000
direction.
So, is this critical point
00:34:47.000 --> 00:34:53.000
stable or unstable?
It's stable if you approach it
00:34:51.000 --> 00:34:57.000
from the left.
So, how, in fact,
00:34:53.000 --> 00:34:59.000
do the curves,
how would the corresponding
00:34:57.000 --> 00:35:03.000
curves look?
Well, there's our long-term
00:35:00.000 --> 00:35:06.000
solution.
This corresponds to that point.
00:35:05.000 --> 00:35:11.000
Let's call this a,
and then this will be the
00:35:10.000 --> 00:35:16.000
value, a.
If I start below it,
00:35:14.000 --> 00:35:20.000
I rise to it.
If I start above it,
00:35:18.000 --> 00:35:24.000
I increase.
So, if I start above it,
00:35:22.000 --> 00:35:28.000
I do this.
Well, now, that's stable on one
00:35:27.000 --> 00:35:33.000
side, and unstable on the other.
And, that's indicated by saying
00:35:35.000 --> 00:35:41.000
it's semi-stable.
That's a brilliant word.
00:35:40.000 --> 00:35:46.000
I wonder how long it to do
think that one up,
00:35:43.000 --> 00:35:49.000
semi-stable critical point:
stable on one side,
00:35:46.000 --> 00:35:52.000
unstable on the other depending
on whether you start below it.
00:35:50.000 --> 00:35:56.000
And, of course,
it could be reversed if I had
00:35:53.000 --> 00:35:59.000
drawn the picture the other way.
I could have approached it from
00:35:57.000 --> 00:36:03.000
the top, and left it from below.
You get the idea of the
00:36:03.000 --> 00:36:09.000
behavior.
Okay, let's now take,
00:36:08.000 --> 00:36:14.000
I'm going to soup up this
logistic equation just a little
00:36:16.000 --> 00:36:22.000
bit more.
So, let's talk about the
00:36:21.000 --> 00:36:27.000
logistic equation.
But, I'm going to add to it
00:36:28.000 --> 00:36:34.000
harvesting, with harvesting.
So, this is a very late 20th
00:36:36.000 --> 00:36:42.000
century concept.
So, we imagine,
00:36:39.000 --> 00:36:45.000
for example,
a bunch of formerly free range
00:36:43.000 --> 00:36:49.000
Atlantic salmon penned in one of
these huge factory farms off the
00:36:49.000 --> 00:36:55.000
coast of Maine or someplace.
They've made salmon much
00:36:54.000 --> 00:37:00.000
cheaper than it used to be,
but at a certain cost to the
00:37:00.000 --> 00:37:06.000
salmon, and possibly to our
environment.
00:37:05.000 --> 00:37:11.000
So, what happens?
Well, the salmon grow,
00:37:08.000 --> 00:37:14.000
and grow, and do what salmon
do.
00:37:11.000 --> 00:37:17.000
And, they are harvested.
That's a word somewhere in the
00:37:15.000 --> 00:37:21.000
category of ethnic cleansing in
my opinion.
00:37:19.000 --> 00:37:25.000
But, it's, again,
a very 20th-century word.
00:37:23.000 --> 00:37:29.000
I think it was Hitler who
discovered that,
00:37:27.000 --> 00:37:33.000
that all you had to do was call
something by a sanitary name,
00:37:32.000 --> 00:37:38.000
and no matter how horrible it
was, good bourgeois people would
00:37:37.000 --> 00:37:43.000
accept it.
So, the harvesting,
00:37:42.000 --> 00:37:48.000
which means,
of course, picking them up and
00:37:47.000 --> 00:37:53.000
killing them,
and putting them in cans and
00:37:52.000 --> 00:37:58.000
stuff like that,
okay, so what's the equation?
00:37:58.000 --> 00:38:04.000
I'm going to assume that the
harvest is at a constant time
00:38:05.000 --> 00:38:11.000
rate.
In other words,
00:38:08.000 --> 00:38:14.000
it's not a certain fraction of
all the salmon that are being
00:38:13.000 --> 00:38:19.000
caught each day and canned.
The factory has a certain
00:38:17.000 --> 00:38:23.000
capacity, so,
400 pounds of salmon each day
00:38:21.000 --> 00:38:27.000
are pulled out and canned.
So, it's a constant time rate.
00:38:25.000 --> 00:38:31.000
That means that the equation is
now going to be dy/dt is equal
00:38:31.000 --> 00:38:37.000
to, well, salmon grow
logistically.
00:38:35.000 --> 00:38:41.000
ay minus b y squared,
so, that part of the
00:38:39.000 --> 00:38:45.000
equation is the same.
But, I need a term to take care
00:38:44.000 --> 00:38:50.000
of this constant harvesting
rate, and that will be h.
00:38:48.000 --> 00:38:54.000
Let's call it h,
not h times y.
00:38:50.000 --> 00:38:56.000
Then, I would be harvesting a
certain fraction of all the
00:38:55.000 --> 00:39:01.000
salmon there,
which is not what I'm doing.
00:39:00.000 --> 00:39:06.000
Okay: our equation.
Now, I want to analyze what the
00:39:03.000 --> 00:39:09.000
critical points of this look
like.
00:39:05.000 --> 00:39:11.000
Now, this is a little more
subtle because there's now a new
00:39:09.000 --> 00:39:15.000
parameter, there.
And, what I want to see is how
00:39:12.000 --> 00:39:18.000
that varies with the new
parameter.
00:39:15.000 --> 00:39:21.000
The best thing to do is,
I mean, the thing not to do is
00:39:19.000 --> 00:39:25.000
make this equal to zero,
fiddle around with the
00:39:22.000 --> 00:39:28.000
quadratic formula,
get some massive expression,
00:39:25.000 --> 00:39:31.000
and then spend the next half
hour scratching your head trying
00:39:29.000 --> 00:39:35.000
to figure out what it means,
and what information you are
00:39:33.000 --> 00:39:39.000
supposed to be getting out of
it.
00:39:37.000 --> 00:39:43.000
Draw pictures instead.
Draw pictures.
00:39:40.000 --> 00:39:46.000
If h is zero,
that's the smallest harvesting
00:39:45.000 --> 00:39:51.000
rate I could have.
The picture looks like our old
00:39:50.000 --> 00:39:56.000
one.
So, if h is zero,
00:39:52.000 --> 00:39:58.000
the picture looks like,
what color did I,
00:39:56.000 --> 00:40:02.000
okay, pink.
Yellow.
00:40:00.000 --> 00:40:06.000
Yellow is the cheapest,
but I can't find it.
00:40:03.000 --> 00:40:09.000
Okay, yellow is commercially
available.
00:40:06.000 --> 00:40:12.000
These are precious.
All right, purple if it's okay,
00:40:11.000 --> 00:40:17.000
purple.
So, this is the one,
00:40:13.000 --> 00:40:19.000
our original one corresponding
to h equals zero.
00:40:17.000 --> 00:40:23.000
Or, in other words,
it's the equation ay minus b y
00:40:21.000 --> 00:40:27.000
squared. h is zero.
00:40:24.000 --> 00:40:30.000
Now, if I want to find,
I now want to increase the
00:40:28.000 --> 00:40:34.000
value of h, well,
if I increase the value of h,
00:40:32.000 --> 00:40:38.000
in other words,
harvest more and more,
00:40:35.000 --> 00:40:41.000
what's happening?
Well, I simply lower this
00:40:41.000 --> 00:40:47.000
function by h.
So, if I lower h somewhat,
00:40:45.000 --> 00:40:51.000
it will come to here.
So, this is some value,
00:40:49.000 --> 00:40:55.000
ay minus b y squared minus h1,
00:40:54.000 --> 00:41:00.000
let's say.
That's this curve.
00:40:57.000 --> 00:41:03.000
If I lower it a lot,
it will look like this.
00:41:03.000 --> 00:41:09.000
So, ay minus b y squared minus
h a lot, h twenty.
00:41:08.000 --> 00:41:14.000
This doesn't mean anything.
00:41:11.000 --> 00:41:17.000
Two.
Obviously, there's one
00:41:14.000 --> 00:41:20.000
interesting value to lower it
by.
00:41:17.000 --> 00:41:23.000
It's a value which would lower
it exactly by this amount.
00:41:22.000 --> 00:41:28.000
Let me put that in special.
If I lower it by just that
00:41:27.000 --> 00:41:33.000
amount, the curve always looks
the same.
00:41:32.000 --> 00:41:38.000
It's just been lowered.
I'm going to say this one is,
00:41:36.000 --> 00:41:42.000
so this one is the same thing,
except that I've subtracted h
00:41:42.000 --> 00:41:48.000
sub m. Where is h sub m on
00:41:46.000 --> 00:41:52.000
the picture?
Well, I lowered it by this
00:41:49.000 --> 00:41:55.000
amount.
So, this height is h sub m.
00:41:53.000 --> 00:41:59.000
In other words,
if I find the maximum height
00:41:57.000 --> 00:42:03.000
here, which is easy to do
because it's a parabola,
00:42:01.000 --> 00:42:07.000
and lower it by exactly that
amount, I will have lowered it
00:42:07.000 --> 00:42:13.000
to this point.
This will be a critical point.
00:42:12.000 --> 00:42:18.000
Now, the question is,
what's happened to the critical
00:42:16.000 --> 00:42:22.000
point as I did this?
I started with the critical
00:42:20.000 --> 00:42:26.000
points here and here.
As I lower h,
00:42:22.000 --> 00:42:28.000
the critical point changed to
this and that.
00:42:26.000 --> 00:42:32.000
And now, it changed to this one
when I got to the purple line.
00:42:32.000 --> 00:42:38.000
And, as I went still further
down, there were no critical
00:42:36.000 --> 00:42:42.000
points.
So, this curve has no critical
00:42:38.000 --> 00:42:44.000
points attached to it.
What are the corresponding
00:42:42.000 --> 00:42:48.000
pictures?
Well, the corresponding
00:42:44.000 --> 00:42:50.000
pictures, well,
we've already drawn,
00:42:47.000 --> 00:42:53.000
the picture for h equals zero
is drawn already.
00:42:51.000 --> 00:42:57.000
The pictures that I'm talking
about are how the solutions
00:42:55.000 --> 00:43:01.000
look.
How would the solution look
00:42:57.000 --> 00:43:03.000
like for this one for h one?
00:43:00.000 --> 00:43:06.000
For h1, the solutions look
like, here is a over b.
00:43:06.000 --> 00:43:12.000
Here is a over b,
but the critical points aren't
00:43:10.000 --> 00:43:16.000
at zero and a over b anymore.
They've moved in a little bit.
00:43:15.000 --> 00:43:21.000
So, they are here and here.
And, otherwise,
00:43:18.000 --> 00:43:24.000
the solutions look just like
they did before,
00:43:22.000 --> 00:43:28.000
and the analysis is the same.
And, similarly,
00:43:25.000 --> 00:43:31.000
if h two goes very far,
if h2 is very large,
00:43:29.000 --> 00:43:35.000
there are no critical points.
h, too large,
00:43:33.000 --> 00:43:39.000
no critical points.
Are the solutions decreasing
00:43:38.000 --> 00:43:44.000
all the time or increasing?
Well, they are always
00:43:42.000 --> 00:43:48.000
decreasing because the function
is always negative.
00:43:46.000 --> 00:43:52.000
Solutions always go down,
always.
00:43:49.000 --> 00:43:55.000
The interesting one is this
last one, where I decreased it
00:43:54.000 --> 00:44:00.000
just to (h)m.
And, what happens there is
00:43:57.000 --> 00:44:03.000
there is this certain,
magic critical point whose
00:44:01.000 --> 00:44:07.000
value we could calculate.
There's one constant solution.
00:44:07.000 --> 00:44:13.000
So, this is one that has the
value.
00:44:10.000 --> 00:44:16.000
Sorry, I'm calculating the
solutions out.
00:44:13.000 --> 00:44:19.000
So, y here and t here,
so here it is value,
00:44:17.000 --> 00:44:23.000
(h)m is the value by which it
has been lowered.
00:44:20.000 --> 00:44:26.000
So, this is the picture for
(h)m.
00:44:23.000 --> 00:44:29.000
And, how do the solutions look?
Well, to the right of that,
00:44:29.000 --> 00:44:35.000
they are decreasing.
And, to the left they are also
00:44:33.000 --> 00:44:39.000
decreasing because this function
is always negative.
00:44:37.000 --> 00:44:43.000
So, the solutions look like
this, if you start above,
00:44:42.000 --> 00:44:48.000
and if you start below,
they decrease.
00:44:45.000 --> 00:44:51.000
And, of course,
they can't get lower than zero
00:44:49.000 --> 00:44:55.000
because these are salmon.
What is the significance of
00:44:53.000 --> 00:44:59.000
(h)m?
(h)m is the maximum rate of
00:44:56.000 --> 00:45:02.000
harvesting.
It's an extremely important
00:44:59.000 --> 00:45:05.000
number for this industry.
If the maximum time rate at
00:45:05.000 --> 00:45:11.000
which you can pull the salmon
daily out of the water,
00:45:09.000 --> 00:45:15.000
and can them without what
happening?
00:45:12.000 --> 00:45:18.000
Without the salmon going to
zero.
00:45:15.000 --> 00:45:21.000
As long as you start above,
and don't harvest it more than
00:45:20.000 --> 00:45:26.000
this rate, it will be following
these curves.
00:45:23.000 --> 00:45:29.000
You will be following these
curves, and you will still have
00:45:28.000 --> 00:45:34.000
salmon.
If you harvest just a little
00:45:31.000 --> 00:45:37.000
bit more, you will be on this
curve that has no critical
00:45:36.000 --> 00:45:42.000
points, and the salmon in the
tank will decrease to zero.