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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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00:05:03,980 --> 00:05:08,400
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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00:05:17,740 --> 00:05:20,850
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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00:05:40,110 --> 00:05:43,170
Do you see what the
pictures are looking like?
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00:05:43,170 --> 00:05:49,350
If it starts at different times,
so these are different times.
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00:05:49,350 --> 00:05:51,850
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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00:09:16,540 --> 00:09:18,690
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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00:10:13,950 --> 00:10:18,180
The slope along
the-- and it doesn't
176
00:10:18,180 --> 00:10:21,080
depend on t in these examples.
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00:10:21,080 --> 00:10:27,420
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.
186
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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
190
00:11:12,070 --> 00:11:15,060
starts a little bit above 0?
191
00:11:15,060 --> 00:11:16,500
It climbs.
192
00:11:16,500 --> 00:11:24,390
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.
194
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And the slope is positive.
195
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And it goes up.
196
00:11:33,710 --> 00:11:35,380
So do you see what it's doing?
197
00:11:35,380 --> 00:11:38,200
The slope will just, if it
starts a little bit above,
198
00:11:38,200 --> 00:11:39,720
it'll have a small slope.
199
00:11:39,720 --> 00:11:42,230
But that slope will
gradually increase.
200
00:11:42,230 --> 00:11:45,860
But then actually at
this point, the slope
201
00:11:45,860 --> 00:11:48,660
is 3/4 minus whatever it is.
202
00:11:48,660 --> 00:11:53,030
It slows down,
still going upwards.
203
00:11:53,030 --> 00:11:55,000
y is still bigger
than y squared.
204
00:11:57,570 --> 00:12:03,210
You recognize what the
curve is going to look like.
205
00:12:03,210 --> 00:12:06,550
So there is an S curve.
206
00:12:06,550 --> 00:12:07,500
It's an S curve.
207
00:12:13,180 --> 00:12:16,500
Which we saw for the
logistic equation, and here
208
00:12:16,500 --> 00:12:18,920
we have a formula for it.
209
00:12:18,920 --> 00:12:21,970
Well, the whole point
of today's video
210
00:12:21,970 --> 00:12:24,180
was we don't need a formula.
211
00:12:24,180 --> 00:12:26,560
So you don't need that.
212
00:12:26,560 --> 00:12:30,920
The arrows will tell you
that it starts up slowly.
213
00:12:30,920 --> 00:12:35,020
It gets only-- that's the
biggest slope it gets.
214
00:12:35,020 --> 00:12:36,560
And then it starts down.
215
00:12:36,560 --> 00:12:38,080
The slope goes down again.
216
00:12:38,080 --> 00:12:40,360
But it's still a positive slope.
217
00:12:40,360 --> 00:12:45,440
Still climbing, climbing,
climbing, and approaching 1.
218
00:12:45,440 --> 00:12:49,350
Now that's sort of a
sandwich in the picture.
219
00:12:49,350 --> 00:12:53,470
But it could start
with a negative.
220
00:12:53,470 --> 00:12:56,850
So what happens if it
starts at y equal minus 1?
221
00:12:56,850 --> 00:13:00,450
The slope, if y is minus
1, we have minus 1,
222
00:13:00,450 --> 00:13:02,820
minus 1, a slope of minus 2.
223
00:13:02,820 --> 00:13:08,310
That's a steeper
serious downward slope.
224
00:13:08,310 --> 00:13:12,950
So the solution that starts
here has-- that's tangent.
225
00:13:12,950 --> 00:13:15,490
You see that it's
tangent to the arrow,
226
00:13:15,490 --> 00:13:18,540
because it has the same
slope as the arrow.
227
00:13:18,540 --> 00:13:19,890
And it comes down.
228
00:13:19,890 --> 00:13:23,870
But as it goes down, the
slopes are getting steeper.
229
00:13:23,870 --> 00:13:26,315
Whoops, not flatter,
but steeper.
230
00:13:29,120 --> 00:13:34,850
For example, if y is minus
2, I have minus 2, minus 4
231
00:13:34,850 --> 00:13:36,340
is minus 6.
232
00:13:36,340 --> 00:13:38,770
So as soon as it
gets down to minus 2,
233
00:13:38,770 --> 00:13:42,500
the slope has jumped
way down to minus 6.
234
00:13:42,500 --> 00:13:46,480
So here is the--
it falls right off.
235
00:13:46,480 --> 00:13:49,690
It's a drop-off curve,
a drop-off curve.
236
00:13:49,690 --> 00:13:52,040
It falls right off
actually to infinity.
237
00:13:52,040 --> 00:13:55,320
It never makes it out
to-- it falls down
238
00:13:55,320 --> 00:14:01,330
to y equal minus infinity in a
fixed time, in a definite time.
239
00:14:01,330 --> 00:14:06,140
And so here's a whole
region of curves going down
240
00:14:06,140 --> 00:14:07,610
to minus infinity.
241
00:14:07,610 --> 00:14:10,540
Here is a whole region.
242
00:14:10,540 --> 00:14:11,970
What happens in this region?
243
00:14:11,970 --> 00:14:16,850
Suppose y starts at plus 2?
244
00:14:16,850 --> 00:14:19,110
Well, I have 2 minus 4.
245
00:14:19,110 --> 00:14:20,730
So the slope is negative.
246
00:14:20,730 --> 00:14:22,180
The slope is negative up here.
247
00:14:22,180 --> 00:14:22,740
Yeah.
248
00:14:22,740 --> 00:14:24,450
And this is the big picture.
249
00:14:24,450 --> 00:14:29,570
The slope, the arrows are
positive below this line.
250
00:14:29,570 --> 00:14:31,120
They're upward.
251
00:14:31,120 --> 00:14:33,120
They were downward here.
252
00:14:33,120 --> 00:14:36,580
They're slowly upward
in this sandwich.
253
00:14:36,580 --> 00:14:40,500
And then up above,
they're downward again.
254
00:14:40,500 --> 00:14:51,920
So if slopes are coming down,
and they drop into actually
255
00:14:51,920 --> 00:14:54,530
it's a symmetric picture.
256
00:14:54,530 --> 00:14:59,210
Really-- no reason not
to go backwards in time.
257
00:14:59,210 --> 00:15:01,000
Where are these coming from?
258
00:15:01,000 --> 00:15:03,000
They're all coming from curves.
259
00:15:03,000 --> 00:15:05,240
The whole plane
is full of curves.
260
00:15:05,240 --> 00:15:08,180
And these start
at plus infinity.
261
00:15:08,180 --> 00:15:11,150
They drop into 2.
262
00:15:11,150 --> 00:15:18,380
These start below 0, and they
drop off to minus infinity.
263
00:15:18,380 --> 00:15:24,580
And then the real interest in
studying population was these.
264
00:15:24,580 --> 00:15:26,250
Can you do one more example?
265
00:15:26,250 --> 00:15:32,760
Let me take a third example
that has a t in the function.
266
00:15:32,760 --> 00:15:36,480
So the arrows won't be the
same along the whole line.
267
00:15:36,480 --> 00:15:40,630
In fact, the arrows
will be the same.
268
00:15:40,630 --> 00:15:47,460
So if I have 1 plus t minus y,
that's the f, equal a constant.
269
00:15:49,990 --> 00:15:55,340
Then that's a curve-- well,
it's actually a straight line.
270
00:15:55,340 --> 00:15:58,880
It's actually a 45-degree
line in this plane.
271
00:15:58,880 --> 00:16:08,610
And along that line the f, this
is the f, the f of t and y,
272
00:16:08,610 --> 00:16:10,090
the arrow slope.
273
00:16:10,090 --> 00:16:13,760
The arrows slopes are
the same along that line.
274
00:16:13,760 --> 00:16:16,740
That line is called an isocline.
275
00:16:16,740 --> 00:16:22,965
This is called an I-S-O, meaning
the same, cline, meaning slope.
276
00:16:25,610 --> 00:16:26,700
So that's an isocline.
277
00:16:26,700 --> 00:16:28,460
Here's an isocline.
278
00:16:28,460 --> 00:16:30,670
It's a 45-degree line.
279
00:16:30,670 --> 00:16:38,170
That's the 45-degree line,
1 plus t minus y equal 1.
280
00:16:38,170 --> 00:16:48,910
Let me draw the 45-degree line
1 plus t minus y equals 0.
281
00:16:48,910 --> 00:16:50,660
So it's a little bit higher.
282
00:16:50,660 --> 00:16:51,160
OK.
283
00:16:54,110 --> 00:16:59,680
Now arrows, and then put
in the curves, the solution
284
00:16:59,680 --> 00:17:02,400
curves that match the arrows.
285
00:17:02,400 --> 00:17:07,920
So the arrows have this
slope along that line.
286
00:17:07,920 --> 00:17:11,380
Along this line, 1 plus t
minus y, they have slope 0.
287
00:17:11,380 --> 00:17:12,829
Oh, interesting.
288
00:17:12,829 --> 00:17:15,329
At every point on the
line the slope is 0.
289
00:17:18,060 --> 00:17:21,210
Because this is the
slope of the arrows.
290
00:17:21,210 --> 00:17:24,650
At every point on
this line, the slope
291
00:17:24,650 --> 00:17:27,440
is 1, also very interesting.
292
00:17:27,440 --> 00:17:29,710
Because that's right
along the line.
293
00:17:29,710 --> 00:17:33,910
So here we have a solution line.
294
00:17:33,910 --> 00:17:35,350
That must be a solution line.
295
00:17:35,350 --> 00:17:39,210
That's the line where y is t.
296
00:17:39,210 --> 00:17:43,860
That's a very big
45-degree important line.
297
00:17:43,860 --> 00:17:48,920
Because if y equals
t, if y equals t
298
00:17:48,920 --> 00:17:51,440
then dy dt should be 1.
299
00:17:51,440 --> 00:17:53,560
And it is 1 for y equals t.
300
00:17:53,560 --> 00:17:57,260
So that's a solution
line with that solution.
301
00:17:57,260 --> 00:18:02,340
Now what about a line
with 1 plus t minus y
302
00:18:02,340 --> 00:18:04,710
equal minus 1, a line?
303
00:18:04,710 --> 00:18:10,410
If 1 plus t minus y is
minus 1, if f is minus 1,
304
00:18:10,410 --> 00:18:14,420
the slope is negative.
305
00:18:14,420 --> 00:18:15,930
So what does that mean?
306
00:18:15,930 --> 00:18:21,110
If 1 plus t minus y is minus
1, the slope is negative.
307
00:18:21,110 --> 00:18:27,300
So at points on this line,
the slope is going downwards.
308
00:18:27,300 --> 00:18:29,550
Oh, interesting.
309
00:18:29,550 --> 00:18:31,380
I wasn't quite expecting that.
310
00:18:31,380 --> 00:18:35,230
Let me just see if I
got a suitable picture.
311
00:18:35,230 --> 00:18:37,700
Why is it not right?
312
00:18:37,700 --> 00:18:44,660
If 1 plus t minus y
is-- oh, I'm sorry.
313
00:18:44,660 --> 00:18:49,540
This is the line,
y equal 1 plus t.
314
00:18:49,540 --> 00:18:52,090
I think what I'm
expecting to see
315
00:18:52,090 --> 00:18:56,260
is I'm expecting to
see it from the formula
316
00:18:56,260 --> 00:19:01,700
too that as time goes
on, this part goes to 0,
317
00:19:01,700 --> 00:19:03,180
and y goes to t.
318
00:19:03,180 --> 00:19:06,140
I believe that all
the solutions will
319
00:19:06,140 --> 00:19:09,803
approach this y equal to t.
320
00:19:09,803 --> 00:19:12,350
I think their
slopes, their slopes
321
00:19:12,350 --> 00:19:15,610
here-- darn, that's not right.
322
00:19:15,610 --> 00:19:20,090
Their slopes should
be coming upwards.
323
00:19:20,090 --> 00:19:22,460
Yeah, let me-- I
can figure that out.
324
00:19:22,460 --> 00:19:25,740
If t is let's say 1, and y is 0.
325
00:19:25,740 --> 00:19:26,340
OK.
326
00:19:26,340 --> 00:19:30,180
If t is 1, and y is 0,
I have a slope of 2.
327
00:19:30,180 --> 00:19:30,930
Good.
328
00:19:30,930 --> 00:19:31,800
OK.
329
00:19:31,800 --> 00:19:35,370
There's a point t equal to 1.
330
00:19:35,370 --> 00:19:37,700
Here is 0, 0.
331
00:19:37,700 --> 00:19:41,050
Here's the point t
equal to 1, y equals 0.
332
00:19:41,050 --> 00:19:43,070
The slope came out to be 2.
333
00:19:43,070 --> 00:19:44,820
It went up that way.
334
00:19:44,820 --> 00:19:48,620
So along that line, the
slopes are going up.
335
00:19:51,260 --> 00:19:54,550
Along this line, the slopes
are right on the line.
336
00:19:54,550 --> 00:19:57,390
On this line the
slopes are flat,
337
00:19:57,390 --> 00:20:01,130
and the curve is
moving toward the line.
338
00:20:01,130 --> 00:20:06,230
I'll just draw the beautiful
picture now of the solution.
339
00:20:06,230 --> 00:20:10,080
So the solutions look like this.
340
00:20:10,080 --> 00:20:14,442
They are-- this is the big line.
341
00:20:14,442 --> 00:20:16,500
You've got to keep
your eye on that line.
342
00:20:16,500 --> 00:20:19,800
Because that's the steady
state line that all solutions
343
00:20:19,800 --> 00:20:22,610
are approaching.
344
00:20:22,610 --> 00:20:28,630
So if you have the idea of
arrows to show the slope,
345
00:20:28,630 --> 00:20:34,070
fitting solution curves
through tangent to the arrows,
346
00:20:34,070 --> 00:20:37,260
and sometimes having
a formula to confirm
347
00:20:37,260 --> 00:20:42,870
that you did it right, you
get a picture like this.
348
00:20:42,870 --> 00:20:47,030
So that's the idea of first
order equations, which
349
00:20:47,030 --> 00:20:50,600
are graphed in the y-t plane.
350
00:20:50,600 --> 00:20:54,650
And the arrows tell
you the derivative.
351
00:20:54,650 --> 00:20:56,200
Thanks.