WEBVTT
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PROFESSOR: OK.
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So we've moved on
into Chapter 3.
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Chapter 1 and 2 were about
equations we could solve,
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first order equations,
chapter one;
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second order equations
in chapter 2, often
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linear, constant
coefficient sometimes.
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Now we take any equation.
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And I'll start with first order.
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First derivative is
some function and not
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a linear function, so I
don't expect a formula.
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A solution will exist.
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But I won't have a
formula for the solution.
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But I can make a
picture of the solution.
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You see what's happening
as time goes on.
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And so that's today's
lecture, is a picture.
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So this function, whatever
it is, gives the slope of y.
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That's the slope.
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And it will be the
slope of the arrows
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that I will draw
in this picture.
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So here's a picture
that started, y, t.
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And the slope of
the arrows is f.
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And here is my example.
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Well, you will see.
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I chose a constant
coefficient linear equation.
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Because I could find a solution.
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So 2 minus y, I know
from that minus sign
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that I'm going to
have exponential decay
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in the null solution.
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And then y equal to 2 is a very
special particular solution,
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a constant.
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And in my picture, y
equal to 2, it jumps out.
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Because when y is 2, when
y is 2 the slope is 0.
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So all my arrows on the y
equal to 2 line, have slope 0.
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So that's a very special line.
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And since the solution
follows the arrows
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that's the whole point.
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The solution follows the arrows.
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Because the arrows
tell the slope.
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So if I'm on that
line, the solution
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just follows those arrows,
and stays on the line.
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y equal to 2 is a fixed
point, fixed point
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of the solution, a fixed
point for the equation.
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And the question is, if I
don't start at y equal to 2,
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do I move toward
2 or away from it?
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OK.
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So I can see from the formula
what the answer is going to be.
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If I start with some other
value, some other value of c
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not 0, then there will
be a null solution part.
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But as t gets large
that goes to 0.
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So I move toward 2.
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Now let's see that
in the picture.
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So let me-- I'm drawing
the arrows first.
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So this is all time starting
at if y is 0, then if y is 0
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then dy dt is 2.
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So I draw arrows with slope
2, along the y equals 0 line.
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This is the y equals 0 line.
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All my arrows have slope 2.
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Now what else?
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So that's a few arrows that
show what will-- so the solution
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if it starts there, will start
in the direction of that arrow.
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But then I have to see
what the other arrows are
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for other values of y.
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Because right away the
solution y will change.
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And the slope will change.
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And that's it needs more arrows.
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Well, actually it
needs way more arrows
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than I can possibly draw.
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Let me draw another
line of arrows
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when y is 1, along that line,
along the line y equal 1.
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When y is 1, 2 minus 1 is 1.
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The slope is 1.
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f is 1.
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And my arrows have slope 1.
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So all along here,
the arrows go up.
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Those went up
steeply with slope 2.
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Now the arrows will go up.
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So I'll have arrows that are
going a 45-degree angle, slope
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1.
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Do you see?
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I hope you begin to
see the picture here.
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The solution might start there.
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It would start with that slope.
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But it will curve down.
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Because the arrows
are not so steep.
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As I go upward, the
arrows are getting flat.
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And so the curve that
follows the arrows
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has to flatten out,
flatten out, flatten out.
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The arrows are
still, at that point
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the arrows are still slope 1.
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But it's flattening out.
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And it's never going
to cross this line.
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And it will run closer
and closer to that line.
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And wherever it
starts, if it starts
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at time t equals to 1 there,
it'll do the same thing.
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And it will stay just
below the other one.
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Do you see what the
pictures are looking like?
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If it starts at different times,
so these are different times.
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These are different starts.
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Yeah, really we're
used to, at t equals 0,
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we're used to giving y of 0.
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So this is starting at 0.
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This is starting at 2.
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Starting at 1 would
be a higher start.
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What about starting at 4?
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Suppose y of 0 is 4.
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That point is t
equals 0, y equal 4.
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So that point is y of 0 equal 4.
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What's the graph of the
solution with that start?
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Actually, I could
figure out what
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the solution would be if y
of 0 was 4, I'd have 2 plus 2
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e to the minus t.
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At t equals 0, that's 4.
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And it fits.
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It solves the equation.
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And it's going to be its graph.
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I should be able learn
that from the arrows.
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So along this line of y equal
4, all the arrows when y is 4,
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the slope is minus 2.
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So these arrows
from these points,
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go down with slope minus 2.
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But the solution starts down.
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So it starts like that.
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But then it has to
follow the new arrows.
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And the new arrows
are not so steep.
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So the new arrows are
I have slope minus 1.
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I hope my picture is
showing the steeper slope 2
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along this line, and
the flatter slope 1,
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or rather minus 1
downwards, along this line.
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So it just follows along here.
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Well of course it's just a
mirror image of that one.
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It's a mirror image of that one.
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I'm trying to show the
graph of all solutions
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from all starts,
the whole plane.
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Actually I could go, t
could go to minus infinity.
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And y could go all the way
from minus infinity up,
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all the way up.
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I could fill the whole board
here with arrows, and then
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with solutions.
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And the solutions would follow
the arrows, the arrows changing
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slope and actually in this
case, all solutions wherever
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you started, would approach 2.
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And that's what
the formula says.
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But we get that information
from the arrows with no formula.
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Let me show you a next example.
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And here's our next example.
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The logistic equation,
it's not linear.
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So it's going to be
more interesting.
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And do you remember
the solution?
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You remember maybe the trick
with the logistic equation
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was 1 over the solution, gave a
linear equation and expression
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like that.
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OK.
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Time to draw arrows.
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OK.
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When y is 0-- so here's y--
when y is 0, the slope is 0.
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So I have a whole line of flat.
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I have a flat horizontal line.
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That's the solution, y
equals 0 fixed point.
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Also we have
another fixed point.
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When y is 1, 1 minus 1 is 0.
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Slope is 0.
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Slope stays 0.
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The arrows all have zero slope
along the line y equal 1.
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So there is another
solution, which
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doesn't do anything exciting.
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It just stays at 1.
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y equal 1 is another
fixed point, steady state,
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whatever words we want to use.
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But again, the real picture
is what about other starts.
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What about a start at 1/2?
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Well if it starts at-- if y is
1/2 half at the starting time,
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what is the slope?
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1/2 minus 1/4 is 1/4.
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So the slope is upwards,
but not very steep.
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The slope along
the-- and it doesn't
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depend on t in these examples.
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So that slope is the
same as long as y is 1/2,
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doesn't matter what the time is.
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y equals 1/2 gives me a 1/2
minus 1/4, which is a 1/4.
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It gives me that slope.
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What about the slope 1/4?
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So 1/4, I have 1/4 minus 1/16.
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I think that's 3/16.
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So it's beginning
to climb upward.
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So it's upwards again.
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But 3/16 is a
little-- I don't know
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if I'm going to get the
picture too brilliantly.
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The slope, as soon it-- if it
just starts a little above 0,
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what happens to
the solution that
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starts a little bit above 0?
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It climbs.
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Because if y is above 0, say
if it starts between 0 and 1,
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if y is between 0 and 1, then
y is bigger than y squared.
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And the slope is positive.
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And it goes up.
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So do you see what it's doing?
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The slope will just, if it
starts a little bit above,
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it'll have a small slope.
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But that slope will
gradually increase.
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But then actually at
this point, the slope
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is 3/4 minus whatever it is.
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It slows down,
still going upwards.
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y is still bigger
than y squared.
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You recognize what the
curve is going to look like.
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So there is an S curve.
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It's an S curve.
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Which we saw for the
logistic equation, and here
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we have a formula for it.
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Well, the whole point
of today's video
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was we don't need a formula.
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So you don't need that.
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The arrows will tell you
that it starts up slowly.
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It gets only-- that's the
biggest slope it gets.
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And then it starts down.
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The slope goes down again.
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But it's still a positive slope.
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Still climbing, climbing,
climbing, and approaching 1.
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Now that's sort of a
sandwich in the picture.
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But it could start
with a negative.
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So what happens if it
starts at y equal minus 1?
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The slope, if y is minus
1, we have minus 1,
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minus 1, a slope of minus 2.
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That's a steeper
serious downward slope.
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So the solution that starts
here has-- that's tangent.
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You see that it's
tangent to the arrow,
00:13:15.490 --> 00:13:18.540
because it has the same
slope as the arrow.
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And it comes down.
00:13:19.890 --> 00:13:23.870
But as it goes down, the
slopes are getting steeper.
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Whoops, not flatter,
but steeper.
00:13:29.120 --> 00:13:34.850
For example, if y is minus
2, I have minus 2, minus 4
00:13:34.850 --> 00:13:36.340
is minus 6.
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So as soon as it
gets down to minus 2,
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the slope has jumped
way down to minus 6.
00:13:42.500 --> 00:13:46.480
So here is the--
it falls right off.
00:13:46.480 --> 00:13:49.690
It's a drop-off curve,
a drop-off curve.
00:13:49.690 --> 00:13:52.040
It falls right off
actually to infinity.
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It never makes it out
to-- it falls down
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to y equal minus infinity in a
fixed time, in a definite time.
00:14:01.330 --> 00:14:06.140
And so here's a whole
region of curves going down
00:14:06.140 --> 00:14:07.610
to minus infinity.
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Here is a whole region.
00:14:10.540 --> 00:14:11.970
What happens in this region?
00:14:11.970 --> 00:14:16.850
Suppose y starts at plus 2?
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Well, I have 2 minus 4.
00:14:19.110 --> 00:14:20.730
So the slope is negative.
00:14:20.730 --> 00:14:22.180
The slope is negative up here.
00:14:22.180 --> 00:14:22.740
Yeah.
00:14:22.740 --> 00:14:24.450
And this is the big picture.
00:14:24.450 --> 00:14:29.570
The slope, the arrows are
positive below this line.
00:14:29.570 --> 00:14:31.120
They're upward.
00:14:31.120 --> 00:14:33.120
They were downward here.
00:14:33.120 --> 00:14:36.580
They're slowly upward
in this sandwich.
00:14:36.580 --> 00:14:40.500
And then up above,
they're downward again.
00:14:40.500 --> 00:14:51.920
So if slopes are coming down,
and they drop into actually
00:14:51.920 --> 00:14:54.530
it's a symmetric picture.
00:14:54.530 --> 00:14:59.210
Really-- no reason not
to go backwards in time.
00:14:59.210 --> 00:15:01.000
Where are these coming from?
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They're all coming from curves.
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The whole plane
is full of curves.
00:15:05.240 --> 00:15:08.180
And these start
at plus infinity.
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They drop into 2.
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These start below 0, and they
drop off to minus infinity.
00:15:18.380 --> 00:15:24.580
And then the real interest in
studying population was these.
00:15:24.580 --> 00:15:26.250
Can you do one more example?
00:15:26.250 --> 00:15:32.760
Let me take a third example
that has a t in the function.
00:15:32.760 --> 00:15:36.480
So the arrows won't be the
same along the whole line.
00:15:36.480 --> 00:15:40.630
In fact, the arrows
will be the same.
00:15:40.630 --> 00:15:47.460
So if I have 1 plus t minus y,
that's the f, equal a constant.
00:15:49.990 --> 00:15:55.340
Then that's a curve-- well,
it's actually a straight line.
00:15:55.340 --> 00:15:58.880
It's actually a 45-degree
line in this plane.
00:15:58.880 --> 00:16:08.610
And along that line the f, this
is the f, the f of t and y,
00:16:08.610 --> 00:16:10.090
the arrow slope.
00:16:10.090 --> 00:16:13.760
The arrows slopes are
the same along that line.
00:16:13.760 --> 00:16:16.740
That line is called an isocline.
00:16:16.740 --> 00:16:22.965
This is called an I-S-O, meaning
the same, cline, meaning slope.
00:16:25.610 --> 00:16:26.700
So that's an isocline.
00:16:26.700 --> 00:16:28.460
Here's an isocline.
00:16:28.460 --> 00:16:30.670
It's a 45-degree line.
00:16:30.670 --> 00:16:38.170
That's the 45-degree line,
1 plus t minus y equal 1.
00:16:38.170 --> 00:16:48.910
Let me draw the 45-degree line
1 plus t minus y equals 0.
00:16:48.910 --> 00:16:50.660
So it's a little bit higher.
00:16:50.660 --> 00:16:51.160
OK.
00:16:54.110 --> 00:16:59.680
Now arrows, and then put
in the curves, the solution
00:16:59.680 --> 00:17:02.400
curves that match the arrows.
00:17:02.400 --> 00:17:07.920
So the arrows have this
slope along that line.
00:17:07.920 --> 00:17:11.380
Along this line, 1 plus t
minus y, they have slope 0.
00:17:11.380 --> 00:17:12.829
Oh, interesting.
00:17:12.829 --> 00:17:15.329
At every point on the
line the slope is 0.
00:17:18.060 --> 00:17:21.210
Because this is the
slope of the arrows.
00:17:21.210 --> 00:17:24.650
At every point on
this line, the slope
00:17:24.650 --> 00:17:27.440
is 1, also very interesting.
00:17:27.440 --> 00:17:29.710
Because that's right
along the line.
00:17:29.710 --> 00:17:33.910
So here we have a solution line.
00:17:33.910 --> 00:17:35.350
That must be a solution line.
00:17:35.350 --> 00:17:39.210
That's the line where y is t.
00:17:39.210 --> 00:17:43.860
That's a very big
45-degree important line.
00:17:43.860 --> 00:17:48.920
Because if y equals
t, if y equals t
00:17:48.920 --> 00:17:51.440
then dy dt should be 1.
00:17:51.440 --> 00:17:53.560
And it is 1 for y equals t.
00:17:53.560 --> 00:17:57.260
So that's a solution
line with that solution.
00:17:57.260 --> 00:18:02.340
Now what about a line
with 1 plus t minus y
00:18:02.340 --> 00:18:04.710
equal minus 1, a line?
00:18:04.710 --> 00:18:10.410
If 1 plus t minus y is
minus 1, if f is minus 1,
00:18:10.410 --> 00:18:14.420
the slope is negative.
00:18:14.420 --> 00:18:15.930
So what does that mean?
00:18:15.930 --> 00:18:21.110
If 1 plus t minus y is minus
1, the slope is negative.
00:18:21.110 --> 00:18:27.300
So at points on this line,
the slope is going downwards.
00:18:27.300 --> 00:18:29.550
Oh, interesting.
00:18:29.550 --> 00:18:31.380
I wasn't quite expecting that.
00:18:31.380 --> 00:18:35.230
Let me just see if I
got a suitable picture.
00:18:35.230 --> 00:18:37.700
Why is it not right?
00:18:37.700 --> 00:18:44.660
If 1 plus t minus y
is-- oh, I'm sorry.
00:18:44.660 --> 00:18:49.540
This is the line,
y equal 1 plus t.
00:18:49.540 --> 00:18:52.090
I think what I'm
expecting to see
00:18:52.090 --> 00:18:56.260
is I'm expecting to
see it from the formula
00:18:56.260 --> 00:19:01.700
too that as time goes
on, this part goes to 0,
00:19:01.700 --> 00:19:03.180
and y goes to t.
00:19:03.180 --> 00:19:06.140
I believe that all
the solutions will
00:19:06.140 --> 00:19:09.803
approach this y equal to t.
00:19:09.803 --> 00:19:12.350
I think their
slopes, their slopes
00:19:12.350 --> 00:19:15.610
here-- darn, that's not right.
00:19:15.610 --> 00:19:20.090
Their slopes should
be coming upwards.
00:19:20.090 --> 00:19:22.460
Yeah, let me-- I
can figure that out.
00:19:22.460 --> 00:19:25.740
If t is let's say 1, and y is 0.
00:19:25.740 --> 00:19:26.340
OK.
00:19:26.340 --> 00:19:30.180
If t is 1, and y is 0,
I have a slope of 2.
00:19:30.180 --> 00:19:30.930
Good.
00:19:30.930 --> 00:19:31.800
OK.
00:19:31.800 --> 00:19:35.370
There's a point t equal to 1.
00:19:35.370 --> 00:19:37.700
Here is 0, 0.
00:19:37.700 --> 00:19:41.050
Here's the point t
equal to 1, y equals 0.
00:19:41.050 --> 00:19:43.070
The slope came out to be 2.
00:19:43.070 --> 00:19:44.820
It went up that way.
00:19:44.820 --> 00:19:48.620
So along that line, the
slopes are going up.
00:19:51.260 --> 00:19:54.550
Along this line, the slopes
are right on the line.
00:19:54.550 --> 00:19:57.390
On this line the
slopes are flat,
00:19:57.390 --> 00:20:01.130
and the curve is
moving toward the line.
00:20:01.130 --> 00:20:06.230
I'll just draw the beautiful
picture now of the solution.
00:20:06.230 --> 00:20:10.080
So the solutions look like this.
00:20:10.080 --> 00:20:14.442
They are-- this is the big line.
00:20:14.442 --> 00:20:16.500
You've got to keep
your eye on that line.
00:20:16.500 --> 00:20:19.800
Because that's the steady
state line that all solutions
00:20:19.800 --> 00:20:22.610
are approaching.
00:20:22.610 --> 00:20:28.630
So if you have the idea of
arrows to show the slope,
00:20:28.630 --> 00:20:34.070
fitting solution curves
through tangent to the arrows,
00:20:34.070 --> 00:20:37.260
and sometimes having
a formula to confirm
00:20:37.260 --> 00:20:42.870
that you did it right, you
get a picture like this.
00:20:42.870 --> 00:20:47.030
So that's the idea of first
order equations, which
00:20:47.030 --> 00:20:50.600
are graphed in the y-t plane.
00:20:50.600 --> 00:20:54.650
And the arrows tell
you the derivative.
00:20:54.650 --> 00:20:56.200
Thanks.