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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,

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of course, the solution is
simply an exponential curve.

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There's nothing more to say
about it.

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Now, let's make it more
interesting.

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Let's suppose there is a shifty
teller at the bank,

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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

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paying any attention to it.
You're off skiing somewhere,

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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

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number of dollars every day,
the same number of dollars

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being withdrawn for the count.
Okay, now, of course,

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you could solve this.
This separates variables

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immediately.
You get the answer,

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and there's no problem with
that.

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Let's analyze the behavior of
the solutions without solving

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the equation by using these two
points.

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So, I want to analyze this
equation using the method of

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critical points.
So, the first thing I should do

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is, so, here's our equation,
is find the critical points.

233
00:16:34,000 --> 00:16:40,000
Notice it's an autonomous
equation all right,

234
00:16:38,000 --> 00:16:44,000
because there's no t on the
right-hand side.

235
00:16:42,000 --> 00:16:48,000
Okay, so, the critical points,
well, that's where ry minus w

236
00:16:48,000 --> 00:16:54,000
equals zero.
In other words,

237
00:16:51,000 --> 00:16:57,000
there's only one critical
point, and that occurs when y is

238
00:16:56,000 --> 00:17:02,000
equal to w over r.
So, that's the only critical

239
00:17:01,000 --> 00:17:07,000
point.
Now, I want to know what's

240
00:17:05,000 --> 00:17:11,000
happening to the solution.
So, in other words,

241
00:17:09,000 --> 00:17:15,000
if I plot, I can write away,
of course, negative values

242
00:17:14,000 --> 00:17:20,000
aren't of particularly
interesting here,

243
00:17:17,000 --> 00:17:23,000
there is definitely a
horizontal solution,

244
00:17:21,000 --> 00:17:27,000
and it has the value,
it's at the height,

245
00:17:25,000 --> 00:17:31,000
w over r.
That's a solution.

246
00:17:30,000 --> 00:17:36,000
The question is,
what do the other solutions

247
00:17:33,000 --> 00:17:39,000
look like?
Now, watch how I make the

248
00:17:35,000 --> 00:17:41,000
analysis because I'm going to
use two now.

249
00:17:39,000 --> 00:17:45,000
So, this is step one,
then step two.

250
00:17:41,000 --> 00:17:47,000
What do I do? Well,
I'm going to graph f of y.

251
00:17:45,000 --> 00:17:51,000
Well, f of y is ry minus w.

252
00:17:47,000 --> 00:17:53,000
What does that look like?

253
00:17:50,000 --> 00:17:56,000
Okay, so, here is the y-axis.
Notice the y-axis is going

254
00:17:55,000 --> 00:18:01,000
horizontally because what I'm
interested in is the graph of

255
00:17:59,000 --> 00:18:05,000
this function.
What do I call the other axis?

256
00:18:04,000 --> 00:18:10,000
I'm going to use the same
terminology that is used on the

257
00:18:08,000 --> 00:18:14,000
little visual that describes
this.

258
00:18:11,000 --> 00:18:17,000
And, that's dy.
You could call this other axis

259
00:18:14,000 --> 00:18:20,000
the f of y axis.
That's not a good name for it.

260
00:18:18,000 --> 00:18:24,000
You could call it the dy / dt
axis because it's,

261
00:18:22,000 --> 00:18:28,000
so to speak,
the other variable.

262
00:18:24,000 --> 00:18:30,000
That's not great either.
But, worst of all would be

263
00:18:28,000 --> 00:18:34,000
introducing yet another letter
for which we would have no use

264
00:18:32,000 --> 00:18:38,000
whatsoever.
So, let's think of it.

265
00:18:37,000 --> 00:18:43,000
We are plotting,
now, the graph of f of y.

266
00:18:40,000 --> 00:18:46,000
f of y is this function,

267
00:18:43,000 --> 00:18:49,000
ry minus w.
Well, that's a line.

268
00:18:48,000 --> 00:18:54,000
Its intercept is down here at
w, and so the graph looks

269
00:18:53,000 --> 00:18:59,000
something like this.
It's a line.

270
00:18:55,000 --> 00:19:01,000
This is the line,
ry minus w.

271
00:18:59,000 --> 00:19:05,000
It has slope r.
Well, what am I going to get

272
00:19:05,000 --> 00:19:11,000
out of that line?
Just exactly this.

273
00:19:08,000 --> 00:19:14,000
What am I interested in about
that line?

274
00:19:13,000 --> 00:19:19,000
Nothing other than where is it
above the axis,

275
00:19:18,000 --> 00:19:24,000
and where is it below?
This function is positive over

276
00:19:23,000 --> 00:19:29,000
here, and therefore,
I'm going to indicate that

277
00:19:28,000 --> 00:19:34,000
symbolically,
this, by putting an arrow here.

278
00:19:35,000 --> 00:19:41,000
The meeting of this arrow is
that y of t is

279
00:19:39,000 --> 00:19:45,000
increasing.
See where it's the right-hand

280
00:19:43,000 --> 00:19:49,000
side of that last board?
y of t is increasing when f of

281
00:19:48,000 --> 00:19:54,000
y is positive.
f of y is positive here,

282
00:19:52,000 --> 00:19:58,000
and therefore,
to the right of this point,

283
00:19:55,000 --> 00:20:01,000
it's increasing.
Here to the left of it,

284
00:19:59,000 --> 00:20:05,000
f of y is negative,
and therefore over here it's

285
00:20:03,000 --> 00:20:09,000
going to be decreasing.
What point is this,

286
00:20:08,000 --> 00:20:14,000
in fact?
Well, that's where it crosses

287
00:20:11,000 --> 00:20:17,000
the axis.
That's exactly the critical

288
00:20:15,000 --> 00:20:21,000
point, w over r.
Therefore, what this says is

289
00:20:19,000 --> 00:20:25,000
that a solution,
once it's bigger than y over r,

290
00:20:24,000 --> 00:20:30,000
it increases,
and it increases faster and

291
00:20:28,000 --> 00:20:34,000
faster because this function is
higher and higher.

292
00:20:34,000 --> 00:20:40,000
And, that represents the rate
of change.

293
00:20:37,000 --> 00:20:43,000
So, in other words,
once the solution,

294
00:20:40,000 --> 00:20:46,000
let's say a solution starts
over here at time zero.

295
00:20:44,000 --> 00:20:50,000
So, this is the t axis.
And, here is the y axis.

296
00:20:48,000 --> 00:20:54,000
So, now, I'm plotting
solutions.

297
00:20:51,000 --> 00:20:57,000
If it starts at t equals zero,
above this line,

298
00:20:55,000 --> 00:21:01,000
that is, starts with the value
w over r,

299
00:20:59,000 --> 00:21:05,000
which is bigger than zero,
a value bigger than w over r,

300
00:21:04,000 --> 00:21:10,000
then it increases,
and increases faster and

301
00:21:07,000 --> 00:21:13,000
faster.
If it starts below that,

302
00:21:12,000 --> 00:21:18,000
it decreases and decreases
faster and faster.

303
00:21:16,000 --> 00:21:22,000
Now, in fact,
I only have to draw two of

304
00:21:20,000 --> 00:21:26,000
those because what do all the
others look like?

305
00:21:24,000 --> 00:21:30,000
They are translations.
All the other curves look

306
00:21:29,000 --> 00:21:35,000
exactly like those.
They are just translations of

307
00:21:35,000 --> 00:21:41,000
them.
This guy, if I start closer,

308
00:21:39,000 --> 00:21:45,000
it's still going to decrease.
Well, that's supposed to be a

309
00:21:47,000 --> 00:21:53,000
translation.
Maybe it is.

310
00:21:50,000 --> 00:21:56,000
So, these guys look like that.
Let's do just a tiny bit more

311
00:21:57,000 --> 00:22:03,000
interpretation of that.
Well, I think I better leave it

312
00:22:06,000 --> 00:22:12,000
there because we've got harder
things to do,

313
00:22:13,000 --> 00:22:19,000
and I want to make sure we've
got time for it.

314
00:22:19,000 --> 00:22:25,000
Sorry.
Okay, next example,

315
00:22:23,000 --> 00:22:29,000
a logistic equation.
Some of you have already solved

316
00:22:31,000 --> 00:22:37,000
this in recitation,
and some of you haven't.

317
00:22:39,000 --> 00:22:45,000
This is a population equation.
This is the one that section

318
00:22:44,000 --> 00:22:50,000
7.1 and section 1.7 is most
heavily concerned with,

319
00:22:48,000 --> 00:22:54,000
this particular equation.
The derivation of it is a

320
00:22:52,000 --> 00:22:58,000
little vague.
It's an equation which

321
00:22:55,000 --> 00:23:01,000
describes how population
increases.

322
00:22:58,000 --> 00:23:04,000
And one minute,
the population behavior of some

323
00:23:02,000 --> 00:23:08,000
population, --
-- let's call it,

324
00:23:06,000 --> 00:23:12,000
y is the only thing I know to
call anything today,

325
00:23:10,000 --> 00:23:16,000
but of course your book uses
capital P for population,

326
00:23:15,000 --> 00:23:21,000
to get you used to different
variables.

327
00:23:18,000 --> 00:23:24,000
Now, the basic population
equation runs dy / dt.

328
00:23:22,000 --> 00:23:28,000
There's a certain growth rate.
Let's call it k y.

329
00:23:26,000 --> 00:23:32,000
So, k is what's called the
growth rate.

330
00:23:29,000 --> 00:23:35,000
It's actually,
sometimes it's talked about in

331
00:23:33,000 --> 00:23:39,000
terms of birthrate.
But, it's the net birth rate.

332
00:23:38,000 --> 00:23:44,000
It's the rate at which people,
or bacteria,

333
00:23:42,000 --> 00:23:48,000
or whatever are being born
minus the rate at which they are

334
00:23:47,000 --> 00:23:53,000
dying.
So, it's a net birthrate.

335
00:23:49,000 --> 00:23:55,000
But, let's just call it the
growth rate.

336
00:23:53,000 --> 00:23:59,000
Now, if this is the equation,
we can think of this,

337
00:23:57,000 --> 00:24:03,000
if k is constant,
that's what's called simple

338
00:24:01,000 --> 00:24:07,000
population growth.
And you are all familiar with

339
00:24:05,000 --> 00:24:11,000
that.
Logistical growth allows for

340
00:24:08,000 --> 00:24:14,000
slightly more complex
situations.

341
00:24:13,000 --> 00:24:19,000
Logistic growth says that
calling k a constant is

342
00:24:18,000 --> 00:24:24,000
unrealistic because the Earth is
not filled entirely with people.

343
00:24:26,000 --> 00:24:32,000
What stops it from having
unlimited growth?

344
00:24:32,000 --> 00:24:38,000
Well, the fact that the
resources, the food,

345
00:24:36,000 --> 00:24:42,000
the organism has to live on
gets depleted.

346
00:24:41,000 --> 00:24:47,000
And, in other words,
the growth rate declines as y

347
00:24:46,000 --> 00:24:52,000
increases.
As the population increases,

348
00:24:50,000 --> 00:24:56,000
one expects the growth rate to
decline because resources are

349
00:24:56,000 --> 00:25:02,000
being used up,
and they are not indefinitely

350
00:25:01,000 --> 00:25:07,000
available.
Well, in other words,

351
00:25:05,000 --> 00:25:11,000
we should replace k by a
function with this behavior.

352
00:25:09,000 --> 00:25:15,000
What's the simplest function
that declines as y increases?

353
00:25:14,000 --> 00:25:20,000
The simplest choice,
and if you are ignorant about

354
00:25:17,000 --> 00:25:23,000
what else to do,
stick with the simplest,

355
00:25:20,000 --> 00:25:26,000
at least you won't work any
harder than you have to,

356
00:25:24,000 --> 00:25:30,000
would be to take k equal to the
simplest declining function of y

357
00:25:29,000 --> 00:25:35,000
there is, which is simply a
linear function,

358
00:25:32,000 --> 00:25:38,000
A minus BY.
So, if I use that as the choice

359
00:25:37,000 --> 00:25:43,000
of the declining growth rate,
the new equation is dy / dt

360
00:25:41,000 --> 00:25:47,000
equals, here's my new k.
The y stays the same,

361
00:25:45,000 --> 00:25:51,000
so the equation becomes a minus
by, the quantity times y,

362
00:25:50,000 --> 00:25:56,000
or in other words,

363
00:25:53,000 --> 00:25:59,000
ay minus b y squared.

364
00:25:55,000 --> 00:26:01,000
This equation is what's called
the logistic equation.

365
00:26:01,000 --> 00:26:07,000
It has many applications,
not just to population growth.

366
00:26:05,000 --> 00:26:11,000
It's applied to the spread of
disease, the spread of a rumor,

367
00:26:11,000 --> 00:26:17,000
the spread of many things.
Yeah, a couple pieces of chalk

368
00:26:16,000 --> 00:26:22,000
here.

369
00:26:28,000 --> 00:26:34,000
Okay, now, those of you who
have solved it know that the

370
00:26:34,000 --> 00:26:40,000
explicit solution involves,
well, you separate variables,

371
00:26:40,000 --> 00:26:46,000
but you will have to use
partial fractions,

372
00:26:44,000 --> 00:26:50,000
ugh, I hope you love partial
fractions.

373
00:26:48,000 --> 00:26:54,000
You're going to need them later
in the term.

374
00:26:53,000 --> 00:26:59,000
But, I could avoid them now by
not solving the equation

375
00:26:59,000 --> 00:27:05,000
explicitly.
But anyway, you get a solution,

376
00:27:03,000 --> 00:27:09,000
which I was going to write on
the board for you,

377
00:27:06,000 --> 00:27:12,000
but you could look it up in
your book.

378
00:27:09,000 --> 00:27:15,000
It's unpleasant enough looking
to make you feel that there must

379
00:27:13,000 --> 00:27:19,000
be an easier way at least to get
the basic information out.

380
00:27:16,000 --> 00:27:22,000
Okay, let's see if we can get
the basic information out.

381
00:27:20,000 --> 00:27:26,000
What are the critical points?
Well, this is pretty easy.

382
00:27:23,000 --> 00:27:29,000
A, I want to set the right-hand
side equal to zero.

383
00:27:26,000 --> 00:27:32,000
So, I'm going to solve the
equation.

384
00:27:30,000 --> 00:27:36,000
I can factor out a y.
It's going to be y times a

385
00:27:35,000 --> 00:27:41,000
minus by equals zero.

386
00:27:39,000 --> 00:27:45,000
And therefore,
the critical points are where y

387
00:27:43,000 --> 00:27:49,000
equals zero. That's one.

388
00:27:47,000 --> 00:27:53,000
And, the other factor is when
this factor is zero,

389
00:27:52,000 --> 00:27:58,000
and that happens when y is
equal to a over b.

390
00:27:59,000 --> 00:28:05,000
So, there are my two critical
points.

391
00:28:03,000 --> 00:28:09,000
Okay, what does,
let's start drawing pictures of

392
00:28:08,000 --> 00:28:14,000
solutions.
Let's put it in those right

393
00:28:12,000 --> 00:28:18,000
away.
Okay, the critical point,

394
00:28:15,000 --> 00:28:21,000
zero, gives me a solution that
looks like this.

395
00:28:18,000 --> 00:28:24,000
And, the critical point,
a over b,

396
00:28:21,000 --> 00:28:27,000
those are positive numbers.
So, that's somewhere up here.

397
00:28:25,000 --> 00:28:31,000
So, those are two solutions,
constant solutions.

398
00:28:29,000 --> 00:28:35,000
In other words,
if the population by dumb luck

399
00:28:32,000 --> 00:28:38,000
started at zero,
it would stay at zero for all

400
00:28:35,000 --> 00:28:41,000
time.
That's not terribly surprising.

401
00:28:39,000 --> 00:28:45,000
But, it's a little less obvious
that if it starts at that magic

402
00:28:43,000 --> 00:28:49,000
number, a over b,
it will also stay at that magic

403
00:28:47,000 --> 00:28:53,000
number for all time without
moving up or down or away from

404
00:28:50,000 --> 00:28:56,000
it.
Now, the question is,

405
00:28:52,000 --> 00:28:58,000
therefore, what happens in
between?

406
00:28:54,000 --> 00:29:00,000
So, for the in between,
I'm going to make that same

407
00:28:58,000 --> 00:29:04,000
analysis that I made before.
And, it's really not very hard.

408
00:29:03,000 --> 00:29:09,000
Look, so here's my dy/dt-axis.
I'll call that y prime,

409
00:29:09,000 --> 00:29:15,000
okay?
And, here's the y-axis.

410
00:29:12,000 --> 00:29:18,000
So, I'm now doing step two.
This was step one.

411
00:29:16,000 --> 00:29:22,000
Okay, the function that I want
to graph is this one,

412
00:29:21,000 --> 00:29:27,000
ay minus b y squared,
or in factor form,

413
00:29:26,000 --> 00:29:32,000
y times a minus by.

414
00:29:29,000 --> 00:29:35,000
Now, this function,
we know, has a zero.

415
00:29:32,000 --> 00:29:38,000
It has a zero here,
and it has a zero at the point

416
00:29:37,000 --> 00:29:43,000
a over b.
At these two critical points,

417
00:29:43,000 --> 00:29:49,000
it has a zero.
What is it doing in between?

418
00:29:46,000 --> 00:29:52,000
Well, in between,
it's a parabola.

419
00:29:49,000 --> 00:29:55,000
It's a quadratic function.
It's a parabola.

420
00:29:52,000 --> 00:29:58,000
Does it go up or does it go
down?

421
00:29:55,000 --> 00:30:01,000
Well, when y is very large,
it's very negative.

422
00:29:59,000 --> 00:30:05,000
That means it must be a
downward-opening parabola.

423
00:30:04,000 --> 00:30:10,000
And therefore,
this curve looks like this.

424
00:30:07,000 --> 00:30:13,000
So, I'm interested in knowing,
where is it positive,

425
00:30:12,000 --> 00:30:18,000
and where is it negative?
Well, it's positive,

426
00:30:16,000 --> 00:30:22,000
here, for this range of values
of y.

427
00:30:19,000 --> 00:30:25,000
Since it's positive there,
it will be increasing there.

428
00:30:24,000 --> 00:30:30,000
Here, it's negative,
and therefore it will be

429
00:30:28,000 --> 00:30:34,000
decreasing.
Here, it's negative,

430
00:30:32,000 --> 00:30:38,000
and therefore,
dy / dt will be negative also,

431
00:30:36,000 --> 00:30:42,000
and therefore the function,
y, will be decreasing here.

432
00:30:41,000 --> 00:30:47,000
So, how do these other
solutions look?

433
00:30:44,000 --> 00:30:50,000
Well, we can put them in.
I'll put them in in white,

434
00:30:49,000 --> 00:30:55,000
okay, because this has got to
last until the end of the term.

435
00:30:54,000 --> 00:31:00,000
So, how are they doing?
They are increasing between the

436
00:30:59,000 --> 00:31:05,000
two curves.
They are not allowed to cross

437
00:31:04,000 --> 00:31:10,000
either of these yellow curves.
But, they are always

438
00:31:08,000 --> 00:31:14,000
increasing.
Well, if they're always

439
00:31:11,000 --> 00:31:17,000
increasing, they must start here
and increase,

440
00:31:15,000 --> 00:31:21,000
and not allowed to cross.
It must do something like that.

441
00:31:19,000 --> 00:31:25,000
This must be a translation of
it.

442
00:31:22,000 --> 00:31:28,000
In other words,
the curves must look like that.

443
00:31:26,000 --> 00:31:32,000
Those are supposed to be
translations of each other.

444
00:31:32,000 --> 00:31:38,000
I know they aren't,
but use your imaginations.

445
00:31:35,000 --> 00:31:41,000
But what's happening above?
So in other words,

446
00:31:38,000 --> 00:31:44,000
if I start with a population
anywhere bigger than zero but

447
00:31:42,000 --> 00:31:48,000
less than a over b,
it increases asymptotically to

448
00:31:46,000 --> 00:31:52,000
the level a over b.
What happens if I start above

449
00:31:50,000 --> 00:31:56,000
that?
Well, then it decreases to it

450
00:31:52,000 --> 00:31:58,000
because, this way,
for the values of y bigger than

451
00:31:56,000 --> 00:32:02,000
a over b,
it decreases as time increases.

452
00:32:00,000 --> 00:32:06,000
So, these guys up here are
doing this.

453
00:32:04,000 --> 00:32:10,000
And, how about the ones below
the axis?

454
00:32:06,000 --> 00:32:12,000
Well, they have no physical
significance.

455
00:32:09,000 --> 00:32:15,000
But let's put them in anyway.
Whether they doing?

456
00:32:12,000 --> 00:32:18,000
They are decreasing away from
zero.

457
00:32:15,000 --> 00:32:21,000
So, these guys don't mean
anything physically,

458
00:32:18,000 --> 00:32:24,000
but mathematically they exist.
Their solutions,

459
00:32:21,000 --> 00:32:27,000
they're going down like that.
Now, you notice from this

460
00:32:25,000 --> 00:32:31,000
picture that there are,
even though both of these are

461
00:32:29,000 --> 00:32:35,000
constant solutions,
they have dramatically

462
00:32:32,000 --> 00:32:38,000
different behavior.
This one, this solution,

463
00:32:37,000 --> 00:32:43,000
is the one that all other
solutions try to approach as

464
00:32:41,000 --> 00:32:47,000
time goes to infinity.
This one, the solution zero,

465
00:32:45,000 --> 00:32:51,000
is repulsive,
as it were.

466
00:32:48,000 --> 00:32:54,000
Any solution that starts near
zero, if it starts at zero,

467
00:32:52,000 --> 00:32:58,000
of course, it stays there for
all time, but if it starts just

468
00:32:58,000 --> 00:33:04,000
a little bit above zero,
it increases to a over b.

469
00:33:02,000 --> 00:33:08,000
This is called a stable

470
00:33:05,000 --> 00:33:11,000
solution because everybody tries
to get closer and closer to it.

471
00:33:11,000 --> 00:33:17,000
This is called,
zero is also a constant

472
00:33:14,000 --> 00:33:20,000
solution, but this is an
unstable solution.

473
00:33:17,000 --> 00:33:23,000
And now, usually,
solution is too general a word.

474
00:33:21,000 --> 00:33:27,000
I think it's better to call it
a stable critical point,

475
00:33:26,000 --> 00:33:32,000
and an unstable critical point.
But, of course,

476
00:33:30,000 --> 00:33:36,000
it also corresponds to a
solution.

477
00:33:34,000 --> 00:33:40,000
So, critical points are not all
the same.

478
00:33:37,000 --> 00:33:43,000
Some are stable,
and some are unstable.

479
00:33:40,000 --> 00:33:46,000
And, you can see which is which
just by looking at this picture.

480
00:33:46,000 --> 00:33:52,000
If the arrows point towards
them, you've got a stable

481
00:33:51,000 --> 00:33:57,000
critical point.
If it arrows point away from

482
00:33:55,000 --> 00:34:01,000
them, you've got an unstable
critical point.

483
00:33:58,000 --> 00:34:04,000
Now, there is a third
possibility.

484
00:34:03,000 --> 00:34:09,000
Okay, I think we'd better
address it because otherwise

485
00:34:09,000 --> 00:34:15,000
you're going to sit there
wondering, hey,

486
00:34:13,000 --> 00:34:19,000
what did he do?
Suppose it looks like this.

487
00:34:18,000 --> 00:34:24,000
Suppose it were just tangent.
Well, this is the picture of

488
00:34:24,000 --> 00:34:30,000
that curve, the pink curve.
What would the arrows look like

489
00:34:31,000 --> 00:34:37,000
then?
What would the arrows look like

490
00:34:35,000 --> 00:34:41,000
then?
Well, since they are positive,

491
00:34:38,000 --> 00:34:44,000
it's always positive,
the arrow goes like this.

492
00:34:41,000 --> 00:34:47,000
And then on the side,
it also goes in the same

493
00:34:45,000 --> 00:34:51,000
direction.
So, is this critical point

494
00:34:47,000 --> 00:34:53,000
stable or unstable?
It's stable if you approach it

495
00:34:51,000 --> 00:34:57,000
from the left.
So, how, in fact,

496
00:34:53,000 --> 00:34:59,000
do the curves,
how would the corresponding

497
00:34:57,000 --> 00:35:03,000
curves look?
Well, there's our long-term

498
00:35:00,000 --> 00:35:06,000
solution.
This corresponds to that point.

499
00:35:05,000 --> 00:35:11,000
Let's call this a,
and then this will be the

500
00:35:10,000 --> 00:35:16,000
value, a.
If I start below it,

501
00:35:14,000 --> 00:35:20,000
I rise to it.
If I start above it,

502
00:35:18,000 --> 00:35:24,000
I increase.
So, if I start above it,

503
00:35:22,000 --> 00:35:28,000
I do this.
Well, now, that's stable on one

504
00:35:27,000 --> 00:35:33,000
side, and unstable on the other.
And, that's indicated by saying

505
00:35:35,000 --> 00:35:41,000
it's semi-stable.
That's a brilliant word.

506
00:35:40,000 --> 00:35:46,000
I wonder how long it to do
think that one up,

507
00:35:43,000 --> 00:35:49,000
semi-stable critical point:
stable on one side,

508
00:35:46,000 --> 00:35:52,000
unstable on the other depending
on whether you start below it.

509
00:35:50,000 --> 00:35:56,000
And, of course,
it could be reversed if I had

510
00:35:53,000 --> 00:35:59,000
drawn the picture the other way.
I could have approached it from

511
00:35:57,000 --> 00:36:03,000
the top, and left it from below.
You get the idea of the

512
00:36:03,000 --> 00:36:09,000
behavior.
Okay, let's now take,

513
00:36:08,000 --> 00:36:14,000
I'm going to soup up this
logistic equation just a little

514
00:36:16,000 --> 00:36:22,000
bit more.
So, let's talk about the

515
00:36:21,000 --> 00:36:27,000
logistic equation.
But, I'm going to add to it

516
00:36:28,000 --> 00:36:34,000
harvesting, with harvesting.
So, this is a very late 20th

517
00:36:36,000 --> 00:36:42,000
century concept.
So, we imagine,

518
00:36:39,000 --> 00:36:45,000
for example,
a bunch of formerly free range

519
00:36:43,000 --> 00:36:49,000
Atlantic salmon penned in one of
these huge factory farms off the

520
00:36:49,000 --> 00:36:55,000
coast of Maine or someplace.
They've made salmon much

521
00:36:54,000 --> 00:37:00,000
cheaper than it used to be,
but at a certain cost to the

522
00:37:00,000 --> 00:37:06,000
salmon, and possibly to our
environment.

523
00:37:05,000 --> 00:37:11,000
So, what happens?
Well, the salmon grow,

524
00:37:08,000 --> 00:37:14,000
and grow, and do what salmon
do.

525
00:37:11,000 --> 00:37:17,000
And, they are harvested.
That's a word somewhere in the

526
00:37:15,000 --> 00:37:21,000
category of ethnic cleansing in
my opinion.

527
00:37:19,000 --> 00:37:25,000
But, it's, again,
a very 20th-century word.

528
00:37:23,000 --> 00:37:29,000
I think it was Hitler who
discovered that,

529
00:37:27,000 --> 00:37:33,000
that all you had to do was call
something by a sanitary name,

530
00:37:32,000 --> 00:37:38,000
and no matter how horrible it
was, good bourgeois people would

531
00:37:37,000 --> 00:37:43,000
accept it.
So, the harvesting,

532
00:37:42,000 --> 00:37:48,000
which means,
of course, picking them up and

533
00:37:47,000 --> 00:37:53,000
killing them,
and putting them in cans and

534
00:37:52,000 --> 00:37:58,000
stuff like that,
okay, so what's the equation?

535
00:37:58,000 --> 00:38:04,000
I'm going to assume that the
harvest is at a constant time

536
00:38:05,000 --> 00:38:11,000
rate.
In other words,

537
00:38:08,000 --> 00:38:14,000
it's not a certain fraction of
all the salmon that are being

538
00:38:13,000 --> 00:38:19,000
caught each day and canned.
The factory has a certain

539
00:38:17,000 --> 00:38:23,000
capacity, so,
400 pounds of salmon each day

540
00:38:21,000 --> 00:38:27,000
are pulled out and canned.
So, it's a constant time rate.

541
00:38:25,000 --> 00:38:31,000
That means that the equation is
now going to be dy/dt is equal

542
00:38:31,000 --> 00:38:37,000
to, well, salmon grow
logistically.

543
00:38:35,000 --> 00:38:41,000
ay minus b y squared,
so, that part of the

544
00:38:39,000 --> 00:38:45,000
equation is the same.
But, I need a term to take care

545
00:38:44,000 --> 00:38:50,000
of this constant harvesting
rate, and that will be h.

546
00:38:48,000 --> 00:38:54,000
Let's call it h,
not h times y.

547
00:38:50,000 --> 00:38:56,000
Then, I would be harvesting a
certain fraction of all the

548
00:38:55,000 --> 00:39:01,000
salmon there,
which is not what I'm doing.

549
00:39:00,000 --> 00:39:06,000
Okay: our equation.
Now, I want to analyze what the

550
00:39:03,000 --> 00:39:09,000
critical points of this look
like.

551
00:39:05,000 --> 00:39:11,000
Now, this is a little more
subtle because there's now a new

552
00:39:09,000 --> 00:39:15,000
parameter, there.
And, what I want to see is how

553
00:39:12,000 --> 00:39:18,000
that varies with the new
parameter.

554
00:39:15,000 --> 00:39:21,000
The best thing to do is,
I mean, the thing not to do is

555
00:39:19,000 --> 00:39:25,000
make this equal to zero,
fiddle around with the

556
00:39:22,000 --> 00:39:28,000
quadratic formula,
get some massive expression,

557
00:39:25,000 --> 00:39:31,000
and then spend the next half
hour scratching your head trying

558
00:39:29,000 --> 00:39:35,000
to figure out what it means,
and what information you are

559
00:39:33,000 --> 00:39:39,000
supposed to be getting out of
it.

560
00:39:37,000 --> 00:39:43,000
Draw pictures instead.
Draw pictures.

561
00:39:40,000 --> 00:39:46,000
If h is zero,
that's the smallest harvesting

562
00:39:45,000 --> 00:39:51,000
rate I could have.
The picture looks like our old

563
00:39:50,000 --> 00:39:56,000
one.
So, if h is zero,

564
00:39:52,000 --> 00:39:58,000
the picture looks like,
what color did I,

565
00:39:56,000 --> 00:40:02,000
okay, pink.
Yellow.

566
00:40:00,000 --> 00:40:06,000
Yellow is the cheapest,
but I can't find it.

567
00:40:03,000 --> 00:40:09,000
Okay, yellow is commercially
available.

568
00:40:06,000 --> 00:40:12,000
These are precious.
All right, purple if it's okay,

569
00:40:11,000 --> 00:40:17,000
purple.
So, this is the one,

570
00:40:13,000 --> 00:40:19,000
our original one corresponding
to h equals zero.

571
00:40:17,000 --> 00:40:23,000
Or, in other words,
it's the equation ay minus b y

572
00:40:21,000 --> 00:40:27,000
squared. h is zero.

573
00:40:24,000 --> 00:40:30,000
Now, if I want to find,
I now want to increase the

574
00:40:28,000 --> 00:40:34,000
value of h, well,
if I increase the value of h,

575
00:40:32,000 --> 00:40:38,000
in other words,
harvest more and more,

576
00:40:35,000 --> 00:40:41,000
what's happening?
Well, I simply lower this

577
00:40:41,000 --> 00:40:47,000
function by h.
So, if I lower h somewhat,

578
00:40:45,000 --> 00:40:51,000
it will come to here.
So, this is some value,

579
00:40:49,000 --> 00:40:55,000
ay minus b y squared minus h1,

580
00:40:54,000 --> 00:41:00,000
let's say.
That's this curve.

581
00:40:57,000 --> 00:41:03,000
If I lower it a lot,
it will look like this.

582
00:41:03,000 --> 00:41:09,000
So, ay minus b y squared minus
h a lot, h twenty.

583
00:41:08,000 --> 00:41:14,000
This doesn't mean anything.

584
00:41:11,000 --> 00:41:17,000
Two.
Obviously, there's one

585
00:41:14,000 --> 00:41:20,000
interesting value to lower it
by.

586
00:41:17,000 --> 00:41:23,000
It's a value which would lower
it exactly by this amount.

587
00:41:22,000 --> 00:41:28,000
Let me put that in special.
If I lower it by just that

588
00:41:27,000 --> 00:41:33,000
amount, the curve always looks
the same.

589
00:41:32,000 --> 00:41:38,000
It's just been lowered.
I'm going to say this one is,

590
00:41:36,000 --> 00:41:42,000
so this one is the same thing,
except that I've subtracted h

591
00:41:42,000 --> 00:41:48,000
sub m. Where is h sub m on

592
00:41:46,000 --> 00:41:52,000
the picture?
Well, I lowered it by this

593
00:41:49,000 --> 00:41:55,000
amount.
So, this height is h sub m.

594
00:41:53,000 --> 00:41:59,000
In other words,
if I find the maximum height

595
00:41:57,000 --> 00:42:03,000
here, which is easy to do
because it's a parabola,

596
00:42:01,000 --> 00:42:07,000
and lower it by exactly that
amount, I will have lowered it

597
00:42:07,000 --> 00:42:13,000
to this point.
This will be a critical point.

598
00:42:12,000 --> 00:42:18,000
Now, the question is,
what's happened to the critical

599
00:42:16,000 --> 00:42:22,000
point as I did this?
I started with the critical

600
00:42:20,000 --> 00:42:26,000
points here and here.
As I lower h,

601
00:42:22,000 --> 00:42:28,000
the critical point changed to
this and that.

602
00:42:26,000 --> 00:42:32,000
And now, it changed to this one
when I got to the purple line.

603
00:42:32,000 --> 00:42:38,000
And, as I went still further
down, there were no critical

604
00:42:36,000 --> 00:42:42,000
points.
So, this curve has no critical

605
00:42:38,000 --> 00:42:44,000
points attached to it.
What are the corresponding

606
00:42:42,000 --> 00:42:48,000
pictures?
Well, the corresponding

607
00:42:44,000 --> 00:42:50,000
pictures, well,
we've already drawn,

608
00:42:47,000 --> 00:42:53,000
the picture for h equals zero
is drawn already.

609
00:42:51,000 --> 00:42:57,000
The pictures that I'm talking
about are how the solutions

610
00:42:55,000 --> 00:43:01,000
look.
How would the solution look

611
00:42:57,000 --> 00:43:03,000
like for this one for h one?

612
00:43:00,000 --> 00:43:06,000
For h1, the solutions look
like, here is a over b.

613
00:43:06,000 --> 00:43:12,000
Here is a over b,
but the critical points aren't

614
00:43:10,000 --> 00:43:16,000
at zero and a over b anymore.
They've moved in a little bit.

615
00:43:15,000 --> 00:43:21,000
So, they are here and here.
And, otherwise,

616
00:43:18,000 --> 00:43:24,000
the solutions look just like
they did before,

617
00:43:22,000 --> 00:43:28,000
and the analysis is the same.
And, similarly,

618
00:43:25,000 --> 00:43:31,000
if h two goes very far,
if h2 is very large,

619
00:43:29,000 --> 00:43:35,000
there are no critical points.
h, too large,

620
00:43:33,000 --> 00:43:39,000
no critical points.
Are the solutions decreasing

621
00:43:38,000 --> 00:43:44,000
all the time or increasing?
Well, they are always

622
00:43:42,000 --> 00:43:48,000
decreasing because the function
is always negative.

623
00:43:46,000 --> 00:43:52,000
Solutions always go down,
always.

624
00:43:49,000 --> 00:43:55,000
The interesting one is this
last one, where I decreased it

625
00:43:54,000 --> 00:44:00,000
just to (h)m.
And, what happens there is

626
00:43:57,000 --> 00:44:03,000
there is this certain,
magic critical point whose

627
00:44:01,000 --> 00:44:07,000
value we could calculate.
There's one constant solution.

628
00:44:07,000 --> 00:44:13,000
So, this is one that has the
value.

629
00:44:10,000 --> 00:44:16,000
Sorry, I'm calculating the
solutions out.

630
00:44:13,000 --> 00:44:19,000
So, y here and t here,
so here it is value,

631
00:44:17,000 --> 00:44:23,000
(h)m is the value by which it
has been lowered.

632
00:44:20,000 --> 00:44:26,000
So, this is the picture for
(h)m.

633
00:44:23,000 --> 00:44:29,000
And, how do the solutions look?
Well, to the right of that,

634
00:44:29,000 --> 00:44:35,000
they are decreasing.
And, to the left they are also

635
00:44:33,000 --> 00:44:39,000
decreasing because this function
is always negative.

636
00:44:37,000 --> 00:44:43,000
So, the solutions look like
this, if you start above,

637
00:44:42,000 --> 00:44:48,000
and if you start below,
they decrease.

638
00:44:45,000 --> 00:44:51,000
And, of course,
they can't get lower than zero

639
00:44:49,000 --> 00:44:55,000
because these are salmon.
What is the significance of

640
00:44:53,000 --> 00:44:59,000
(h)m?
(h)m is the maximum rate of

641
00:44:56,000 --> 00:45:02,000
harvesting.
It's an extremely important

642
00:44:59,000 --> 00:45:05,000
number for this industry.
If the maximum time rate at

643
00:45:05,000 --> 00:45:11,000
which you can pull the salmon
daily out of the water,

644
00:45:09,000 --> 00:45:15,000
and can them without what
happening?

645
00:45:12,000 --> 00:45:18,000
Without the salmon going to
zero.

646
00:45:15,000 --> 00:45:21,000
As long as you start above,
and don't harvest it more than

647
00:45:20,000 --> 00:45:26,000
this rate, it will be following
these curves.

648
00:45:23,000 --> 00:45:29,000
You will be following these
curves, and you will still have

649
00:45:28,000 --> 00:45:34,000
salmon.
If you harvest just a little

650
00:45:31,000 --> 00:45:37,000
bit more, you will be on this
curve that has no critical

651
00:45:36,000 --> 00:45:42,000
points, and the salmon in the
tank will decrease to zero.