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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.
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Notice it's an autonomous
equation all right,
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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,
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there's only one critical
point, and that occurs when y is
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equal to w over r.
So, that's the only critical
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point.
Now, I want to know what's
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happening to the solution.
So, in other words,
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if I plot, I can write away,
of course, negative values
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aren't of particularly
interesting here,
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there is definitely a
horizontal solution,
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and it has the value,
it's at the height,
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w over r.
That's a solution.
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The question is,
what do the other solutions
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look like?
Now, watch how I make the
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analysis because I'm going to
use two now.
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So, this is step one,
then step two.
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What do I do? Well,
I'm going to graph f of y.
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Well, f of y is ry minus w.
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What does that look like?
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Okay, so, here is the y-axis.
Notice the y-axis is going
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horizontally because what I'm
interested in is the graph of
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this function.
What do I call the other axis?
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I'm going to use the same
terminology that is used on the
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little visual that describes
this.
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And, that's dy.
You could call this other axis
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the f of y axis.
That's not a good name for it.
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You could call it the dy / dt
axis because it's,
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so to speak,
the other variable.
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00:18:24 --> 00:18:30
That's not great either.
But, worst of all would be
264
00:18:28 --> 00:18:34
introducing yet another letter
for which we would have no use
265
00:18:32 --> 00:18:38
whatsoever.
So, let's think of it.
266
00:18:37 --> 00:18:43
We are plotting,
now, the graph of f of y.
267
00:18:40 --> 00:18:46
f of y is this function,
268
00:18:43 --> 00:18:49
ry minus w.
Well, that's a line.
269
00:18:48 --> 00:18:54
Its intercept is down here at
w, and so the graph looks
270
00:18:53 --> 00:18:59
something like this.
It's a line.
271
00:18:55 --> 00:19:01
This is the line,
ry minus w.
272
00:18:59 --> 00:19:05
It has slope r.
Well, what am I going to get
273
00:19:05 --> 00:19:11
out of that line?
Just exactly this.
274
00:19:08 --> 00:19:14
What am I interested in about
that line?
275
00:19:13 --> 00:19:19
Nothing other than where is it
above the axis,
276
00:19:18 --> 00:19:24
and where is it below?
This function is positive over
277
00:19:23 --> 00:19:29
here, and therefore,
I'm going to indicate that
278
00:19:28 --> 00:19:34
symbolically,
this, by putting an arrow here.
279
00:19:35 --> 00:19:41
The meeting of this arrow is
that y of t is
280
00:19:39 --> 00:19:45
increasing.
See where it's the right-hand
281
00:19:43 --> 00:19:49
side of that last board?
y of t is increasing when f of
282
00:19:48 --> 00:19:54
y is positive.
f of y is positive here,
283
00:19:52 --> 00:19:58
and therefore,
to the right of this point,
284
00:19:55 --> 00:20:01
it's increasing.
Here to the left of it,
285
00:19:59 --> 00:20:05
f of y is negative,
and therefore over here it's
286
00:20:03 --> 00:20:09
going to be decreasing.
What point is this,
287
00:20:08 --> 00:20:14
in fact?
Well, that's where it crosses
288
00:20:11 --> 00:20:17
the axis.
That's exactly the critical
289
00:20:15 --> 00:20:21
point, w over r.
Therefore, what this says is
290
00:20:19 --> 00:20:25
that a solution,
once it's bigger than y over r,
291
00:20:24 --> 00:20:30
it increases,
and it increases faster and
292
00:20:28 --> 00:20:34
faster because this function is
higher and higher.
293
00:20:34 --> 00:20:40
And, that represents the rate
of change.
294
00:20:37 --> 00:20:43
So, in other words,
once the solution,
295
00:20:40 --> 00:20:46
let's say a solution starts
over here at time zero.
296
00:20:44 --> 00:20:50
So, this is the t axis.
And, here is the y axis.
297
00:20:48 --> 00:20:54
So, now, I'm plotting
solutions.
298
00:20:51 --> 00:20:57
If it starts at t equals zero,
above this line,
299
00:20:55 --> 00:21:01
that is, starts with the value
w over r,
300
00:20:59 --> 00:21:05
which is bigger than zero,
a value bigger than w over r,
301
00:21:04 --> 00:21:10
then it increases,
and increases faster and
302
00:21:07 --> 00:21:13
faster.
If it starts below that,
303
00:21:12 --> 00:21:18
it decreases and decreases
faster and faster.
304
00:21:16 --> 00:21:22
Now, in fact,
I only have to draw two of
305
00:21:20 --> 00:21:26
those because what do all the
others look like?
306
00:21:24 --> 00:21:30
They are translations.
All the other curves look
307
00:21:29 --> 00:21:35
exactly like those.
They are just translations of
308
00:21:35 --> 00:21:41
them.
This guy, if I start closer,
309
00:21:39 --> 00:21:45
it's still going to decrease.
Well, that's supposed to be a
310
00:21:47 --> 00:21:53
translation.
Maybe it is.
311
00:21:50 --> 00:21:56
So, these guys look like that.
Let's do just a tiny bit more
312
00:21:57 --> 00:22:03
interpretation of that.
Well, I think I better leave it
313
00:22:06 --> 00:22:12
there because we've got harder
things to do,
314
00:22:13 --> 00:22:19
and I want to make sure we've
got time for it.
315
00:22:19 --> 00:22:25
Sorry.
Okay, next example,
316
00:22:23 --> 00:22:29
a logistic equation.
Some of you have already solved
317
00:22:31 --> 00:22:37
this in recitation,
and some of you haven't.
318
00:22:39 --> 00:22:45
This is a population equation.
This is the one that section
319
00:22:44 --> 00:22:50
7.1 and section 1.7 is most
heavily concerned with,
320
00:22:48 --> 00:22:54
this particular equation.
The derivation of it is a
321
00:22:52 --> 00:22:58
little vague.
It's an equation which
322
00:22:55 --> 00:23:01
describes how population
increases.
323
00:22:58 --> 00:23:04
And one minute,
the population behavior of some
324
00:23:02 --> 00:23:08
population, --
-- let's call it,
325
00:23:06 --> 00:23:12
y is the only thing I know to
call anything today,
326
00:23:10 --> 00:23:16
but of course your book uses
capital P for population,
327
00:23:15 --> 00:23:21
to get you used to different
variables.
328
00:23:18 --> 00:23:24
Now, the basic population
equation runs dy / dt.
329
00:23:22 --> 00:23:28
There's a certain growth rate.
Let's call it k y.
330
00:23:26 --> 00:23:32
So, k is what's called the
growth rate.
331
00:23:29 --> 00:23:35
It's actually,
sometimes it's talked about in
332
00:23:33 --> 00:23:39
terms of birthrate.
But, it's the net birth rate.
333
00:23:38 --> 00:23:44
It's the rate at which people,
or bacteria,
334
00:23:42 --> 00:23:48
or whatever are being born
minus the rate at which they are
335
00:23:47 --> 00:23:53
dying.
So, it's a net birthrate.
336
00:23:49 --> 00:23:55
But, let's just call it the
growth rate.
337
00:23:53 --> 00:23:59
Now, if this is the equation,
we can think of this,
338
00:23:57 --> 00:24:03
if k is constant,
that's what's called simple
339
00:24:01 --> 00:24:07
population growth.
And you are all familiar with
340
00:24:05 --> 00:24:11
that.
Logistical growth allows for
341
00:24:08 --> 00:24:14
slightly more complex
situations.
342
00:24:13 --> 00:24:19
Logistic growth says that
calling k a constant is
343
00:24:18 --> 00:24:24
unrealistic because the Earth is
not filled entirely with people.
344
00:24:26 --> 00:24:32
What stops it from having
unlimited growth?
345
00:24:32 --> 00:24:38
Well, the fact that the
resources, the food,
346
00:24:36 --> 00:24:42
the organism has to live on
gets depleted.
347
00:24:41 --> 00:24:47
And, in other words,
the growth rate declines as y
348
00:24:46 --> 00:24:52
increases.
As the population increases,
349
00:24:50 --> 00:24:56
one expects the growth rate to
decline because resources are
350
00:24:56 --> 00:25:02
being used up,
and they are not indefinitely
351
00:25:01 --> 00:25:07
available.
Well, in other words,
352
00:25:05 --> 00:25:11
we should replace k by a
function with this behavior.
353
00:25:09 --> 00:25:15
What's the simplest function
that declines as y increases?
354
00:25:14 --> 00:25:20
The simplest choice,
and if you are ignorant about
355
00:25:17 --> 00:25:23
what else to do,
stick with the simplest,
356
00:25:20 --> 00:25:26
at least you won't work any
harder than you have to,
357
00:25:24 --> 00:25:30
would be to take k equal to the
simplest declining function of y
358
00:25:29 --> 00:25:35
there is, which is simply a
linear function,
359
00:25:32 --> 00:25:38
A minus BY.
So, if I use that as the choice
360
00:25:37 --> 00:25:43
of the declining growth rate,
the new equation is dy / dt
361
00:25:41 --> 00:25:47
equals, here's my new k.
The y stays the same,
362
00:25:45 --> 00:25:51
so the equation becomes a minus
by, the quantity times y,
363
00:25:50 --> 00:25:56
or in other words,
364
00:25:53 --> 00:25:59
ay minus b y squared.
365
00:25:55 --> 00:26:01
This equation is what's called
the logistic equation.
366
00:26:01 --> 00:26:07
It has many applications,
not just to population growth.
367
00:26:05 --> 00:26:11
It's applied to the spread of
disease, the spread of a rumor,
368
00:26:11 --> 00:26:17
the spread of many things.
Yeah, a couple pieces of chalk
369
00:26:16 --> 00:26:22
here.
370
00:26:17 --> 00:26:23
371
372
373
00:26:28 --> 00:26:34
Okay, now, those of you who
have solved it know that the
374
00:26:34 --> 00:26:40
explicit solution involves,
well, you separate variables,
375
00:26:40 --> 00:26:46
but you will have to use
partial fractions,
376
00:26:44 --> 00:26:50
ugh, I hope you love partial
fractions.
377
00:26:48 --> 00:26:54
You're going to need them later
in the term.
378
00:26:53 --> 00:26:59
But, I could avoid them now by
not solving the equation
379
00:26:59 --> 00:27:05
explicitly.
But anyway, you get a solution,
380
00:27:03 --> 00:27:09
which I was going to write on
the board for you,
381
00:27:06 --> 00:27:12
but you could look it up in
your book.
382
00:27:09 --> 00:27:15
It's unpleasant enough looking
to make you feel that there must
383
00:27:13 --> 00:27:19
be an easier way at least to get
the basic information out.
384
00:27:16 --> 00:27:22
Okay, let's see if we can get
the basic information out.
385
00:27:20 --> 00:27:26
What are the critical points?
Well, this is pretty easy.
386
00:27:23 --> 00:27:29
A, I want to set the right-hand
side equal to zero.
387
00:27:26 --> 00:27:32
So, I'm going to solve the
equation.
388
00:27:30 --> 00:27:36
I can factor out a y.
It's going to be y times a
389
00:27:35 --> 00:27:41
minus by equals zero.
390
00:27:39 --> 00:27:45
And therefore,
the critical points are where y
391
00:27:43 --> 00:27:49
equals zero. That's one.
392
00:27:47 --> 00:27:53
And, the other factor is when
this factor is zero,
393
00:27:52 --> 00:27:58
and that happens when y is
equal to a over b.
394
00:27:59 --> 00:28:05
So, there are my two critical
points.
395
00:28:03 --> 00:28:09
Okay, what does,
let's start drawing pictures of
396
00:28:08 --> 00:28:14
solutions.
Let's put it in those right
397
00:28:12 --> 00:28:18
away.
Okay, the critical point,
398
00:28:15 --> 00:28:21
zero, gives me a solution that
looks like this.
399
00:28:18 --> 00:28:24
And, the critical point,
a over b,
400
00:28:21 --> 00:28:27
those are positive numbers.
So, that's somewhere up here.
401
00:28:25 --> 00:28:31
So, those are two solutions,
constant solutions.
402
00:28:29 --> 00:28:35
In other words,
if the population by dumb luck
403
00:28:32 --> 00:28:38
started at zero,
it would stay at zero for all
404
00:28:35 --> 00:28:41
time.
That's not terribly surprising.
405
00:28:39 --> 00:28:45
But, it's a little less obvious
that if it starts at that magic
406
00:28:43 --> 00:28:49
number, a over b,
it will also stay at that magic
407
00:28:47 --> 00:28:53
number for all time without
moving up or down or away from
408
00:28:50 --> 00:28:56
it.
Now, the question is,
409
00:28:52 --> 00:28:58
therefore, what happens in
between?
410
00:28:54 --> 00:29:00
So, for the in between,
I'm going to make that same
411
00:28:58 --> 00:29:04
analysis that I made before.
And, it's really not very hard.
412
00:29:03 --> 00:29:09
Look, so here's my dy/dt-axis.
I'll call that y prime,
413
00:29:09 --> 00:29:15
okay?
And, here's the y-axis.
414
00:29:12 --> 00:29:18
So, I'm now doing step two.
This was step one.
415
00:29:16 --> 00:29:22
Okay, the function that I want
to graph is this one,
416
00:29:21 --> 00:29:27
ay minus b y squared,
or in factor form,
417
00:29:26 --> 00:29:32
y times a minus by.
418
00:29:29 --> 00:29:35
Now, this function,
we know, has a zero.
419
00:29:32 --> 00:29:38
It has a zero here,
and it has a zero at the point
420
00:29:37 --> 00:29:43
a over b.
At these two critical points,
421
00:29:43 --> 00:29:49
it has a zero.
What is it doing in between?
422
00:29:46 --> 00:29:52
Well, in between,
it's a parabola.
423
00:29:49 --> 00:29:55
It's a quadratic function.
It's a parabola.
424
00:29:52 --> 00:29:58
Does it go up or does it go
down?
425
00:29:55 --> 00:30:01
Well, when y is very large,
it's very negative.
426
00:29:59 --> 00:30:05
That means it must be a
downward-opening parabola.
427
00:30:04 --> 00:30:10
And therefore,
this curve looks like this.
428
00:30:07 --> 00:30:13
So, I'm interested in knowing,
where is it positive,
429
00:30:12 --> 00:30:18
and where is it negative?
Well, it's positive,
430
00:30:16 --> 00:30:22
here, for this range of values
of y.
431
00:30:19 --> 00:30:25
Since it's positive there,
it will be increasing there.
432
00:30:24 --> 00:30:30
Here, it's negative,
and therefore it will be
433
00:30:28 --> 00:30:34
decreasing.
Here, it's negative,
434
00:30:32 --> 00:30:38
and therefore,
dy / dt will be negative also,
435
00:30:36 --> 00:30:42
and therefore the function,
y, will be decreasing here.
436
00:30:41 --> 00:30:47
So, how do these other
solutions look?
437
00:30:44 --> 00:30:50
Well, we can put them in.
I'll put them in in white,
438
00:30:49 --> 00:30:55
okay, because this has got to
last until the end of the term.
439
00:30:54 --> 00:31:00
So, how are they doing?
They are increasing between the
440
00:30:59 --> 00:31:05
two curves.
They are not allowed to cross
441
00:31:04 --> 00:31:10
either of these yellow curves.
But, they are always
442
00:31:08 --> 00:31:14
increasing.
Well, if they're always
443
00:31:11 --> 00:31:17
increasing, they must start here
and increase,
444
00:31:15 --> 00:31:21
and not allowed to cross.
It must do something like that.
445
00:31:19 --> 00:31:25
This must be a translation of
it.
446
00:31:22 --> 00:31:28
In other words,
the curves must look like that.
447
00:31:26 --> 00:31:32
Those are supposed to be
translations of each other.
448
00:31:32 --> 00:31:38
I know they aren't,
but use your imaginations.
449
00:31:35 --> 00:31:41
But what's happening above?
So in other words,
450
00:31:38 --> 00:31:44
if I start with a population
anywhere bigger than zero but
451
00:31:42 --> 00:31:48
less than a over b,
it increases asymptotically to
452
00:31:46 --> 00:31:52
the level a over b.
What happens if I start above
453
00:31:50 --> 00:31:56
that?
Well, then it decreases to it
454
00:31:52 --> 00:31:58
because, this way,
for the values of y bigger than
455
00:31:56 --> 00:32:02
a over b,
it decreases as time increases.
456
00:32:00 --> 00:32:06
So, these guys up here are
doing this.
457
00:32:04 --> 00:32:10
And, how about the ones below
the axis?
458
00:32:06 --> 00:32:12
Well, they have no physical
significance.
459
00:32:09 --> 00:32:15
But let's put them in anyway.
Whether they doing?
460
00:32:12 --> 00:32:18
They are decreasing away from
zero.
461
00:32:15 --> 00:32:21
So, these guys don't mean
anything physically,
462
00:32:18 --> 00:32:24
but mathematically they exist.
Their solutions,
463
00:32:21 --> 00:32:27
they're going down like that.
Now, you notice from this
464
00:32:25 --> 00:32:31
picture that there are,
even though both of these are
465
00:32:29 --> 00:32:35
constant solutions,
they have dramatically
466
00:32:32 --> 00:32:38
different behavior.
This one, this solution,
467
00:32:37 --> 00:32:43
is the one that all other
solutions try to approach as
468
00:32:41 --> 00:32:47
time goes to infinity.
This one, the solution zero,
469
00:32:45 --> 00:32:51
is repulsive,
as it were.
470
00:32:48 --> 00:32:54
Any solution that starts near
zero, if it starts at zero,
471
00:32:52 --> 00:32:58
of course, it stays there for
all time, but if it starts just
472
00:32:58 --> 00:33:04
a little bit above zero,
it increases to a over b.
473
00:33:02 --> 00:33:08
This is called a stable
474
00:33:05 --> 00:33:11
solution because everybody tries
to get closer and closer to it.
475
00:33:11 --> 00:33:17
This is called,
zero is also a constant
476
00:33:14 --> 00:33:20
solution, but this is an
unstable solution.
477
00:33:17 --> 00:33:23
And now, usually,
solution is too general a word.
478
00:33:21 --> 00:33:27
I think it's better to call it
a stable critical point,
479
00:33:26 --> 00:33:32
and an unstable critical point.
But, of course,
480
00:33:30 --> 00:33:36
it also corresponds to a
solution.
481
00:33:34 --> 00:33:40
So, critical points are not all
the same.
482
00:33:37 --> 00:33:43
Some are stable,
and some are unstable.
483
00:33:40 --> 00:33:46
And, you can see which is which
just by looking at this picture.
484
00:33:46 --> 00:33:52
If the arrows point towards
them, you've got a stable
485
00:33:51 --> 00:33:57
critical point.
If it arrows point away from
486
00:33:55 --> 00:34:01
them, you've got an unstable
critical point.
487
00:33:58 --> 00:34:04
Now, there is a third
possibility.
488
00:34:03 --> 00:34:09
Okay, I think we'd better
address it because otherwise
489
00:34:09 --> 00:34:15
you're going to sit there
wondering, hey,
490
00:34:13 --> 00:34:19
what did he do?
Suppose it looks like this.
491
00:34:18 --> 00:34:24
Suppose it were just tangent.
Well, this is the picture of
492
00:34:24 --> 00:34:30
that curve, the pink curve.
What would the arrows look like
493
00:34:31 --> 00:34:37
then?
What would the arrows look like
494
00:34:35 --> 00:34:41
then?
Well, since they are positive,
495
00:34:38 --> 00:34:44
it's always positive,
the arrow goes like this.
496
00:34:41 --> 00:34:47
And then on the side,
it also goes in the same
497
00:34:45 --> 00:34:51
direction.
So, is this critical point
498
00:34:47 --> 00:34:53
stable or unstable?
It's stable if you approach it
499
00:34:51 --> 00:34:57
from the left.
So, how, in fact,
500
00:34:53 --> 00:34:59
do the curves,
how would the corresponding
501
00:34:57 --> 00:35:03
curves look?
Well, there's our long-term
502
00:35:00 --> 00:35:06
solution.
This corresponds to that point.
503
00:35:05 --> 00:35:11
Let's call this a,
and then this will be the
504
00:35:10 --> 00:35:16
value, a.
If I start below it,
505
00:35:14 --> 00:35:20
I rise to it.
If I start above it,
506
00:35:18 --> 00:35:24
I increase.
So, if I start above it,
507
00:35:22 --> 00:35:28
I do this.
Well, now, that's stable on one
508
00:35:27 --> 00:35:33
side, and unstable on the other.
And, that's indicated by saying
509
00:35:35 --> 00:35:41
it's semi-stable.
That's a brilliant word.
510
00:35:40 --> 00:35:46
I wonder how long it to do
think that one up,
511
00:35:43 --> 00:35:49
semi-stable critical point:
stable on one side,
512
00:35:46 --> 00:35:52
unstable on the other depending
on whether you start below it.
513
00:35:50 --> 00:35:56
And, of course,
it could be reversed if I had
514
00:35:53 --> 00:35:59
drawn the picture the other way.
I could have approached it from
515
00:35:57 --> 00:36:03
the top, and left it from below.
You get the idea of the
516
00:36:03 --> 00:36:09
behavior.
Okay, let's now take,
517
00:36:08 --> 00:36:14
I'm going to soup up this
logistic equation just a little
518
00:36:16 --> 00:36:22
bit more.
So, let's talk about the
519
00:36:21 --> 00:36:27
logistic equation.
But, I'm going to add to it
520
00:36:28 --> 00:36:34
harvesting, with harvesting.
So, this is a very late 20th
521
00:36:36 --> 00:36:42
century concept.
So, we imagine,
522
00:36:39 --> 00:36:45
for example,
a bunch of formerly free range
523
00:36:43 --> 00:36:49
Atlantic salmon penned in one of
these huge factory farms off the
524
00:36:49 --> 00:36:55
coast of Maine or someplace.
They've made salmon much
525
00:36:54 --> 00:37:00
cheaper than it used to be,
but at a certain cost to the
526
00:37:00 --> 00:37:06
salmon, and possibly to our
environment.
527
00:37:05 --> 00:37:11
So, what happens?
Well, the salmon grow,
528
00:37:08 --> 00:37:14
and grow, and do what salmon
do.
529
00:37:11 --> 00:37:17
And, they are harvested.
That's a word somewhere in the
530
00:37:15 --> 00:37:21
category of ethnic cleansing in
my opinion.
531
00:37:19 --> 00:37:25
But, it's, again,
a very 20th-century word.
532
00:37:23 --> 00:37:29
I think it was Hitler who
discovered that,
533
00:37:27 --> 00:37:33
that all you had to do was call
something by a sanitary name,
534
00:37:32 --> 00:37:38
and no matter how horrible it
was, good bourgeois people would
535
00:37:37 --> 00:37:43
accept it.
So, the harvesting,
536
00:37:42 --> 00:37:48
which means,
of course, picking them up and
537
00:37:47 --> 00:37:53
killing them,
and putting them in cans and
538
00:37:52 --> 00:37:58
stuff like that,
okay, so what's the equation?
539
00:37:58 --> 00:38:04
I'm going to assume that the
harvest is at a constant time
540
00:38:05 --> 00:38:11
rate.
In other words,
541
00:38:08 --> 00:38:14
it's not a certain fraction of
all the salmon that are being
542
00:38:13 --> 00:38:19
caught each day and canned.
The factory has a certain
543
00:38:17 --> 00:38:23
capacity, so,
400 pounds of salmon each day
544
00:38:21 --> 00:38:27
are pulled out and canned.
So, it's a constant time rate.
545
00:38:25 --> 00:38:31
That means that the equation is
now going to be dy/dt is equal
546
00:38:31 --> 00:38:37
to, well, salmon grow
logistically.
547
00:38:35 --> 00:38:41
ay minus b y squared,
so, that part of the
548
00:38:39 --> 00:38:45
equation is the same.
But, I need a term to take care
549
00:38:44 --> 00:38:50
of this constant harvesting
rate, and that will be h.
550
00:38:48 --> 00:38:54
Let's call it h,
not h times y.
551
00:38:50 --> 00:38:56
Then, I would be harvesting a
certain fraction of all the
552
00:38:55 --> 00:39:01
salmon there,
which is not what I'm doing.
553
00:39:00 --> 00:39:06
Okay: our equation.
Now, I want to analyze what the
554
00:39:03 --> 00:39:09
critical points of this look
like.
555
00:39:05 --> 00:39:11
Now, this is a little more
subtle because there's now a new
556
00:39:09 --> 00:39:15
parameter, there.
And, what I want to see is how
557
00:39:12 --> 00:39:18
that varies with the new
parameter.
558
00:39:15 --> 00:39:21
The best thing to do is,
I mean, the thing not to do is
559
00:39:19 --> 00:39:25
make this equal to zero,
fiddle around with the
560
00:39:22 --> 00:39:28
quadratic formula,
get some massive expression,
561
00:39:25 --> 00:39:31
and then spend the next half
hour scratching your head trying
562
00:39:29 --> 00:39:35
to figure out what it means,
and what information you are
563
00:39:33 --> 00:39:39
supposed to be getting out of
it.
564
00:39:37 --> 00:39:43
Draw pictures instead.
Draw pictures.
565
00:39:40 --> 00:39:46
If h is zero,
that's the smallest harvesting
566
00:39:45 --> 00:39:51
rate I could have.
The picture looks like our old
567
00:39:50 --> 00:39:56
one.
So, if h is zero,
568
00:39:52 --> 00:39:58
the picture looks like,
what color did I,
569
00:39:56 --> 00:40:02
okay, pink.
Yellow.
570
00:40:00 --> 00:40:06
Yellow is the cheapest,
but I can't find it.
571
00:40:03 --> 00:40:09
Okay, yellow is commercially
available.
572
00:40:06 --> 00:40:12
These are precious.
All right, purple if it's okay,
573
00:40:11 --> 00:40:17
purple.
So, this is the one,
574
00:40:13 --> 00:40:19
our original one corresponding
to h equals zero.
575
00:40:17 --> 00:40:23
Or, in other words,
it's the equation ay minus b y
576
00:40:21 --> 00:40:27
squared. h is zero.
577
00:40:24 --> 00:40:30
Now, if I want to find,
I now want to increase the
578
00:40:28 --> 00:40:34
value of h, well,
if I increase the value of h,
579
00:40:32 --> 00:40:38
in other words,
harvest more and more,
580
00:40:35 --> 00:40:41
what's happening?
Well, I simply lower this
581
00:40:41 --> 00:40:47
function by h.
So, if I lower h somewhat,
582
00:40:45 --> 00:40:51
it will come to here.
So, this is some value,
583
00:40:49 --> 00:40:55
ay minus b y squared minus h1,
584
00:40:54 --> 00:41:00
let's say.
That's this curve.
585
00:40:57 --> 00:41:03
If I lower it a lot,
it will look like this.
586
00:41:03 --> 00:41:09
So, ay minus b y squared minus
h a lot, h twenty.
587
00:41:08 --> 00:41:14
This doesn't mean anything.
588
00:41:11 --> 00:41:17
Two.
Obviously, there's one
589
00:41:14 --> 00:41:20
interesting value to lower it
by.
590
00:41:17 --> 00:41:23
It's a value which would lower
it exactly by this amount.
591
00:41:22 --> 00:41:28
Let me put that in special.
If I lower it by just that
592
00:41:27 --> 00:41:33
amount, the curve always looks
the same.
593
00:41:32 --> 00:41:38
It's just been lowered.
I'm going to say this one is,
594
00:41:36 --> 00:41:42
so this one is the same thing,
except that I've subtracted h
595
00:41:42 --> 00:41:48
sub m. Where is h sub m on
596
00:41:46 --> 00:41:52
the picture?
Well, I lowered it by this
597
00:41:49 --> 00:41:55
amount.
So, this height is h sub m.
598
00:41:53 --> 00:41:59
In other words,
if I find the maximum height
599
00:41:57 --> 00:42:03
here, which is easy to do
because it's a parabola,
600
00:42:01 --> 00:42:07
and lower it by exactly that
amount, I will have lowered it
601
00:42:07 --> 00:42:13
to this point.
This will be a critical point.
602
00:42:12 --> 00:42:18
Now, the question is,
what's happened to the critical
603
00:42:16 --> 00:42:22
point as I did this?
I started with the critical
604
00:42:20 --> 00:42:26
points here and here.
As I lower h,
605
00:42:22 --> 00:42:28
the critical point changed to
this and that.
606
00:42:26 --> 00:42:32
And now, it changed to this one
when I got to the purple line.
607
00:42:32 --> 00:42:38
And, as I went still further
down, there were no critical
608
00:42:36 --> 00:42:42
points.
So, this curve has no critical
609
00:42:38 --> 00:42:44
points attached to it.
What are the corresponding
610
00:42:42 --> 00:42:48
pictures?
Well, the corresponding
611
00:42:44 --> 00:42:50
pictures, well,
we've already drawn,
612
00:42:47 --> 00:42:53
the picture for h equals zero
is drawn already.
613
00:42:51 --> 00:42:57
The pictures that I'm talking
about are how the solutions
614
00:42:55 --> 00:43:01
look.
How would the solution look
615
00:42:57 --> 00:43:03
like for this one for h one?
616
00:43:00 --> 00:43:06
For h1, the solutions look
like, here is a over b.
617
00:43:06 --> 00:43:12
Here is a over b,
but the critical points aren't
618
00:43:10 --> 00:43:16
at zero and a over b anymore.
They've moved in a little bit.
619
00:43:15 --> 00:43:21
So, they are here and here.
And, otherwise,
620
00:43:18 --> 00:43:24
the solutions look just like
they did before,
621
00:43:22 --> 00:43:28
and the analysis is the same.
And, similarly,
622
00:43:25 --> 00:43:31
if h two goes very far,
if h2 is very large,
623
00:43:29 --> 00:43:35
there are no critical points.
h, too large,
624
00:43:33 --> 00:43:39
no critical points.
Are the solutions decreasing
625
00:43:38 --> 00:43:44
all the time or increasing?
Well, they are always
626
00:43:42 --> 00:43:48
decreasing because the function
is always negative.
627
00:43:46 --> 00:43:52
Solutions always go down,
always.
628
00:43:49 --> 00:43:55
The interesting one is this
last one, where I decreased it
629
00:43:54 --> 00:44:00
just to (h)m.
And, what happens there is
630
00:43:57 --> 00:44:03
there is this certain,
magic critical point whose
631
00:44:01 --> 00:44:07
value we could calculate.
There's one constant solution.
632
00:44:07 --> 00:44:13
So, this is one that has the
value.
633
00:44:10 --> 00:44:16
Sorry, I'm calculating the
solutions out.
634
00:44:13 --> 00:44:19
So, y here and t here,
so here it is value,
635
00:44:17 --> 00:44:23
(h)m is the value by which it
has been lowered.
636
00:44:20 --> 00:44:26
So, this is the picture for
(h)m.
637
00:44:23 --> 00:44:29
And, how do the solutions look?
Well, to the right of that,
638
00:44:29 --> 00:44:35
they are decreasing.
And, to the left they are also
639
00:44:33 --> 00:44:39
decreasing because this function
is always negative.
640
00:44:37 --> 00:44:43
So, the solutions look like
this, if you start above,
641
00:44:42 --> 00:44:48
and if you start below,
they decrease.
642
00:44:45 --> 00:44:51
And, of course,
they can't get lower than zero
643
00:44:49 --> 00:44:55
because these are salmon.
What is the significance of
644
00:44:53 --> 00:44:59
(h)m?
(h)m is the maximum rate of
645
00:44:56 --> 00:45:02
harvesting.
It's an extremely important
646
00:44:59 --> 00:45:05
number for this industry.
If the maximum time rate at
647
00:45:05 --> 00:45:11
which you can pull the salmon
daily out of the water,
648
00:45:09 --> 00:45:15
and can them without what
happening?
649
00:45:12 --> 00:45:18
Without the salmon going to
zero.
650
00:45:15 --> 00:45:21
As long as you start above,
and don't harvest it more than
651
00:45:20 --> 00:45:26
this rate, it will be following
these curves.
652
00:45:23 --> 00:45:29
You will be following these
curves, and you will still have
653
00:45:28 --> 00:45:34
salmon.
If you harvest just a little
654
00:45:31 --> 00:45:37
bit more, you will be on this
curve that has no critical
655
00:45:36 --> 00:45:42
points, and the salmon in the
tank will decrease to zero.