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00:00:00,500 --> 00:00:01,400
PROFESSOR: OK.

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00:00:01,400 --> 00:00:06,040
So we've moved on
into Chapter 3.

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00:00:06,040 --> 00:00:10,540
Chapter 1 and 2 were about
equations we could solve,

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00:00:10,540 --> 00:00:12,400
first order equations,
chapter one;

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00:00:12,400 --> 00:00:15,380
second order equations
in chapter 2, often

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00:00:15,380 --> 00:00:19,100
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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00:00:27,720 --> 00:00:34,600
a linear function, so I
don't expect a formula.

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A solution will exist.

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00:00:36,290 --> 00:00:38,950
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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00:00:51,590 --> 00:00:56,880
So this function, whatever
it is, gives the slope of y.

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That's the slope.

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00:00:58,280 --> 00:01:00,890
And it will be the
slope of the arrows

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00:01:00,890 --> 00:01:03,230
that I will draw
in this picture.

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00:01:03,230 --> 00:01:07,520
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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00:01:28,580 --> 00:01:30,730
in the null solution.

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And then y equal to 2 is a very
special particular solution,

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00:01:36,550 --> 00:01:37,510
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

36
00:02:02,300 --> 00:02:03,860
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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00:02:24,230 --> 00:02:27,990
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.

46
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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

48
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not 0, then there will
be a null solution part.

49
00:02:54,710 --> 00:02:58,550
But as t gets large
that goes to 0.

50
00:02:58,550 --> 00:03:01,050
So I move toward 2.

51
00:03:01,050 --> 00:03:03,360
Now let's see that
in the picture.

52
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So let me-- I'm drawing
the arrows first.

53
00:03:07,780 --> 00:03:18,990
So this is all time starting
at if y is 0, then if y is 0

54
00:03:18,990 --> 00:03:21,010
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.

58
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Now what else?

59
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So that's a few arrows that
show what will-- so the solution

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00:03:40,280 --> 00:03:45,580
if it starts there, will start
in the direction of that arrow.

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00:03:45,580 --> 00:03:51,840
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.

65
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And that's it needs more arrows.

66
00:04:01,610 --> 00:04:03,969
Well, actually it
needs way more arrows

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than I can possibly draw.

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00:04:05,010 --> 00:04:09,230
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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00:04:15,410 --> 00:04:19,209
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.

76
00:04:30,860 --> 00:04:32,650
Now the arrows will go up.

77
00:04:32,650 --> 00:04:39,660
So I'll have arrows that are
going a 45-degree angle, slope

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00:04:39,660 --> 00:04:40,160
1.

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Do you see?

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00:04:45,770 --> 00:04:49,340
I hope you begin to
see the picture here.

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00:04:49,340 --> 00:04:52,230
The solution might start there.

82
00:04:52,230 --> 00:04:53,870
It would start with that slope.

83
00:04:53,870 --> 00:04:56,900
But it will curve down.

84
00:04:56,900 --> 00:04:59,510
Because the arrows
are not so steep.

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00:04:59,510 --> 00:05:03,980
As I go upward, the
arrows are getting flat.

86
00:05:03,980 --> 00:05:08,400
And so the curve that
follows the arrows

87
00:05:08,400 --> 00:05:11,120
has to flatten out,
flatten out, flatten out.

88
00:05:11,120 --> 00:05:13,490
The arrows are
still, at that point

89
00:05:13,490 --> 00:05:15,620
the arrows are still slope 1.

90
00:05:15,620 --> 00:05:17,740
But it's flattening out.

91
00:05:17,740 --> 00:05:20,850
And it's never going
to cross this line.

92
00:05:20,850 --> 00:05:25,240
And it will run closer
and closer to that line.

93
00:05:25,240 --> 00:05:28,370
And wherever it
starts, if it starts

94
00:05:28,370 --> 00:05:35,390
at time t equals to 1 there,
it'll do the same thing.

95
00:05:35,390 --> 00:05:40,110
And it will stay just
below the other one.

96
00:05:40,110 --> 00:05:43,170
Do you see what the
pictures are looking like?

97
00:05:43,170 --> 00:05:49,350
If it starts at different times,
so these are different times.

98
00:05:49,350 --> 00:05:51,850
These are different starts.

99
00:05:51,850 --> 00:05:56,680
Yeah, really we're
used to, at t equals 0,

100
00:05:56,680 --> 00:05:59,530
we're used to giving y of 0.

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00:05:59,530 --> 00:06:01,230
So this is starting at 0.

102
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This is starting at 2.

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00:06:02,740 --> 00:06:06,410
Starting at 1 would
be a higher start.

104
00:06:06,410 --> 00:06:09,130
What about starting at 4?

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00:06:09,130 --> 00:06:11,295
Suppose y of 0 is 4.

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That point is t
equals 0, y equal 4.

107
00:06:15,210 --> 00:06:19,780
So that point is y of 0 equal 4.

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00:06:19,780 --> 00:06:23,690
What's the graph of the
solution with that start?

109
00:06:23,690 --> 00:06:26,160
Actually, I could
figure out what

110
00:06:26,160 --> 00:06:31,870
the solution would be if y
of 0 was 4, I'd have 2 plus 2

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00:06:31,870 --> 00:06:33,860
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.

121
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But the solution starts down.

122
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So it starts like that.

123
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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.

125
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So the new arrows are
I have slope minus 1.

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00:07:22,720 --> 00:07:27,520
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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00:08:02,435 --> 00:08:03,060
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.

139
00:08:08,620 --> 00:08:12,590
And the solutions would follow
the arrows, the arrows changing

140
00:08:12,590 --> 00:08:17,570
slope and actually in this
case, all solutions wherever

141
00:08:17,570 --> 00:08:20,820
you started, would approach 2.

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And that's what
the formula says.

143
00:08:24,540 --> 00:08:28,220
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.

147
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So it's going to be
more interesting.

148
00:08:38,000 --> 00:08:39,860
And do you remember
the solution?

149
00:08:39,860 --> 00:08:43,710
You remember maybe the trick
with the logistic equation

150
00:08:43,710 --> 00:08:52,700
was 1 over the solution, gave a
linear equation and expression

151
00:08:52,700 --> 00:08:53,680
like that.

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OK.

153
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Time to draw arrows.

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OK.

155
00:08:57,180 --> 00:09:03,670
When y is 0-- so here's y--
when y is 0, the slope is 0.

156
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So I have a whole line of flat.

157
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I have a flat horizontal line.

158
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That's the solution, y
equals 0 fixed point.

159
00:09:16,540 --> 00:09:18,690
Also we have
another fixed point.

160
00:09:18,690 --> 00:09:22,280
When y is 1, 1 minus 1 is 0.

161
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Slope is 0.

162
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Slope stays 0.

163
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The arrows all have zero slope
along the line y equal 1.

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00:09:30,710 --> 00:09:34,490
So there is another
solution, which

165
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doesn't do anything exciting.

166
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It just stays at 1.

167
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y equal 1 is another
fixed point, steady state,

168
00:09:47,360 --> 00:09:50,240
whatever words we want to use.

169
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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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00:09:58,830 --> 00:10:03,460
Well if it starts at-- if y is
1/2 half at the starting time,

172
00:10:03,460 --> 00:10:05,540
what is the slope?

173
00:10:05,540 --> 00:10:10,250
1/2 minus 1/4 is 1/4.

174
00:10:10,250 --> 00:10:13,950
So the slope is upwards,
but not very steep.

175
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.

177
00:10:21,080 --> 00:10:27,420
So that slope is the
same as long as y is 1/2,

178
00:10:27,420 --> 00:10:29,530
doesn't matter what the time is.

179
00:10:29,530 --> 00:10:33,470
y equals 1/2 gives me a 1/2
minus 1/4, which is a 1/4.

180
00:10:33,470 --> 00:10:34,890
It gives me that slope.

181
00:10:34,890 --> 00:10:38,450
What about the slope 1/4?

182
00:10:38,450 --> 00:10:43,420
So 1/4, I have 1/4 minus 1/16.

183
00:10:43,420 --> 00:10:45,960
I think that's 3/16.

184
00:10:45,960 --> 00:10:52,390
So it's beginning
to climb upward.

185
00:10:52,390 --> 00:10:53,970
So it's upwards again.

186
00:10:53,970 --> 00:10:56,570
But 3/16 is a
little-- I don't know

187
00:10:56,570 --> 00:10:59,930
if I'm going to get the
picture too brilliantly.

188
00:10:59,930 --> 00:11:10,300
The slope, as soon it-- if it
just starts a little above 0,

189
00:11:10,300 --> 00:11:12,070
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,

193
00:11:24,390 --> 00:11:30,390
if y is between 0 and 1, then
y is bigger than y squared.

194
00:11:30,390 --> 00:11:32,120
And the slope is positive.

195
00:11:32,120 --> 00:11:33,710
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.