1
00:00:00,080 --> 00:00:01,670
The following
content is provided

2
00:00:01,670 --> 00:00:03,820
under a Creative
Commons license.

3
00:00:03,820 --> 00:00:06,540
Your support will help MIT
OpenCourseWare continue

4
00:00:06,540 --> 00:00:10,150
to offer high quality
educational resources for free.

5
00:00:10,150 --> 00:00:12,690
To make a donation, or to
view additional materials

6
00:00:12,690 --> 00:00:16,600
from hundreds of MIT courses,
visit MIT OpenCourseWare

7
00:00:16,600 --> 00:00:17,307
at ocw.mit.edu.

8
00:00:22,460 --> 00:00:26,020
PROFESSOR: Monday's lecture
was all linear equations.

9
00:00:26,020 --> 00:00:28,070
And I thought I
would start today

10
00:00:28,070 --> 00:00:32,330
with nonlinear equations,
still first order.

11
00:00:32,330 --> 00:00:36,620
And we can't deal with
every nonlinear equation.

12
00:00:36,620 --> 00:00:38,990
That's too much to ask.

13
00:00:38,990 --> 00:00:43,020
These are going
to be made easier

14
00:00:43,020 --> 00:00:45,910
by a property
called "separable."

15
00:00:45,910 --> 00:00:54,280
So these will be separable
nonlinear equations.

16
00:00:54,280 --> 00:00:57,950
And let me start with
a couple of examples

17
00:00:57,950 --> 00:01:01,360
and then you'll
see the whole idea.

18
00:01:01,360 --> 00:01:10,570
So one example would be the
simplest nonlinear equation

19
00:01:10,570 --> 00:01:13,980
I can think of,
with a y squared.

20
00:01:13,980 --> 00:01:17,180
So how to get there?

21
00:01:19,910 --> 00:01:21,470
Here's the trick.

22
00:01:21,470 --> 00:01:24,340
This is the separable idea.

23
00:01:24,340 --> 00:01:26,750
You're going to
see it in one shot.

24
00:01:26,750 --> 00:01:30,230
We can separate, put
the Ys on one side

25
00:01:30,230 --> 00:01:31,730
and the Ds on the other.

26
00:01:31,730 --> 00:01:39,210
So I write this as dy
over y squared equal dt.

27
00:01:39,210 --> 00:01:41,820
I put the dt up and
brought the y squared down.

28
00:01:41,820 --> 00:01:44,100
So now they're
separated, in a kind

29
00:01:44,100 --> 00:01:48,210
of hookie way with
infinitesimals.

30
00:01:48,210 --> 00:01:51,540
But I'll makes sense out
of that by integrating.

31
00:01:51,540 --> 00:01:53,610
I'll integrate both sides.

32
00:01:53,610 --> 00:01:58,790
I'll integrate time from 0 to t.

33
00:01:58,790 --> 00:02:05,570
And I have an initial
condition, y of 0, always.

34
00:02:05,570 --> 00:02:11,700
And since this one is about y,
when t starts at 0, this guy

35
00:02:11,700 --> 00:02:16,550
starts at y of 0, up
to, this ends at t,

36
00:02:16,550 --> 00:02:20,810
so this ends at y of t.

37
00:02:20,810 --> 00:02:21,850
OK.

38
00:02:21,850 --> 00:02:25,570
Now the point is, also
the problem was nonlinear,

39
00:02:25,570 --> 00:02:29,740
we've got two separate
ordinary integrals to do.

40
00:02:29,740 --> 00:02:32,330
And we can do them.

41
00:02:32,330 --> 00:02:36,010
We can certainly do
the right hand side.

42
00:02:36,010 --> 00:02:37,790
I get t.

43
00:02:37,790 --> 00:02:40,350
And on the left hand
side, what do I get?

44
00:02:40,350 --> 00:02:43,280
Well, that maybe I better
leave a little space

45
00:02:43,280 --> 00:02:45,480
to figure out this one.

46
00:02:48,220 --> 00:02:51,610
But the point is we can
integrate 1 over y squared.

47
00:02:51,610 --> 00:02:54,960
And I guess we get
minus 1 over y.

48
00:02:54,960 --> 00:03:02,460
So I get minus 1 over y
between y of 0 and y of t.

49
00:03:02,460 --> 00:03:06,770
In other words, I'm
getting let's see,

50
00:03:06,770 --> 00:03:11,430
so what the right, the
derivative of the integral of 1

51
00:03:11,430 --> 00:03:14,980
over y squared is minus
1 over y because I always

52
00:03:14,980 --> 00:03:18,700
check the derivative
gives me that back.

53
00:03:18,700 --> 00:03:21,680
So now I'm ready to
plug-in those limits.

54
00:03:21,680 --> 00:03:26,330
So I'll do the
bottom limit first

55
00:03:26,330 --> 00:03:30,160
because it comes with a minus
sign, canceling that minus, 1

56
00:03:30,160 --> 00:03:35,590
over y of 0 minus 1 over y of t.

57
00:03:35,590 --> 00:03:36,740
Got it.

58
00:03:36,740 --> 00:03:40,640
And that equals the other
integral, which is just t.

59
00:03:43,740 --> 00:03:51,240
So that's the answer as it
comes directly from integration.

60
00:03:51,240 --> 00:03:53,510
And we can do more.

61
00:03:53,510 --> 00:04:01,130
You can see that finding the
solution when these things are

62
00:04:01,130 --> 00:04:04,930
separable has boiled
down to two integrals.

63
00:04:04,930 --> 00:04:16,485
And we could have a
function of t here, too.

64
00:04:16,485 --> 00:04:19,269
And that would be
allowed, a function of t

65
00:04:19,269 --> 00:04:21,339
multiplying this
guy, because then I

66
00:04:21,339 --> 00:04:24,430
would leave the function
of t on that side.

67
00:04:24,430 --> 00:04:27,020
And I would have
to integrate that.

68
00:04:27,020 --> 00:04:28,240
And I would bring the y.

69
00:04:28,240 --> 00:04:30,620
You see, I've just
separated the y.

70
00:04:30,620 --> 00:04:37,000
In general, these
equations look like dy, dt

71
00:04:37,000 --> 00:04:43,240
is some function of t divided
by some function of y.

72
00:04:43,240 --> 00:04:46,560
Maybe the book
calls the top one g,

73
00:04:46,560 --> 00:04:51,440
I think, and the bottom one f.

74
00:04:51,440 --> 00:04:56,050
And everybody in this room
sees that I can put the f of y

75
00:04:56,050 --> 00:04:57,290
up there.

76
00:04:57,290 --> 00:05:00,000
I can put the dt up there.

77
00:05:00,000 --> 00:05:01,310
And I've separated it.

78
00:05:01,310 --> 00:05:02,030
OK.

79
00:05:02,030 --> 00:05:04,030
So that's sort of the
general situation.

80
00:05:05,340 --> 00:05:10,290
This is a kind of nice example,
nice example, dy, dt equals

81
00:05:10,290 --> 00:05:11,700
y squared.

82
00:05:11,700 --> 00:05:14,200
Can we just play with
this a little bit?

83
00:05:14,200 --> 00:05:19,900
Let me take y of 0 to be 1,
just to make the numbers easy.

84
00:05:19,900 --> 00:05:24,140
So if y of 0 is 1,
then I have, I'll

85
00:05:24,140 --> 00:05:26,940
just keep going a little bit.

86
00:05:26,940 --> 00:05:29,090
You do have to keep
going a little bit

87
00:05:29,090 --> 00:05:32,600
because when you finish
the integral right there,

88
00:05:32,600 --> 00:05:35,370
you haven't got y equal.

89
00:05:35,370 --> 00:05:38,600
You've got some equation
that involves y,

90
00:05:38,600 --> 00:05:40,880
but you have to solve for y.

91
00:05:40,880 --> 00:05:45,070
So I have to solve
that equation for y.

92
00:05:45,070 --> 00:05:46,620
Let me just do it.

93
00:05:46,620 --> 00:05:48,250
So how would I solve it?

94
00:05:48,250 --> 00:05:50,670
And let me take y of 0 to be 1.

95
00:05:50,670 --> 00:05:56,680
So now, if I just write
it below, I'm at 1 minus 1

96
00:05:56,680 --> 00:05:59,960
over y equals t.

97
00:05:59,960 --> 00:06:02,590
Good?

98
00:06:02,590 --> 00:06:09,850
So I'm going to put the 1 over
y of t on that side and the t

99
00:06:09,850 --> 00:06:10,680
on that side.

100
00:06:10,680 --> 00:06:16,120
So if I just continue here,
I've got 1 over y of t

101
00:06:16,120 --> 00:06:23,050
on this side and, do I have
1 minus t on that side?

102
00:06:23,050 --> 00:06:23,990
Yeah.

103
00:06:23,990 --> 00:06:25,795
Looking good.

104
00:06:25,795 --> 00:06:33,430
So solution starting from
y of 0 equal 1 is y of t

105
00:06:33,430 --> 00:06:35,950
equal 1 over 1 minus t.

106
00:06:38,870 --> 00:06:41,640
You could do that.

107
00:06:41,640 --> 00:06:44,230
You could do that.

108
00:06:44,230 --> 00:06:49,610
And I can always, like, mentally
I check the algebra at t

109
00:06:49,610 --> 00:06:50,300
equals 0.

110
00:06:50,300 --> 00:06:52,430
That gives me the answer, 1.

111
00:06:52,430 --> 00:06:56,320
But let's step back and
look at that answer.

112
00:06:56,320 --> 00:06:58,330
I mean, that's part of
differential equations

113
00:06:58,330 --> 00:07:03,390
is to do some algebra, if
possible, and get to a formula.

114
00:07:03,390 --> 00:07:05,940
But if we don't think
about the formula,

115
00:07:05,940 --> 00:07:07,240
we haven't learned anything.

116
00:07:07,240 --> 00:07:08,310
Right there, yes.

117
00:07:08,310 --> 00:07:09,720
Good.

118
00:07:09,720 --> 00:07:14,650
So what happens?

119
00:07:14,650 --> 00:07:24,060
I want to compare with
the linear case that

120
00:07:24,060 --> 00:07:26,870
was like e to the t.

121
00:07:26,870 --> 00:07:30,480
This was y prime equal y, right?

122
00:07:30,480 --> 00:07:32,030
And that led to e to the t.

123
00:07:32,030 --> 00:07:37,370
Y prime equals y squared
leads to that one.

124
00:07:37,370 --> 00:07:40,260
So first observation.

125
00:07:40,260 --> 00:07:44,010
I haven't got exponentials
anymore in that solution.

126
00:07:44,010 --> 00:07:46,920
Exponentials are
just like perfection

127
00:07:46,920 --> 00:07:49,170
for linear equations.

128
00:07:49,170 --> 00:07:52,760
For nonlinear equations,
we get other functions.

129
00:07:52,760 --> 00:07:56,345
Professor Fry had a
hyperbolic tangent function

130
00:07:56,345 --> 00:07:58,010
in his first lecture.

131
00:07:58,010 --> 00:07:59,550
Other things happen.

132
00:07:59,550 --> 00:08:00,160
OK.

133
00:08:00,160 --> 00:08:06,890
Now, how do those
compare if I graph those?

134
00:08:06,890 --> 00:08:09,140
It's just like, why not try?

135
00:08:09,140 --> 00:08:12,510
So they both started at 1.

136
00:08:12,510 --> 00:08:16,610
And e to the t went
up exponentially.

137
00:08:16,610 --> 00:08:17,480
E to the t.

138
00:08:20,520 --> 00:08:23,810
And, I don't know,
we use exponential.

139
00:08:23,810 --> 00:08:26,815
In our minds, we think,
that's pretty fast growth.

140
00:08:26,815 --> 00:08:31,690
! mean, that's the common
expression, grew exponentially.

141
00:08:31,690 --> 00:08:35,330
But here, this guy
is going to grow

142
00:08:35,330 --> 00:08:39,850
faster because y is going
to be bigger than 1.

143
00:08:39,850 --> 00:08:44,220
So y squared is going
to be bigger than y.

144
00:08:44,220 --> 00:08:46,800
That one's going to grow faster.

145
00:08:46,800 --> 00:08:49,030
Faster than exponential.

146
00:08:49,030 --> 00:08:51,290
This has the exponential growth.

147
00:08:51,290 --> 00:08:53,070
Pretty fast.

148
00:08:53,070 --> 00:08:56,690
Polynomial, of course,
some parabola or something

149
00:08:56,690 --> 00:09:01,590
would be hanging way down
here, left behind in the dust.

150
00:09:01,590 --> 00:09:04,080
But this 1 over
1 minus t, that's

151
00:09:04,080 --> 00:09:05,540
going to grow really fast.

152
00:09:05,540 --> 00:09:11,080
And what's more, it's
going to go to infinity.

153
00:09:11,080 --> 00:09:16,435
So that y prime equal
y squared, the solution

154
00:09:16,435 --> 00:09:21,770
to that doesn't
just-- e to the t

155
00:09:21,770 --> 00:09:26,720
goes to infinity
at time infinity.

156
00:09:26,720 --> 00:09:30,270
At any finite time,
we get an answer.

157
00:09:30,270 --> 00:09:37,970
Eventually, at t equal infinity,
it's gone above every bound.

158
00:09:37,970 --> 00:09:43,360
But this one, 1 over 1 minus
t is what I want to graph now.

159
00:09:43,360 --> 00:09:50,410
I believe that that takes
off and at a certain point,

160
00:09:50,410 --> 00:09:54,310
capital T, it's
going to infinity.

161
00:09:54,310 --> 00:09:55,240
It's blown up.

162
00:09:55,240 --> 00:09:58,570
So it's blow up in finite time.

163
00:09:58,570 --> 00:10:00,000
Blow up in finite time.

164
00:10:00,000 --> 00:10:01,500
And what is that time?

165
00:10:01,500 --> 00:10:07,010
What's the time at which the
y prime equal y squared has

166
00:10:07,010 --> 00:10:10,220
taken off, gone off the charts?

167
00:10:10,220 --> 00:10:11,360
T equal--

168
00:10:11,360 --> 00:10:11,860
AUDIENCE: 1.

169
00:10:11,860 --> 00:10:13,220
PROFESSOR: --1.

170
00:10:13,220 --> 00:10:17,470
Because when t reaches
1, I have 1 over 0,

171
00:10:17,470 --> 00:10:22,600
and I'm dividing by 0,
and so that's the blow up.

172
00:10:22,600 --> 00:10:25,590
Finite time blow up.

173
00:10:25,590 --> 00:10:27,060
OK.

174
00:10:27,060 --> 00:10:31,540
So this can happen for
some nonlinear equations.

175
00:10:31,540 --> 00:10:33,610
It wouldn't happen
for a linear equation.

176
00:10:33,610 --> 00:10:37,390
For a linear equation,
exponentials are in control.

177
00:10:37,390 --> 00:10:39,410
OK.

178
00:10:39,410 --> 00:10:40,780
So that's one nice example.

179
00:10:40,780 --> 00:10:43,030
Oh, another nice thing
about that example.

180
00:10:47,190 --> 00:10:51,120
Well, I say nice if you're
OK with infinite series.

181
00:10:51,120 --> 00:10:53,050
I just want to compare.

182
00:10:53,050 --> 00:10:59,140
The book mentions the
infinite series for these guys

183
00:10:59,140 --> 00:11:03,110
because that's an old way to
solve differential equations is

184
00:11:03,110 --> 00:11:05,040
term-by-term in an
infinite series.

185
00:11:06,010 --> 00:11:08,730
It's sort of fun to
see the two series.

186
00:11:08,730 --> 00:11:15,620
Well, because they're the two
most important series in math.

187
00:11:15,620 --> 00:11:18,870
Actually, they're the two series
that everybody should know.

188
00:11:18,870 --> 00:11:23,040
The power series,
Taylor series--

189
00:11:23,040 --> 00:11:26,720
whatever word you want to
give it for those two guys.

190
00:11:26,720 --> 00:11:28,700
So let me do them.

191
00:11:28,700 --> 00:11:35,010
E to the t, I'll put that one
first, and 1 over 1 minus t.

192
00:11:35,010 --> 00:11:39,290
These are the great
series of math.

193
00:11:39,290 --> 00:11:45,160
Shall I just write them down
and sort of talk through them?

194
00:11:45,160 --> 00:11:49,540
Because this is not a lecture
on infinite series by any means.

195
00:11:49,540 --> 00:11:52,470
But having these
two in front of us,

196
00:11:52,470 --> 00:11:55,020
coming from these two
beautiful equations,

197
00:11:55,020 --> 00:11:58,010
y prime equal y squared
and y prime equal y,

198
00:11:58,010 --> 00:12:05,370
I can't resist seeing what
they look like this way.

199
00:12:05,370 --> 00:12:08,410
So e to the t, do you
remember e to the t?

200
00:12:08,410 --> 00:12:11,200
It starts at 1.

201
00:12:11,200 --> 00:12:14,800
What's the slope of e to the t?

202
00:12:14,800 --> 00:12:16,370
At t equals 0.

203
00:12:16,370 --> 00:12:18,720
So I'm doing everything--
this series is

204
00:12:18,720 --> 00:12:20,750
going to be, both
of the series are

205
00:12:20,750 --> 00:12:26,190
going to be, around t equals 0.

206
00:12:26,190 --> 00:12:31,450
That's my, like, starting point.

207
00:12:31,450 --> 00:12:39,100
So this e to the t
thing has a tangent.

208
00:12:39,100 --> 00:12:41,160
It has a slope there.

209
00:12:41,160 --> 00:12:45,455
And what's the slope of
e to the t at t equals 0?

210
00:12:45,455 --> 00:12:45,954
AUDIENCE: 1.

211
00:12:45,954 --> 00:12:47,760
PROFESSOR: 1.

212
00:12:47,760 --> 00:12:49,590
It's derivative.

213
00:12:49,590 --> 00:12:52,240
The derivative of e to
the t is e to the t.

214
00:12:52,240 --> 00:12:53,750
The slope is 1.

215
00:12:53,750 --> 00:12:58,880
So that tangent line
has coefficient 1.

216
00:12:58,880 --> 00:13:01,760
That's how it starts.

217
00:13:01,760 --> 00:13:03,620
That's the linear approximation.

218
00:13:03,620 --> 00:13:07,350
That's the heart of
calculus, is this.

219
00:13:07,350 --> 00:13:09,980
But we're going to go better.

220
00:13:09,980 --> 00:13:13,140
We're going to
get the next term.

221
00:13:13,140 --> 00:13:15,940
So what's the next term?

222
00:13:15,940 --> 00:13:17,750
That gave us the tangent line.

223
00:13:18,740 --> 00:13:22,170
Now I'm going to move
to the tangent parabola.

224
00:13:22,170 --> 00:13:26,130
So the parabola has
got another is still

225
00:13:26,130 --> 00:13:29,360
going to be below
the real thing.

226
00:13:29,360 --> 00:13:34,210
Can I squeeze in the
words "line" and "parab,"

227
00:13:34,210 --> 00:13:37,650
for "parabola?"

228
00:13:39,040 --> 00:13:42,180
Parabola has bending.

229
00:13:42,180 --> 00:13:44,680
I'm really explaining
the Taylor series

230
00:13:44,680 --> 00:13:46,520
in what I hope is
a sensible way.

231
00:13:49,090 --> 00:13:51,300
Here is the starting point.

232
00:13:51,300 --> 00:13:53,100
This has the slope.

233
00:13:53,100 --> 00:13:55,530
The next term has the bending.

234
00:13:55,530 --> 00:13:59,630
The bending comes
from what derivative?

235
00:13:59,630 --> 00:14:02,670
What derivative tells
us about bending?

236
00:14:02,670 --> 00:14:04,041
Second derivative.

237
00:14:04,041 --> 00:14:06,340
Second derivative tells
us how much it curves.

238
00:14:08,100 --> 00:14:12,650
Well, the second derivative of
e to the t is still e to the t.

239
00:14:12,650 --> 00:14:15,300
So the bending is 1.

240
00:14:15,300 --> 00:14:17,310
The bending is also 1.

241
00:14:17,310 --> 00:14:20,700
Now that comes in with
a factor of a 1/2.

242
00:14:23,930 --> 00:14:27,020
There is the tangent parabola.

243
00:14:27,020 --> 00:14:32,490
And you will see what
these numbers become.

244
00:14:32,490 --> 00:14:35,580
Let me just go to,
the third derivative

245
00:14:35,580 --> 00:14:38,720
would be responsible
for the t cubed term.

246
00:14:38,720 --> 00:14:42,100
And its coefficient would
be 1 over 3 factorial.

247
00:14:42,100 --> 00:14:45,490
So 2 is the same as 2 factorial.

248
00:14:45,490 --> 00:14:49,330
3 factorial is 3
times 2 times 16.

249
00:14:49,330 --> 00:14:59,300
So the numbers here go 1, 6, 24,
120, whatever the next one is.

250
00:14:59,300 --> 00:15:01,470
720 or something.

251
00:15:01,470 --> 00:15:03,530
They grow fast.

252
00:15:03,530 --> 00:15:07,660
So that series always
gives a finite answer.

253
00:15:07,660 --> 00:15:09,530
It does grow with t.

254
00:15:09,530 --> 00:15:14,200
But it doesn't spike with t.

255
00:15:14,200 --> 00:15:17,800
Now compare that famous series.

256
00:15:17,800 --> 00:15:21,970
And of course, this
is 1 over 1 factorial,

257
00:15:21,970 --> 00:15:23,860
everything consistent.

258
00:15:23,860 --> 00:15:28,810
Compare that with the
series for 1 over 1 minus t.

259
00:15:28,810 --> 00:15:32,540
That's the other famous series
that they learned in algebra.

260
00:15:32,540 --> 00:15:33,950
I'll just write it.

261
00:15:33,950 --> 00:15:39,415
That's 1 plus t plus t
squared plus t cubed plus 1

262
00:15:39,415 --> 00:15:41,180
so on, with coefficient 1.

263
00:15:43,860 --> 00:15:48,060
This had 1 over n factorials.

264
00:15:48,060 --> 00:15:50,550
Those make the series converge.

265
00:15:50,550 --> 00:15:52,980
These don't have
the n factorials.

266
00:15:52,980 --> 00:15:55,410
This is 1, 1, 1, 1.

267
00:15:55,410 --> 00:15:57,980
And, well, I could
check that formula.

268
00:15:57,980 --> 00:16:01,790
But do you remember the
name for that series?

269
00:16:01,790 --> 00:16:06,690
1 plus t plus t squared
plus t cubed plus so on?

270
00:16:06,690 --> 00:16:11,900
Algebra is taught differently
in many high schools now.

271
00:16:11,900 --> 00:16:15,340
And maybe that never got a name.

272
00:16:15,340 --> 00:16:18,960
I guess I would call it
the Geometric series.

273
00:16:18,960 --> 00:16:20,630
Geometric series.

274
00:16:20,630 --> 00:16:23,090
And you see, it's beautiful.

275
00:16:23,090 --> 00:16:24,680
It's the other important series.

276
00:16:24,680 --> 00:16:26,650
But it's quite
different from this one

277
00:16:26,650 --> 00:16:32,255
because, what's the
difference about this series?

278
00:16:32,255 --> 00:16:32,755
Yeah?

279
00:16:32,755 --> 00:16:33,976
AUDIENCE: It goes to infinity.

280
00:16:33,976 --> 00:16:34,760
PROFESSOR: It's a--

281
00:16:34,760 --> 00:16:36,010
AUDIENCE: It goes to infinity.

282
00:16:36,010 --> 00:16:37,690
PROFESSOR: It goes to infinity.

283
00:16:37,690 --> 00:16:38,740
But where?

284
00:16:38,740 --> 00:16:39,860
At what time?

285
00:16:39,860 --> 00:16:45,380
At what value of t is this
sum going to fall apart?

286
00:16:45,380 --> 00:16:47,180
Blow up?

287
00:16:47,180 --> 00:16:48,670
At t equal 1.

288
00:16:48,670 --> 00:16:53,390
When have 1 plus 1 plus 1
plus 1, I'm getting infinity.

289
00:16:53,390 --> 00:16:55,220
So this blows up.

290
00:16:55,220 --> 00:16:59,830
And of course, we see that it
should because this blows up.

291
00:16:59,830 --> 00:17:03,265
Left side blows up at t equal
1, the right side blows up a t

292
00:17:03,265 --> 00:17:04,190
equal 1.

293
00:17:04,190 --> 00:17:09,200
Where the exponential
series, which

294
00:17:09,200 --> 00:17:12,950
is the heart of ordinary
differential equations,

295
00:17:12,950 --> 00:17:18,710
never blows up because of these
big numbers in the denominator.

296
00:17:18,710 --> 00:17:24,660
OK, I'm good for this first
simple example, y prime equals

297
00:17:24,660 --> 00:17:26,410
y squared.

298
00:17:26,410 --> 00:17:30,390
It has so much in it,
it's worth thinking about.

299
00:17:30,390 --> 00:17:33,522
I'm ready, you OK
for a second example?

300
00:17:33,522 --> 00:17:41,040
A second important
separable equation.

301
00:17:41,040 --> 00:17:43,810
I'm going to pick one.

302
00:17:43,810 --> 00:17:49,870
So I'm going to pick an
equation that starts out

303
00:17:49,870 --> 00:17:54,110
with our familiar linear growth.

304
00:17:54,110 --> 00:17:56,670
This could be, you
know, last time

305
00:17:56,670 --> 00:17:59,380
it was growth of
money in a bank.

306
00:17:59,380 --> 00:18:02,330
It could be growth
of population.

307
00:18:02,330 --> 00:18:10,230
The number of, to a sum
first approximation,

308
00:18:10,230 --> 00:18:14,670
the rate of growth
of the population

309
00:18:14,670 --> 00:18:18,990
comes from, like,
births minus deaths.

310
00:18:18,990 --> 00:18:26,550
And with modern medicine,
births are a larger number

311
00:18:26,550 --> 00:18:27,750
than deaths.

312
00:18:27,750 --> 00:18:31,030
So a is positive,
and that grows.

313
00:18:31,030 --> 00:18:35,240
But if we're talking
about the, I mean,

314
00:18:35,240 --> 00:18:37,900
the United Nations
tries to predict,

315
00:18:37,900 --> 00:18:40,960
everybody tries to
predict, population

316
00:18:40,960 --> 00:18:45,980
of the world in future years.

317
00:18:45,980 --> 00:18:51,470
And so this could be called
the Population Equation.

318
00:18:51,470 --> 00:18:59,020
But just to leave it as pure
exponential is obviously wrong.

319
00:18:59,020 --> 00:19:01,190
The world can't grow forever.

320
00:19:01,190 --> 00:19:05,050
The population
can't grow forever.

321
00:19:05,050 --> 00:19:10,320
And the, I guess I hope it
doesn't grow like 1 over 1

322
00:19:10,320 --> 00:19:11,260
minus t.

323
00:19:11,260 --> 00:19:15,110
So this is at least
a little slower.

324
00:19:15,110 --> 00:19:21,520
But somehow competition for
space, competition for food,

325
00:19:21,520 --> 00:19:25,050
for oil, for water-- which
is going to be the big one--

326
00:19:25,050 --> 00:19:26,770
is in here.

327
00:19:26,770 --> 00:19:33,740
Competition here, of
people versus people,

328
00:19:33,740 --> 00:19:38,930
a reasonable term, a
first approximation,

329
00:19:38,930 --> 00:19:42,830
is a y squared, with a
minus, is a y squared

330
00:19:42,830 --> 00:19:45,710
and with some coefficient.

331
00:19:45,710 --> 00:19:47,310
That's a very famous equation.

332
00:19:51,110 --> 00:19:54,970
A first model of
population is it grows.

333
00:19:54,970 --> 00:20:01,020
But this is a competition
term, y against y.

334
00:20:01,020 --> 00:20:07,300
And so, the same
would be true if we

335
00:20:07,300 --> 00:20:09,900
were talking about epidemics.

336
00:20:09,900 --> 00:20:12,340
That's a big subject with
ordinary differential

337
00:20:12,340 --> 00:20:14,850
equations, epidemiology.

338
00:20:14,850 --> 00:20:17,890
Or say, flu.

339
00:20:17,890 --> 00:20:21,460
How does flu spread?

340
00:20:21,460 --> 00:20:22,980
And how does it get cured?

341
00:20:23,740 --> 00:20:26,810
So partly, people are
getting over the flu.

342
00:20:26,810 --> 00:20:31,640
But then y against y is
telling us how many infected,

343
00:20:31,640 --> 00:20:33,120
how many new infections.

344
00:20:33,120 --> 00:20:35,175
So we would like to
solve that equation.

345
00:20:37,910 --> 00:20:40,190
And it's separable.

346
00:20:40,190 --> 00:20:45,480
I can do what I did
before, dy over ay minus by

347
00:20:45,480 --> 00:20:48,650
squared equal dt.

348
00:20:48,650 --> 00:20:53,480
And I can integrate,
starting from year of 0.

349
00:20:53,480 --> 00:20:58,650
Well, why don't we
start from year 2014,

350
00:20:58,650 --> 00:21:03,460
with the population y at
now-- the present population?

351
00:21:07,840 --> 00:21:13,930
That would be a model that
the UN would consider using.

352
00:21:13,930 --> 00:21:17,500
That other people with
very important interest

353
00:21:17,500 --> 00:21:25,080
in measuring population and
measuring every resource

354
00:21:25,080 --> 00:21:27,990
would need equations like this.

355
00:21:27,990 --> 00:21:30,770
And then they would
put on more terms,

356
00:21:30,770 --> 00:21:36,650
like a term for immigration.

357
00:21:36,650 --> 00:21:41,155
All sorts, many improvements
have to go into this equation.

358
00:21:43,910 --> 00:21:45,655
Let me just look
at this as it is.

359
00:21:48,580 --> 00:21:53,480
Well, I've got two choices here.

360
00:21:53,480 --> 00:21:59,200
Well, it's this integral
that I'm looking at.

361
00:21:59,200 --> 00:22:01,220
That is a doable integral.

362
00:22:01,220 --> 00:22:06,160
It's the type of integral that
we saw in the Rocket problem.

363
00:22:06,160 --> 00:22:10,560
The Rocket problem was
more constant minus.

364
00:22:10,560 --> 00:22:15,030
This was a drag term, when
we were looking at rockets.

365
00:22:15,030 --> 00:22:20,410
And this was a
constant, say, gravity.

366
00:22:20,410 --> 00:22:24,510
So it was still a second degree.

367
00:22:24,510 --> 00:22:26,480
Still second degree,
but a little different.

368
00:22:26,480 --> 00:22:32,410
This has the linear in
second degree terms.

369
00:22:32,410 --> 00:22:34,350
If you look up that
integral, you'll find it.

370
00:22:37,310 --> 00:22:40,010
Or there's a systematic
way to do it.

371
00:22:40,010 --> 00:22:47,320
That's in 1801, I guess,
called partial fractions.

372
00:22:47,320 --> 00:22:50,710
It's not a lot of fun.

373
00:22:50,710 --> 00:22:53,190
I don't plan to do it.

374
00:22:53,190 --> 00:22:54,630
It's in the book.

375
00:22:54,630 --> 00:22:57,540
Has to be because that's
the way you can integr-- you

376
00:22:57,540 --> 00:23:02,150
can integrate polynomials
over polynomials

377
00:23:02,150 --> 00:23:04,770
by partial fractions.

378
00:23:04,770 --> 00:23:09,160
That's what they're
for, but there's

379
00:23:09,160 --> 00:23:11,280
a neat way to do this one.

380
00:23:11,280 --> 00:23:16,600
There's a neat trick
that Bernoulli discovered

381
00:23:16,600 --> 00:23:22,010
to solve that equation, to
turn it into a linear equation.

382
00:23:22,010 --> 00:23:24,500
And of course, if we can turn
it into a linear equation,

383
00:23:24,500 --> 00:23:25,400
we're on our way.

384
00:23:25,400 --> 00:23:31,480
So the neat trick is
let z be 1 over y.

385
00:23:35,960 --> 00:23:38,720
You can put this in the
category of lucky accidents,

386
00:23:38,720 --> 00:23:41,810
if you like.

387
00:23:41,810 --> 00:23:45,410
So now I want an equation for z.

388
00:23:45,410 --> 00:23:51,920
So I know that dz, dt if I
take the derivative of that,

389
00:23:51,920 --> 00:23:54,140
that's y to the minus 1.

390
00:23:54,140 --> 00:24:01,850
So it's minus 1 y to
the minus 2 dy, dt.

391
00:24:01,850 --> 00:24:04,700
That's the chain rule.

392
00:24:04,700 --> 00:24:08,850
Take the derivative of 1 over 1,
you get minus 1 over y squared.

393
00:24:08,850 --> 00:24:14,100
Multiply by the derivative
of what's inside.

394
00:24:14,100 --> 00:24:15,331
That's the chain rule.

395
00:24:15,331 --> 00:24:15,830
OK.

396
00:24:15,830 --> 00:24:21,120
So I plan to substitute
those in here.

397
00:24:21,120 --> 00:24:24,430
So dy, dt, let's see.

398
00:24:24,430 --> 00:24:25,320
Can you see me?

399
00:24:25,320 --> 00:24:29,090
You can probably do
it better than me.

400
00:24:29,090 --> 00:24:33,240
So dy, dt is minus.

401
00:24:33,240 --> 00:24:34,390
I'll bring that up.

402
00:24:34,390 --> 00:24:39,620
Dy, dt I'm going to
put-- I hope this'll

403
00:24:39,620 --> 00:24:44,445
work all right-- for dy,
dt, I'm going to put in dz.

404
00:24:47,300 --> 00:24:52,600
Using this, I'm going to
put minus y squared dz, dt.

405
00:24:52,600 --> 00:24:54,070
Did that look right?

406
00:24:54,070 --> 00:24:55,740
I don't think I'm
necessarily doing

407
00:24:55,740 --> 00:24:57,920
this the most brilliant way.

408
00:24:57,920 --> 00:25:03,910
But dy, dt-- I put this up here
and I got that-- equals ay.

409
00:25:06,540 --> 00:25:08,883
So that's a over z.

410
00:25:08,883 --> 00:25:10,820
Oh, y Is 1 over z.

411
00:25:10,820 --> 00:25:17,040
So get this, I want all Zs now.

412
00:25:17,040 --> 00:25:19,540
So that's this part.

413
00:25:19,540 --> 00:25:23,080
And ay is over z
minus by squared

414
00:25:23,080 --> 00:25:26,980
is minus b over z squared.

415
00:25:26,980 --> 00:25:28,755
Would you say OK to that?

416
00:25:31,310 --> 00:25:34,180
I've got Zs now, instead of Ys.

417
00:25:34,180 --> 00:25:40,041
I just took every term and
replaced y by 1 over z.

418
00:25:40,041 --> 00:25:45,130
Y is 1 over z and dy,
dt I can get that way.

419
00:25:45,130 --> 00:25:45,936
OK.

420
00:25:45,936 --> 00:25:47,840
Yeah.

421
00:25:47,840 --> 00:25:49,270
Now what?

422
00:25:49,270 --> 00:25:52,440
Now look what happens, if I
multiply through by z squared

423
00:25:52,440 --> 00:25:54,190
or by minus z squared.

424
00:25:54,190 --> 00:25:56,680
Let me multiply through
by minus z squared.

425
00:25:56,680 --> 00:25:59,530
I get dz, dt.

426
00:25:59,530 --> 00:26:04,290
Multiplying by minus z
squared gives me a minus az.

427
00:26:04,290 --> 00:26:05,910
And what do I get
when I multiply

428
00:26:05,910 --> 00:26:07,795
this one by minus z squared?

429
00:26:10,620 --> 00:26:11,600
AUDIENCE: Plus b.

430
00:26:11,600 --> 00:26:12,690
PROFESSOR: I get plus b.

431
00:26:16,630 --> 00:26:18,980
Look what happened.

432
00:26:18,980 --> 00:26:22,160
By this, like, some magic trick.

433
00:26:25,279 --> 00:26:26,320
You could say, all right.

434
00:26:26,320 --> 00:26:28,920
That was just a one time shot.

435
00:26:28,920 --> 00:26:31,560
But it was a good one.

436
00:26:31,560 --> 00:26:35,730
We ended up with a
linear equation for z.

437
00:26:35,730 --> 00:26:37,230
A linear equation for z.

438
00:26:37,230 --> 00:26:41,940
And we solved that
equation last time.

439
00:26:41,940 --> 00:26:45,550
So let me squeeze in
the solution for z,

440
00:26:45,550 --> 00:26:47,150
and then elsewhere.

441
00:26:47,150 --> 00:26:50,900
So what was the
solution for z of t?

442
00:26:50,900 --> 00:26:56,030
It was some multiple
of, no, yeah.

443
00:26:56,030 --> 00:26:59,830
This is perfect
review of last time.

444
00:26:59,830 --> 00:27:03,580
We have a constant times z.

445
00:27:03,580 --> 00:27:07,250
And so that's going to
go into the exponential.

446
00:27:07,250 --> 00:27:13,380
This will be the, it's
a minus a, notice.

447
00:27:13,380 --> 00:27:18,490
That will be the, what
part of the solution

448
00:27:18,490 --> 00:27:21,150
is that one called?

449
00:27:21,150 --> 00:27:22,975
That's the null solution.

450
00:27:22,975 --> 00:27:26,120
The null solution, when b is o.

451
00:27:26,120 --> 00:27:29,380
And now I add in a
particular solution.

452
00:27:32,000 --> 00:27:33,330
A particular solution.

453
00:27:33,330 --> 00:27:39,670
And one good particular
solution is choose the z

454
00:27:39,670 --> 00:27:41,660
to be a constant.

455
00:27:41,660 --> 00:27:43,220
Then that'll be 0.

456
00:27:43,220 --> 00:27:45,320
So I want that to be 0.

457
00:27:45,320 --> 00:27:49,230
So what constant z makes that 0?

458
00:27:49,230 --> 00:27:51,520
I think it's b
over a, don't you?

459
00:27:51,520 --> 00:27:52,517
I think b over a.

460
00:27:52,517 --> 00:27:53,350
Does that work good?

461
00:27:57,920 --> 00:28:04,320
That's every null solution
plus one particular solution.

462
00:28:04,320 --> 00:28:08,850
Let me say now,
and I'll say again,

463
00:28:08,850 --> 00:28:13,920
looking for solutions which
are steady states, b over

464
00:28:13,920 --> 00:28:20,630
a-- of this particular solution,
that particular solution made

465
00:28:20,630 --> 00:28:22,830
this 0.

466
00:28:22,830 --> 00:28:25,620
So it made this 0.

467
00:28:25,620 --> 00:28:28,760
So it's a solution that's
not going anywhere.

468
00:28:28,760 --> 00:28:29,940
It's a constant solution.

469
00:28:29,940 --> 00:28:34,460
It's a solution that
can live for all time.

470
00:28:34,460 --> 00:28:35,440
OK.

471
00:28:35,440 --> 00:28:36,050
B over a.

472
00:28:39,230 --> 00:28:41,590
Let me put that word
there, steady state.

473
00:28:46,680 --> 00:28:48,300
OK.

474
00:28:48,300 --> 00:28:52,520
And now I would want to match
the initial conditions using c.

475
00:28:56,290 --> 00:28:56,790
Yeah.

476
00:28:56,790 --> 00:28:58,210
I'd better do that.

477
00:28:58,210 --> 00:28:59,620
OK.

478
00:28:59,620 --> 00:29:04,420
And I have to get back to y.

479
00:29:04,420 --> 00:29:09,200
I have y is 1 over z.

480
00:29:09,200 --> 00:29:12,065
So I'm going to have to
flip this upside down.

481
00:29:12,065 --> 00:29:15,440
I'm going to have to
flip this upside down is

482
00:29:15,440 --> 00:29:18,870
what will actually happen.

483
00:29:18,870 --> 00:29:22,680
Let me make it easy to flip.

484
00:29:22,680 --> 00:29:26,100
Let me, I'll change c,
which is just some constant

485
00:29:26,100 --> 00:29:29,070
to some constant d over a.

486
00:29:29,070 --> 00:29:33,400
So then it's a is
everywhere down below.

487
00:29:33,400 --> 00:29:36,250
And I just write it
here in the middle.

488
00:29:36,250 --> 00:29:37,760
That makes it easier to flip.

489
00:29:37,760 --> 00:29:42,120
So finally I get their solution.

490
00:29:42,120 --> 00:29:47,500
Solution to the
population equation.

491
00:29:47,500 --> 00:29:53,330
But that's the famous word
for it, the Logistic equation.

492
00:29:53,330 --> 00:29:59,000
This is section 1.7 of the
text on the differential

493
00:29:59,000 --> 00:30:02,130
equations in linear algebra.

494
00:30:02,130 --> 00:30:05,710
It's a very, very
much studied example.

495
00:30:05,710 --> 00:30:07,930
It's a great example.

496
00:30:07,930 --> 00:30:17,680
It fits the growth of
human population with some,

497
00:30:17,680 --> 00:30:21,710
it's our first
level approximation

498
00:30:21,710 --> 00:30:30,270
to growth of or other
populations or other things.

499
00:30:30,270 --> 00:30:36,000
It's a linear term giving
us exponential growth,

500
00:30:36,000 --> 00:30:40,270
and a quadratic term of
competition slowing it down.

501
00:30:40,270 --> 00:30:42,450
And let's see that slow down.

502
00:30:42,450 --> 00:30:48,120
So now that was
a bit of algebra.

503
00:30:48,120 --> 00:30:50,700
Much nicer than
partial fractions.

504
00:30:50,700 --> 00:30:56,030
The bit of algebra just came
from this idea of going to z.

505
00:30:56,030 --> 00:30:57,750
And now I want to go back to y.

506
00:30:57,750 --> 00:31:01,750
So y is 1 over z.

507
00:31:01,750 --> 00:31:09,630
So it's a over d e to
the minus at plus b.

508
00:31:12,760 --> 00:31:14,930
That's our solution.

509
00:31:14,930 --> 00:31:19,930
A and b came out
of the equation.

510
00:31:19,930 --> 00:31:23,110
And d is going to
be the number that

511
00:31:23,110 --> 00:31:26,110
makes the initial value correct.

512
00:31:26,110 --> 00:31:33,210
So at t equals 0, I would
have y of 0, whatever

513
00:31:33,210 --> 00:31:38,395
the initial population
is, is a over d.

514
00:31:38,395 --> 00:31:42,440
T Is 0, so that's just 1 plus b.

515
00:31:42,440 --> 00:31:44,856
So that tells me what d is.

516
00:31:44,856 --> 00:31:46,520
D equals something.

517
00:31:49,640 --> 00:31:52,440
It comes from y of 0.

518
00:31:52,440 --> 00:31:54,735
So the answer, let me
circle that answer.

519
00:31:58,460 --> 00:32:02,370
That answer has three
numbers in it, a, b, and d.

520
00:32:02,370 --> 00:32:04,240
a and b come from the equation.

521
00:32:04,240 --> 00:32:08,680
D also involves the
initial starting thing,

522
00:32:08,680 --> 00:32:10,548
which is exactly what it showed.

523
00:32:13,620 --> 00:32:15,550
So you could say
we've solved it.

524
00:32:15,550 --> 00:32:19,660
But if you ever solve
an equation like this,

525
00:32:19,660 --> 00:32:21,220
you want to graph it.

526
00:32:21,220 --> 00:32:23,130
You want to graph it.

527
00:32:23,130 --> 00:32:25,810
So let me draw its graph.

528
00:32:25,810 --> 00:32:27,680
This is important picture.

529
00:32:30,190 --> 00:32:32,789
So here is time.

530
00:32:32,789 --> 00:32:33,580
Here is population.

531
00:32:36,120 --> 00:32:38,910
Here's, maybe it started there.

532
00:32:38,910 --> 00:32:40,160
This is times 0.

533
00:32:46,730 --> 00:32:50,220
And now I want to graph this.

534
00:32:50,220 --> 00:32:54,230
I want to graph that function.

535
00:32:56,390 --> 00:33:01,420
Really, this is where
we're going somewhere.

536
00:33:01,420 --> 00:33:06,790
What happens for a long time?

537
00:33:06,790 --> 00:33:09,890
At t equal infinity, what
happens to the population?

538
00:33:12,340 --> 00:33:15,500
Does it grow, like e to the t?

539
00:33:15,500 --> 00:33:20,450
Just remember the examples here.

540
00:33:20,450 --> 00:33:23,940
We had a growth like e to the t.

541
00:33:23,940 --> 00:33:29,540
We had a growth faster than e
to the t that actually blew up.

542
00:33:29,540 --> 00:33:31,750
What about this guy?

543
00:33:31,750 --> 00:33:38,920
What will happen as t goes to
infinity with that population?

544
00:33:38,920 --> 00:33:39,950
It goes to?

545
00:33:39,950 --> 00:33:41,320
AUDIENCE: A over b.

546
00:33:41,320 --> 00:33:42,320
PROFESSOR: A over b.

547
00:33:42,320 --> 00:33:43,840
A over b.

548
00:33:43,840 --> 00:33:46,080
That's the key number
in the whole thing.

549
00:33:49,660 --> 00:33:55,130
It keeps growing, but it
never passes a over b.

550
00:33:55,130 --> 00:33:59,290
This is y at infinity.

551
00:33:59,290 --> 00:34:01,830
That's the final population.

552
00:34:01,830 --> 00:34:03,500
So how does it do this?

553
00:34:03,500 --> 00:34:09,320
If I draw this graph-- and
what about negative time?

554
00:34:09,320 --> 00:34:11,199
Let's go backwards in time.

555
00:34:14,020 --> 00:34:18,880
What is it at t
equals minus infinity?

556
00:34:18,880 --> 00:34:22,690
Then you really see
the whole curve.

557
00:34:22,690 --> 00:34:27,495
At t equal minus infinity,
what is this doing?

558
00:34:27,495 --> 00:34:28,370
AUDIENCE: 0 infinity.

559
00:34:28,370 --> 00:34:29,148
PROFESSOR: It's 0.

560
00:34:29,148 --> 00:34:29,648
Good.

561
00:34:29,648 --> 00:34:30,148
Good.

562
00:34:30,148 --> 00:34:31,260
Good.

563
00:34:31,260 --> 00:34:34,889
T equal minus infinity,
this is enormous.

564
00:34:34,889 --> 00:34:36,449
This is blowing up.

565
00:34:36,449 --> 00:34:38,110
It's in the denominator.

566
00:34:38,110 --> 00:34:39,540
We're dividing by it.

567
00:34:39,540 --> 00:34:41,710
So the whole thing
is going to 0.

568
00:34:41,710 --> 00:34:45,920
So here's what the
logistic curve looks like.

569
00:34:45,920 --> 00:34:47,370
It creeps up.

570
00:34:47,370 --> 00:34:51,580
And it's beautifully, there's
a point of symmetry here.

571
00:34:51,580 --> 00:34:54,489
The growth is increasing here.

572
00:34:54,489 --> 00:34:58,140
And then, as a point of
inflection you could say,

573
00:34:58,140 --> 00:35:01,370
growth is bending
upwards for a while.

574
00:35:01,370 --> 00:35:05,150
At this point, it starts
bending downwards.

575
00:35:05,150 --> 00:35:08,440
From that point on, ooh,
let's see if I can draw it.

576
00:35:08,440 --> 00:35:11,100
It'll get closer, and
exponentially close.

577
00:35:11,660 --> 00:35:13,160
That wasn't a bad picture.

578
00:35:17,370 --> 00:35:20,090
The population here is half way.

579
00:35:20,090 --> 00:35:24,280
Here, the population, the
final population, is a over b.

580
00:35:24,280 --> 00:35:28,720
And just by beautiful symmetry,
the population here is a 1/2

581
00:35:28,720 --> 00:35:31,200
of a over b.

582
00:35:31,200 --> 00:35:32,460
At this point.

583
00:35:35,440 --> 00:35:38,210
If this was the actual
population of the world

584
00:35:38,210 --> 00:35:46,400
we live in-- I think we're
pretty close to this point.

585
00:35:46,400 --> 00:35:49,575
I believe, well, of course,
nobody knows the numbers,

586
00:35:49,575 --> 00:35:54,730
unfortunately, because
the model isn't perfect.

587
00:35:54,730 --> 00:36:02,160
If the model was perfect, then
we could just takes the census

588
00:36:02,160 --> 00:36:04,320
and we would know a and b.

589
00:36:04,320 --> 00:36:11,470
But the model isn't that great.

590
00:36:11,470 --> 00:36:17,330
But it's sort of, we're at
a very interesting time,

591
00:36:17,330 --> 00:36:20,390
close to a very
interesting time.

592
00:36:20,390 --> 00:36:24,780
I believe that with reasonable
numbers, this a over b

593
00:36:24,780 --> 00:36:27,310
might be maybe 12 billion.

594
00:36:31,590 --> 00:36:36,300
And we might be, I think we're
a little above six billion.

595
00:36:36,300 --> 00:36:37,410
I think so.

596
00:36:37,410 --> 00:36:39,870
So we're a little bit past it.

597
00:36:39,870 --> 00:36:42,730
This is now.

598
00:36:42,730 --> 00:36:45,880
This is halfway.

599
00:36:45,880 --> 00:36:47,510
That's the halfway point.

600
00:36:47,510 --> 00:36:49,490
It's perfectly symmetric.

601
00:36:49,490 --> 00:36:51,035
It's called an S curve.

602
00:36:55,350 --> 00:37:05,890
And many, many equations in
math biology involve S curves.

603
00:37:05,890 --> 00:37:12,324
So math biology often gives
rise, with simple models,

604
00:37:12,324 --> 00:37:15,470
to a kind of problem
we've had here

605
00:37:15,470 --> 00:37:19,380
with a quadratic term
slowing things down.

606
00:37:19,380 --> 00:37:24,040
Enzymes, all kinds of.

607
00:37:24,040 --> 00:37:26,730
Ordinary differential
equations are core ideas

608
00:37:26,730 --> 00:37:33,230
in a lot of topics, lot
of areas of science.

609
00:37:33,230 --> 00:37:35,490
OK.

610
00:37:35,490 --> 00:37:39,550
Do I want to say more about
the logistic equation?

611
00:37:39,550 --> 00:37:44,450
I guess I do want to
distinguish one thing.

612
00:37:44,450 --> 00:37:46,000
Yeah.

613
00:37:46,000 --> 00:37:49,730
One thing about logistic
equations and will of course

614
00:37:49,730 --> 00:37:51,970
come back to this.

615
00:37:51,970 --> 00:37:52,470
OK.

616
00:37:52,470 --> 00:37:54,222
Let me look at that
logistic equation.

617
00:37:56,900 --> 00:37:58,110
Here's my equation.

618
00:38:00,680 --> 00:38:02,600
So I've managed to solve it.

619
00:38:02,600 --> 00:38:03,130
Fine.

620
00:38:03,130 --> 00:38:04,100
Great.

621
00:38:04,100 --> 00:38:05,460
Even graph it.

622
00:38:05,460 --> 00:38:09,000
But let me come back
to the question,

623
00:38:09,000 --> 00:38:12,520
suppose I just look at.

624
00:38:12,520 --> 00:38:19,800
I can see two constant
solutions, two steady states,

625
00:38:19,800 --> 00:38:24,210
two solutions where
the derivative is 0.

626
00:38:24,210 --> 00:38:26,260
So nothing will happen.

627
00:38:26,260 --> 00:38:33,910
So in other words, I want to set
this thing set to 0 equal to 0

628
00:38:33,910 --> 00:38:37,390
to find steady solutions.

629
00:38:37,390 --> 00:38:39,850
Steady means the
derivative is 0.

630
00:38:39,850 --> 00:38:41,930
So this side has to be 0.

631
00:38:41,930 --> 00:38:50,820
So what are the two possible
steady states where, if y of 0

632
00:38:50,820 --> 00:38:53,990
is there, it'll stay there?

633
00:38:53,990 --> 00:38:54,830
AUDIENCE: 0.

634
00:38:54,830 --> 00:38:55,830
PROFESSOR: 0.

635
00:38:55,830 --> 00:38:58,510
Y equals 0 is one.

636
00:38:58,510 --> 00:38:59,730
And the other?

637
00:38:59,730 --> 00:39:00,960
AUDIENCE: A over b.

638
00:39:00,960 --> 00:39:02,880
PROFESSOR: Is a over b.

639
00:39:02,880 --> 00:39:04,105
So steady equal to 0.

640
00:39:04,105 --> 00:39:05,830
And I get two steady states.

641
00:39:05,830 --> 00:39:08,890
Let me call them
capital Y equals 0

642
00:39:08,890 --> 00:39:12,330
because that's certainly 0
of, if we have 0 population,

643
00:39:12,330 --> 00:39:14,750
we'll never move.

644
00:39:14,750 --> 00:39:20,090
Or setting this to
0, ay is by squared

645
00:39:20,090 --> 00:39:25,480
cancel y's divide by b a over b.

646
00:39:25,480 --> 00:39:29,350
So the two steady
states are here.

647
00:39:29,350 --> 00:39:33,220
That's a steady state and
that's a steady state.

648
00:39:35,960 --> 00:39:39,020
Those are the only
two in this problem.

649
00:39:39,020 --> 00:39:41,760
You see how easy that was
to find the steady states?

650
00:39:41,760 --> 00:39:44,510
That's an important thing to do.

651
00:39:44,510 --> 00:39:48,640
And then the other important
thing to do is to decide,

652
00:39:48,640 --> 00:39:53,690
are those steady states stable?

653
00:39:53,690 --> 00:39:56,470
When the population's
near a steady state,

654
00:39:56,470 --> 00:40:00,010
does it approach that, does
it go toward that steady state

655
00:40:00,010 --> 00:40:02,480
or away?

656
00:40:02,480 --> 00:40:03,860
So what's the answer?

657
00:40:03,860 --> 00:40:10,690
For this steady state, that
steady state, y is a over b.

658
00:40:10,690 --> 00:40:13,060
Is that stable or unstable?

659
00:40:13,060 --> 00:40:15,420
So I'll write the word stable.

660
00:40:15,420 --> 00:40:20,267
And I'm prepared to put in "un,"
unstable, if you want me to.

661
00:40:24,470 --> 00:40:26,840
This is a key, key idea.

662
00:40:26,840 --> 00:40:28,700
And with nonlinear
equations, you

663
00:40:28,700 --> 00:40:34,140
can answer this stability
stuff without formulas.

664
00:40:34,140 --> 00:40:35,030
Without formulas.

665
00:40:35,030 --> 00:40:36,000
That's the nice thing.

666
00:40:36,000 --> 00:40:39,090
And then that comes
in a later class.

667
00:40:39,090 --> 00:40:41,490
But here's a perfect example.

668
00:40:41,490 --> 00:40:47,630
So do we approach this
answer or do we leave it?

669
00:40:47,630 --> 00:40:52,090
We approach it, the solutions.

670
00:40:52,090 --> 00:40:54,930
This is stable, yes.

671
00:40:54,930 --> 00:40:59,220
And here's the other
stationary point,

672
00:40:59,220 --> 00:41:07,590
capital Y. The other steady
state is that nothing happens.

673
00:41:07,590 --> 00:41:17,790
So now if I'm close to that, if
y is a little number, like 2,

674
00:41:17,790 --> 00:41:23,100
will that 2 drop to 0, will
it approach this steady state,

675
00:41:23,100 --> 00:41:24,914
or will it leave it?

676
00:41:24,914 --> 00:41:25,840
AUDIENCE: Leave it.

677
00:41:25,840 --> 00:41:26,740
PROFESSOR: Leave it.

678
00:41:26,740 --> 00:41:28,245
So this steady state is.

679
00:41:28,245 --> 00:41:29,110
AUDIENCE: Unstable.

680
00:41:29,110 --> 00:41:31,075
PROFESSOR: Unstable.

681
00:41:31,075 --> 00:41:31,575
Unstable.

682
00:41:36,610 --> 00:41:37,580
Right.

683
00:41:37,580 --> 00:41:39,470
Right.

684
00:41:39,470 --> 00:41:41,610
With linear equations,
we really only

685
00:41:41,610 --> 00:41:44,590
had one steady state, like 0.

686
00:41:44,590 --> 00:41:47,480
Once it started, it
took off forever.

687
00:41:47,480 --> 00:41:50,400
Here, it doesn't
go infinitely high.

688
00:41:53,640 --> 00:41:58,700
It bends down again
to that limit,

689
00:41:58,700 --> 00:42:02,071
that carrying capacity is
what it's called, a over b.

690
00:42:04,777 --> 00:42:08,560
I guess I hope you
think a nonlinear

691
00:42:08,560 --> 00:42:12,720
equation like got a
little more to it.

692
00:42:12,720 --> 00:42:15,760
Little more interesting, but
a little more complicated,

693
00:42:15,760 --> 00:42:17,533
than linear equations.

694
00:42:17,533 --> 00:42:18,032
Yep.

695
00:42:18,032 --> 00:42:18,886
Yep.

696
00:42:18,886 --> 00:42:19,385
Yep.

697
00:42:20,170 --> 00:42:24,900
And similarly, the
rocket equation, we

698
00:42:24,900 --> 00:42:29,200
could at the right time soon in
the course, ask the same thing,

699
00:42:29,200 --> 00:42:32,550
a rocket equation was
something like that.

700
00:42:32,550 --> 00:42:34,140
What are the steady states?

701
00:42:34,140 --> 00:42:35,060
Are they stable?

702
00:42:35,060 --> 00:42:36,720
Are they unstable?

703
00:42:36,720 --> 00:42:38,780
Can you find a formula?

704
00:42:38,780 --> 00:42:39,280
Here.

705
00:42:39,280 --> 00:42:40,800
This.

706
00:42:40,800 --> 00:42:41,530
We got a formula.

707
00:42:44,330 --> 00:42:48,440
And there are other nonlinear
equations, which we'll see.

708
00:42:51,650 --> 00:42:52,150
OK.

709
00:42:52,150 --> 00:42:54,480
I could create more
separable equations,

710
00:42:54,480 --> 00:42:59,950
but I guess I hope that you
see with separable equations,

711
00:42:59,950 --> 00:43:03,500
you just separate them
and integrate a y integral

712
00:43:03,500 --> 00:43:04,600
and a t integral.

713
00:43:05,380 --> 00:43:11,300
Is that OK any question on
this nonlinear separable stuff?

714
00:43:16,490 --> 00:43:19,920
Differential equations
courses and the subject

715
00:43:19,920 --> 00:43:25,630
tends to be types of
equations as can solve.

716
00:43:25,630 --> 00:43:30,190
And then there are a hell of a
lot of equations that are not

717
00:43:30,190 --> 00:43:33,880
on anybody's list, where
you could maybe solve them

718
00:43:33,880 --> 00:43:39,320
by an infinite series, but
not by functions that we know.

719
00:43:39,320 --> 00:43:40,720
OK.

720
00:43:40,720 --> 00:43:46,380
I'm ready to do the
other topic for today.

721
00:43:46,380 --> 00:43:52,620
It's the topic that I
left incomplete on Monday.

722
00:43:52,620 --> 00:43:54,890
So I'm staying with
first order equations,

723
00:43:54,890 --> 00:43:59,060
but actually this topic is
essential for second order

724
00:43:59,060 --> 00:43:59,630
equations.

725
00:43:59,630 --> 00:44:05,020
So I'm going to
topic two for today.

726
00:44:05,020 --> 00:44:10,070
So topic two will
involve complex numbers.

727
00:44:10,070 --> 00:44:14,740
So we have to deal
with complex numbers.

728
00:44:14,740 --> 00:44:22,370
And the purpose of introducing
these complex numbers

729
00:44:22,370 --> 00:44:28,370
is to deal with what we met
last time when the right hand

730
00:44:28,370 --> 00:44:32,350
side, the forcing
term, was a cosine.

731
00:44:32,350 --> 00:44:37,640
Typical alternating current,
oscillating, rotating,

732
00:44:37,640 --> 00:44:38,330
rotation.

733
00:44:38,330 --> 00:44:42,110
All these things
produce trig functions.

734
00:44:42,110 --> 00:44:46,960
Maybe rotation is more of
a mechanical engineering

735
00:44:46,960 --> 00:44:49,220
phenomenon.

736
00:44:49,220 --> 00:44:52,100
Alternating current more
of an EE phenomenon.

737
00:44:52,100 --> 00:44:53,350
But they're always there.

738
00:44:54,231 --> 00:44:58,000
And what was the point?

739
00:44:58,000 --> 00:45:04,530
The point was we had
some linear equation,

740
00:45:04,530 --> 00:45:13,860
and we had some forcing by
something like cos omega t.

741
00:45:13,860 --> 00:45:19,510
Or it could be A cos omega
t and B sine omega t.

742
00:45:25,530 --> 00:45:31,970
Either just cosine alone, or
maybe these come together.

743
00:45:31,970 --> 00:45:39,290
And then the
solution was y equals

744
00:45:39,290 --> 00:45:41,960
some combination
of those same guys.

745
00:45:49,690 --> 00:45:53,980
In other words,
what I'm saying is

746
00:45:53,980 --> 00:46:00,080
cosines are nice right
hand forcing functions.

747
00:46:00,080 --> 00:46:03,530
Fortunately, because we
see them all the time.

748
00:46:03,530 --> 00:46:07,710
But they do lead to
cosines and sines.

749
00:46:07,710 --> 00:46:10,020
I emphasized that last time.

750
00:46:10,020 --> 00:46:13,050
If we just have cosines
in the forcing function,

751
00:46:13,050 --> 00:46:16,190
we can't expect that
there's any damping,

752
00:46:16,190 --> 00:46:18,210
we can't expect only cosines.

753
00:46:18,210 --> 00:46:20,170
We have to expect some sines.

754
00:46:20,170 --> 00:46:26,540
In other words, we have to
deal with combinations of them.

755
00:46:26,540 --> 00:46:32,720
And the question is, how do you
understand cos omega t plus 3.

756
00:46:32,720 --> 00:46:35,540
Or let me take a first example.

757
00:46:35,540 --> 00:46:38,598
Example-- cos t plus sine t.

758
00:46:43,410 --> 00:46:46,100
That's a perfect example.

759
00:46:46,100 --> 00:46:50,900
So what is omega here in this
example that I'm starting with?

760
00:46:50,900 --> 00:46:51,400
AUDIENCE: 1.

761
00:46:51,400 --> 00:46:52,840
PROFESSOR: 1.

762
00:46:52,840 --> 00:46:55,360
So I just read off
the coefficient of t

763
00:46:55,360 --> 00:46:59,550
is 1, 1 hertz here.

764
00:46:59,550 --> 00:47:01,800
But we have got
this combination.

765
00:47:01,800 --> 00:47:06,780
And the question is, how do
we understand that cosine

766
00:47:06,780 --> 00:47:07,590
plus sine?

767
00:47:07,590 --> 00:47:10,410
Two very simple functions, but
they're added, unfortunately.

768
00:47:14,240 --> 00:47:16,930
And there's a much
better way to write this

769
00:47:16,930 --> 00:47:19,540
so you really see it.

770
00:47:19,540 --> 00:47:21,630
You really see this.

771
00:47:21,630 --> 00:47:24,510
That's called a sinusoid.

772
00:47:24,510 --> 00:47:28,380
And the rule that
want to focus on now

773
00:47:28,380 --> 00:47:34,250
is that everything of that kind,
of this kind, of this kind,

774
00:47:34,250 --> 00:47:40,240
of a cosine plus a sine, can
be compressed into one term.

775
00:47:40,240 --> 00:47:42,120
One term.

776
00:47:42,120 --> 00:47:46,640
Of course, it's got
to have two constants

777
00:47:46,640 --> 00:47:50,090
to choose because
that had an a and a b.

778
00:47:50,090 --> 00:47:51,540
This had an m and an n.

779
00:47:51,540 --> 00:47:53,700
This had a 1 and a 1.

780
00:47:53,700 --> 00:47:58,750
But the term I'm looking
for is some number R

781
00:47:58,750 --> 00:48:09,060
times a pure cosine of omega
t, but with a phase shift.

782
00:48:09,060 --> 00:48:11,510
So you see there are two
numbers here to choose.

783
00:48:11,510 --> 00:48:15,595
It's really like going
from rectangular to polar.

784
00:48:19,570 --> 00:48:21,590
Say in complex
numbers, let's just

785
00:48:21,590 --> 00:48:25,820
remember the first fact
about a complex number.

786
00:48:25,820 --> 00:48:32,460
If the real part is 3, and the
imaginary part is, let's say 2,

787
00:48:32,460 --> 00:48:36,760
then here's a complex
number, 3 plus 2 i.

788
00:48:39,320 --> 00:48:41,940
So this was the real axis.

789
00:48:41,940 --> 00:48:44,450
This was the imaginary axis.

790
00:48:44,450 --> 00:48:48,530
I went along 3, I went up
2, I got to that number.

791
00:48:48,530 --> 00:48:50,450
There it is.

792
00:48:50,450 --> 00:48:54,282
I plotted the number 3 plus
2 i in the complex plane.

793
00:48:57,060 --> 00:49:00,820
And for me, that number
3 plus and so on,

794
00:49:00,820 --> 00:49:02,360
really saying
something important.

795
00:49:02,360 --> 00:49:06,150
And maybe it's not entirely new.

796
00:49:06,150 --> 00:49:09,360
I'm saying something important
about complex numbers,

797
00:49:09,360 --> 00:49:11,570
this is their rectangular form.

798
00:49:11,570 --> 00:49:12,750
Something plus something.

799
00:49:16,410 --> 00:49:20,700
That form is nice to add
to another complex number.

800
00:49:20,700 --> 00:49:27,115
If I added 3 plus 2 i to 1
plus i, what would I get?

801
00:49:27,115 --> 00:49:28,480
AUDIENCE: 4 plus 3 i.

802
00:49:28,480 --> 00:49:31,020
PROFESSOR: 4 plus 3 i.

803
00:49:31,020 --> 00:49:36,560
But if I multiply,
multiply, 3 plus 2 i times,

804
00:49:36,560 --> 00:49:38,450
let's say I square it.

805
00:49:38,450 --> 00:49:43,260
I multiply 3 plus
2 i by 3 plus 2 i.

806
00:49:43,260 --> 00:49:45,940
What do I get?

807
00:49:45,940 --> 00:49:50,280
If I do it with this
rectangular form, I get a mess.

808
00:49:50,280 --> 00:49:53,600
I can't see what's happening.

809
00:49:53,600 --> 00:49:55,970
It's the same over here.

810
00:49:55,970 --> 00:49:59,880
This is like having a 1
and a 1, with an addition.

811
00:49:59,880 --> 00:50:05,320
This is like a polar
form where it's one term.

812
00:50:05,320 --> 00:50:06,110
OK.

813
00:50:06,110 --> 00:50:08,440
So let me answer the
question here and then let

814
00:50:08,440 --> 00:50:09,930
me answer the question there.

815
00:50:09,930 --> 00:50:15,740
And then you've got a good shot
at what complex numbers can do,

816
00:50:15,740 --> 00:50:21,024
and why we like the polar form
for squaring, for multiplying,

817
00:50:21,024 --> 00:50:21,565
for dividing.

818
00:50:24,510 --> 00:50:26,820
What's the polar form?

819
00:50:26,820 --> 00:50:28,830
Well, I'm using
that word "polar"

820
00:50:28,830 --> 00:50:31,850
in the same way we
use polar coordinates.

821
00:50:31,850 --> 00:50:35,170
What are the polar
coordinates of this point?

822
00:50:35,170 --> 00:50:39,202
They're the radial
distance, which is what?

823
00:50:39,202 --> 00:50:40,670
So what's that distance?

824
00:50:40,670 --> 00:50:43,590
That's the R you could say.

825
00:50:43,590 --> 00:50:46,790
It corresponds to this R here.

826
00:50:46,790 --> 00:50:49,790
So I'm just using Pythagoras.

827
00:50:49,790 --> 00:50:54,132
That hypotenuse is what?

828
00:50:54,132 --> 00:50:55,380
AUDIENCE: Square root of 13.

829
00:50:55,380 --> 00:50:56,350
PROFESSOR: Square root of 13.

830
00:50:56,350 --> 00:50:56,960
Thanks.

831
00:50:56,960 --> 00:51:00,340
9 plus 4, square root of 13.

832
00:51:00,340 --> 00:51:03,350
And what's the
other number that's

833
00:51:03,350 --> 00:51:06,830
locating this in
polar coordinates?

834
00:51:06,830 --> 00:51:08,290
The angle.

835
00:51:08,290 --> 00:51:09,230
And the angle.

836
00:51:09,230 --> 00:51:11,500
What can we say
about that angle?

837
00:51:11,500 --> 00:51:19,250
Let's call it phi
is-- what's the angle?

838
00:51:21,920 --> 00:51:26,310
Well, it's some number.

839
00:51:26,310 --> 00:51:31,220
It's between 0 and pi
over 2, I'm sure of that.

840
00:51:31,220 --> 00:51:33,130
What do I know about that angle?

841
00:51:41,780 --> 00:51:45,270
I know that this
is 2 and this is 3.

842
00:51:45,270 --> 00:51:47,270
So that's telling me the angle.

843
00:51:47,270 --> 00:51:50,170
Well, what is that really
telling me immediately?

844
00:51:50,170 --> 00:51:51,590
It's telling me the.

845
00:51:51,590 --> 00:51:52,340
AUDIENCE: Tangent.

846
00:51:52,340 --> 00:51:54,250
PROFESSOR: Tangent of the angle.

847
00:51:54,250 --> 00:52:02,730
So the tangent of the
angle is 2 over 3.

848
00:52:02,730 --> 00:52:05,745
And the magnitude is
square root of 13.

849
00:52:08,320 --> 00:52:08,820
OK.

850
00:52:11,710 --> 00:52:16,130
So those beautiful
numbers, 2 and 3,

851
00:52:16,130 --> 00:52:19,500
have become a little weirder.

852
00:52:19,500 --> 00:52:23,120
Square root of 13,
inverse tangent of 2/3.

853
00:52:23,120 --> 00:52:24,880
You could say, well,
that's not so nice.

854
00:52:29,364 --> 00:52:30,600
What was I going to do?

855
00:52:30,600 --> 00:52:34,030
I was going to try
squaring that number.

856
00:52:34,030 --> 00:52:40,090
So if I square 3 plus 2 i, or
if I take the 10th power of 3

857
00:52:40,090 --> 00:52:47,380
plus 2 i, or the exponential,
all these things,

858
00:52:47,380 --> 00:52:49,800
then I'm happy with
polar coordinates.

859
00:52:49,800 --> 00:52:55,450
Like, what would be the
magnitude of the square?

860
00:52:55,450 --> 00:52:59,860
And where will the square
of that number, so I want

861
00:52:59,860 --> 00:53:04,820
to put in 3 plus 2 i squared,
which I can figure out

862
00:53:04,820 --> 00:53:10,730
in rectangular, of
course-- a 9, and 6 i,

863
00:53:10,730 --> 00:53:14,020
or 12 i, or 4 i squared,
stuff like that.

864
00:53:16,910 --> 00:53:18,710
It's not pleasant.

865
00:53:18,710 --> 00:53:24,100
What's the magnitude,
what's the R for this guy?

866
00:53:24,100 --> 00:53:27,761
What's the size of
that number squared?

867
00:53:27,761 --> 00:53:28,260
Yes?

868
00:53:28,260 --> 00:53:29,269
Say that again.

869
00:53:29,269 --> 00:53:29,810
AUDIENCE: 13.

870
00:53:29,810 --> 00:53:31,210
PROFESSOR: 13.

871
00:53:31,210 --> 00:53:33,000
Right.

872
00:53:33,000 --> 00:53:37,160
I just have to square this
square root so I get 13.

873
00:53:37,160 --> 00:53:43,935
And the angle will be, what's
the angle for the square there?

874
00:53:43,935 --> 00:53:45,080
I don't want a number.

875
00:53:50,510 --> 00:53:52,720
I guess I'm just doing this.

876
00:53:52,720 --> 00:53:58,880
R e to the i phi
squared is R squared.

877
00:53:58,880 --> 00:54:00,540
And what's the angle here?

878
00:54:05,680 --> 00:54:11,370
E to the i phi squared
is e to the 2 i phi.

879
00:54:11,370 --> 00:54:13,740
It's the angle doubled.

880
00:54:13,740 --> 00:54:16,870
E to the 2 i phi.

881
00:54:16,870 --> 00:54:20,680
The angle just went
from phi to 2 phi.

882
00:54:20,680 --> 00:54:23,790
The lengths went from
square root of 13 to 13.

883
00:54:26,320 --> 00:54:30,580
Squaring, multiplying is
nice with complex numbers.

884
00:54:30,580 --> 00:54:35,440
Maybe can I before I go on
and on about complex numbers,

885
00:54:35,440 --> 00:54:40,430
I should ask you, how many
know all this already?

886
00:54:40,430 --> 00:54:42,761
Complex numbers are familiar?

887
00:54:42,761 --> 00:54:43,260
Mostly.

888
00:54:46,450 --> 00:54:48,131
Correctly, with a wiggle.

889
00:54:48,131 --> 00:54:48,630
OK.

890
00:54:51,800 --> 00:54:56,470
I won't go more about
complex numbers.

891
00:54:56,470 --> 00:54:59,070
Let me come back to
my question here.

892
00:55:02,110 --> 00:55:05,520
Let me come back
to the application.

893
00:55:05,520 --> 00:55:08,390
So here it is with
complex numbers.

894
00:55:08,390 --> 00:55:10,270
Here it is with sinusoids.

895
00:55:10,270 --> 00:55:13,850
And the little
beautiful bit of math

896
00:55:13,850 --> 00:55:18,060
is that the sinusoid
question goes completely

897
00:55:18,060 --> 00:55:22,020
parallel to the complex
number question.

898
00:55:22,020 --> 00:55:25,260
So you have an idea on
those complex numbers.

899
00:55:25,260 --> 00:55:27,050
We'll see them again.

900
00:55:27,050 --> 00:55:29,160
Let me go to this.

901
00:55:29,160 --> 00:55:34,290
So I want this to be
the same as this, OK.

902
00:55:34,290 --> 00:55:37,792
Maybe I'm going to have to
use a new board for this.

903
00:55:37,792 --> 00:55:40,820
Can I start a new board?

904
00:55:40,820 --> 00:55:46,660
So I want cos t
plus sine t to be

905
00:55:46,660 --> 00:55:53,910
some number R times
cosine of t 1.

906
00:55:53,910 --> 00:55:59,990
I can see omega's 1, so I just
put to 1 minus some angle.

907
00:55:59,990 --> 00:56:02,100
OK.

908
00:56:02,100 --> 00:56:05,830
And I want to choose R and
phi to make that right.

909
00:56:05,830 --> 00:56:08,100
You see what I like about it?

910
00:56:08,100 --> 00:56:13,360
This tells me the magnitude
of the oscillation.

911
00:56:13,360 --> 00:56:19,080
It tells me how
loud the station is.

912
00:56:19,080 --> 00:56:22,140
When I see cos t and,
separately, sine t,

913
00:56:22,140 --> 00:56:24,990
or I might see 3
cos t and 2 sine t.

914
00:56:27,990 --> 00:56:31,790
3 cos t is a cosine curve.

915
00:56:31,790 --> 00:56:35,810
2 sine t is a sine
curve shifted by 90.

916
00:56:35,810 --> 00:56:38,395
I put them together, it
bumps, it bumps, bumps.

917
00:56:41,100 --> 00:56:43,570
Not completely clear.

918
00:56:43,570 --> 00:56:48,260
It seems to me just beautiful
that if I put together

919
00:56:48,260 --> 00:56:54,870
a cosine curve that we
know, that starts at 0,

920
00:56:54,870 --> 00:57:00,380
with a sine curve
that starts at 0,

921
00:57:00,380 --> 00:57:04,340
the combination
is a cosine curve.

922
00:57:04,340 --> 00:57:06,230
Isn't that nice?

923
00:57:06,230 --> 00:57:07,940
I mean, you know,
that sometimes math

924
00:57:07,940 --> 00:57:10,210
gets worse and worse
whatever you do.

925
00:57:10,210 --> 00:57:14,750
But this is really nice that
we can put the two into one.

926
00:57:14,750 --> 00:57:20,470
But you see, it's going
to-- well, let's do it.

927
00:57:20,470 --> 00:57:24,440
What would R and phi be here?

928
00:57:24,440 --> 00:57:29,280
So I'll use a trig fact here.

929
00:57:29,280 --> 00:57:32,160
A cosine of a
difference of angles,

930
00:57:32,160 --> 00:57:37,536
so this is R times cosine
t, do you member this?

931
00:57:37,536 --> 00:57:41,870
This was the whole point
of going to high school.

932
00:57:41,870 --> 00:57:45,650
Plus sine t sine phi.

933
00:57:52,810 --> 00:57:59,290
So now, how do I get R and phi?

934
00:57:59,290 --> 00:58:03,840
I use the same idea
that worked last time.

935
00:58:03,840 --> 00:58:08,220
I match the cosine terms
and I match the sine terms.

936
00:58:08,220 --> 00:58:10,680
So the cosine t has a 1.

937
00:58:10,680 --> 00:58:14,530
1 cosine t is R cos phi.

938
00:58:18,120 --> 00:58:21,360
That's what's
multiplying cosine t.

939
00:58:21,360 --> 00:58:24,540
And the sine t has a 1.

940
00:58:24,540 --> 00:58:28,250
And that has to agree
with R sine phi.

941
00:58:34,770 --> 00:58:41,690
So I'm in business if I
solve those two equations.

942
00:58:44,770 --> 00:58:48,390
And well, they're
not linear equations.

943
00:58:48,390 --> 00:58:49,790
But I can solve them.

944
00:58:49,790 --> 00:58:53,740
Of course, the one fact
that you never forget

945
00:58:53,740 --> 00:58:56,590
is that sine squared
plus cosine squared is 1.

946
00:58:56,590 --> 00:58:57,840
Right?

947
00:58:57,840 --> 00:59:01,510
So if I square that one, and
square that one, and add,

948
00:59:01,510 --> 00:59:02,670
what will I get?

949
00:59:02,670 --> 00:59:07,110
1 squared and 1 squared will
be 2, on the left hand side.

950
00:59:07,110 --> 00:59:11,400
On the right hand side, I'll
have R squared cos squared,

951
00:59:11,400 --> 00:59:17,870
R squared cos squared, and
plus R squared sine squared.

952
00:59:17,870 --> 00:59:18,964
And what's that?

953
00:59:21,646 --> 00:59:25,436
What's R squared cosine squared
plus r squared sine squared?

954
00:59:25,436 --> 00:59:26,390
AUDIENCE: R squared.

955
00:59:26,390 --> 00:59:28,610
PROFESSOR: It's just R squared.

956
00:59:28,610 --> 00:59:33,470
So all that added
up to R squared.

957
00:59:33,470 --> 00:59:37,370
In other words, it's just
like polar coordinates.

958
00:59:37,370 --> 00:59:39,010
R is the square root of 2.

959
00:59:43,900 --> 00:59:50,870
That's telling us the
magnitude of the response.

960
00:59:50,870 --> 00:59:53,060
Square root of 2.

961
00:59:53,060 --> 00:59:57,390
You see, it's just
like complex numbers.

962
00:59:57,390 --> 01:00:00,170
It's like the cosine
gave us a real part

963
01:00:00,170 --> 01:00:02,800
and the sine gave us
an imaginary part.

964
01:00:02,800 --> 01:00:08,200
And R was the hypotenuse.

965
01:00:08,200 --> 01:00:12,400
And that's really nice.

966
01:00:12,400 --> 01:00:13,930
So R is the square root of 2.

967
01:00:13,930 --> 01:00:14,640
OK.

968
01:00:14,640 --> 01:00:18,650
Now, the angle is
never quite as nice.

969
01:00:18,650 --> 01:00:22,450
But how can we get something
about an angle out of there?

970
01:00:22,450 --> 01:00:25,510
All we could get
in this case here

971
01:00:25,510 --> 01:00:27,710
was the tangent of the angle.

972
01:00:27,710 --> 01:00:30,350
And I'll be happy
with that again here

973
01:00:30,350 --> 01:00:35,750
because it's a totally
parallel question.

974
01:00:35,750 --> 01:00:39,430
How am I going to get
the tangent of the angle?

975
01:00:45,220 --> 01:00:48,120
What do I have?

976
01:00:48,120 --> 01:00:51,110
From these two equations,
I want to eliminate R.

977
01:00:51,110 --> 01:00:52,390
So how do I eliminate R?

978
01:00:57,540 --> 01:00:58,960
What do I do?

979
01:00:58,960 --> 01:00:59,940
Divide.

980
01:00:59,940 --> 01:01:00,970
Divide.

981
01:01:00,970 --> 01:01:04,180
I guess if I want
tangent sine over cosine,

982
01:01:04,180 --> 01:01:08,100
I'll divide this one in the
top by this one in the bottom.

983
01:01:08,100 --> 01:01:09,780
So I take the ratio.

984
01:01:09,780 --> 01:01:11,900
That'll cancel the Rs perfectly.

985
01:01:11,900 --> 01:01:13,650
It'll leave me with 10 phi.

986
01:01:13,650 --> 01:01:16,570
And here it happens to be 1.

987
01:01:16,570 --> 01:01:17,070
OK.

988
01:01:19,890 --> 01:01:21,840
So what have I learned?

989
01:01:21,840 --> 01:01:29,870
I've learned that when
these two add up together,

990
01:01:29,870 --> 01:01:33,250
they equal what?

991
01:01:33,250 --> 01:01:36,220
R square root of 2.

992
01:01:36,220 --> 01:01:37,350
You see how easy it is.

993
01:01:37,350 --> 01:01:39,600
Square root of 2 came from
the square root of 1 plus.

994
01:01:39,600 --> 01:01:42,350
It's like Pythagoras.

995
01:01:42,350 --> 01:01:45,290
Pythagoras going
in circles, really.

996
01:01:45,290 --> 01:01:52,190
Times the cosine of t minus.

997
01:01:52,190 --> 01:01:53,210
And what is phi?

998
01:01:55,750 --> 01:01:59,307
Its tangent is 1, so
what's the angle phi?

999
01:01:59,307 --> 01:02:00,140
AUDIENCE: Pi over 4.

1000
01:02:00,140 --> 01:02:01,911
PROFESSOR: Pi over 4.

1001
01:02:01,911 --> 01:02:02,411
Right.

1002
01:02:06,310 --> 01:02:09,490
So that's the
sinusoidal identity

1003
01:02:09,490 --> 01:02:13,380
when the numbers are 1 and 1.

1004
01:02:13,380 --> 01:02:15,135
But you saw the general rule.

1005
01:02:19,870 --> 01:02:23,640
Let me just take it.

1006
01:02:23,640 --> 01:02:31,200
Suppose this is the output, and
cos omega t plus n sine omega

1007
01:02:31,200 --> 01:02:32,440
t.

1008
01:02:32,440 --> 01:02:33,940
What is the gain?

1009
01:02:33,940 --> 01:02:36,510
What's the magnitude,
the amplitude,

1010
01:02:36,510 --> 01:02:41,780
the loudness of the volume in
this when I'm tuning the radio?

1011
01:02:41,780 --> 01:02:44,790
What's the R for this guy?

1012
01:02:44,790 --> 01:02:45,545
What's this R?

1013
01:02:49,310 --> 01:02:53,360
If we just follow the same idea.

1014
01:02:53,360 --> 01:02:58,010
So if we have m times a
cosine and n times a sine,

1015
01:02:58,010 --> 01:02:59,200
what's your guess?

1016
01:02:59,200 --> 01:03:05,190
What's your guess
for R, the magnitude?

1017
01:03:05,190 --> 01:03:08,320
I'm guessing a
square root of what?

1018
01:03:12,560 --> 01:03:13,180
Yeah?

1019
01:03:13,180 --> 01:03:14,030
You got.

1020
01:03:14,030 --> 01:03:15,600
What is it?

1021
01:03:15,600 --> 01:03:16,440
n squared--

1022
01:03:16,440 --> 01:03:17,387
[INTERPOSING VOICES]

1023
01:03:17,387 --> 01:03:18,470
PROFESSOR: Plus n squared.

1024
01:03:18,470 --> 01:03:19,450
Way to go.

1025
01:03:19,450 --> 01:03:20,810
M squared plus N squared.

1026
01:03:23,370 --> 01:03:27,590
And the angle is
like the phase shift.

1027
01:03:27,590 --> 01:03:30,040
I'm not great at
graphing, but let

1028
01:03:30,040 --> 01:03:34,280
me try to go back to
my simple example.

1029
01:03:34,280 --> 01:03:40,250
If I tried to add up on
the same graph cosine

1030
01:03:40,250 --> 01:03:46,450
t, which would start from 1 and
drop to 0, go like that, right?

1031
01:03:46,450 --> 01:03:48,920
Something like that
would be cosine.

1032
01:03:48,920 --> 01:03:51,630
And now I want to
add sine t to that.

1033
01:03:51,630 --> 01:03:56,830
So that climbs up to
1 back to 0, down.

1034
01:03:56,830 --> 01:04:00,440
And now if I add those
two, this formula

1035
01:04:00,440 --> 01:04:04,730
is telling me that
it comes out neat.

1036
01:04:04,730 --> 01:04:06,490
Neatly.

1037
01:04:06,490 --> 01:04:11,620
That one plus that one is
another sinusoid with height

1038
01:04:11,620 --> 01:04:12,740
square root of 2.

1039
01:04:12,740 --> 01:04:17,020
If I had different
chalk, I've got

1040
01:04:17,020 --> 01:04:20,690
at least a little bit different.

1041
01:04:20,690 --> 01:04:21,800
But does it start here?

1042
01:04:21,800 --> 01:04:22,600
Of course not.

1043
01:04:22,600 --> 01:04:24,570
It starts here, I guess.

1044
01:04:24,570 --> 01:04:26,120
But it goes up, right?

1045
01:04:26,120 --> 01:04:28,040
Because this comes down,
but this is going up.

1046
01:04:28,040 --> 01:04:32,430
All together, it's up
to, where is the peak?

1047
01:04:32,430 --> 01:04:34,360
Where is the peak on the sum?

1048
01:04:34,360 --> 01:04:37,500
So I'm adding, everybody
sees what I'm doing?

1049
01:04:37,500 --> 01:04:40,780
I'm adding a cosine
curve and a sine curve.

1050
01:04:40,780 --> 01:04:42,210
And it goes up.

1051
01:04:42,210 --> 01:04:44,760
And where does it peak?

1052
01:04:44,760 --> 01:04:48,565
What angle is it
going to peek at?

1053
01:04:48,565 --> 01:04:52,056
What's the biggest
value this gets to?

1054
01:04:52,056 --> 01:04:54,450
AUDIENCE: [INAUDIBLE].

1055
01:04:54,450 --> 01:04:56,880
PROFESSOR: At pi
over 4, it'll peak.

1056
01:04:56,880 --> 01:05:00,430
At pi over 4, it'll be the
cosine of 0, which is 1.

1057
01:05:00,430 --> 01:05:04,720
It's height'll be the magnitude,
the gain, square root of 2.

1058
01:05:04,720 --> 01:05:07,570
So it'll peak at
pi over 4, which

1059
01:05:07,570 --> 01:05:09,540
is probably about there, right?

1060
01:05:09,540 --> 01:05:17,220
Peak at pi over 4 and, I don't
know if I got it right frankly.

1061
01:05:17,220 --> 01:05:20,240
I did my best.

1062
01:05:20,240 --> 01:05:23,470
That's the sum.

1063
01:05:23,470 --> 01:05:24,680
That right there.

1064
01:05:27,750 --> 01:05:33,380
The first key point is
it's a perfect cosine.

1065
01:05:33,380 --> 01:05:38,990
The second key point is
it's a shifted cosine.

1066
01:05:38,990 --> 01:05:41,680
The third key point
is its magnitude

1067
01:05:41,680 --> 01:05:45,990
is the square root of 1 squared
plus 1 squared, or n squared

1068
01:05:45,990 --> 01:05:49,940
plus n squared, or a
squared plus b squared.

1069
01:05:49,940 --> 01:05:53,812
So that's the
sinusoidal identity.

1070
01:05:53,812 --> 01:06:02,890
A key identity and being able to
deal with forcing terms, source

1071
01:06:02,890 --> 01:06:05,744
terms, that are sinusoids.

1072
01:06:05,744 --> 01:06:06,244
OK.

1073
01:06:09,690 --> 01:06:13,070
Now, I'm going to take
one more step since we

1074
01:06:13,070 --> 01:06:18,440
have just like 10
minutes left, and let

1075
01:06:18,440 --> 01:06:21,950
the number i get
in here properly.

1076
01:06:21,950 --> 01:06:25,170
Get a complex number
to show up here.

1077
01:06:25,170 --> 01:06:25,670
OK.

1078
01:06:49,860 --> 01:06:51,570
Before I start on
this, let me recap.

1079
01:06:58,660 --> 01:07:00,485
Let me recap today's lecture.

1080
01:07:03,540 --> 01:07:08,480
It started with nonlinear
separable equations.

1081
01:07:08,480 --> 01:07:13,710
And a great example was the
logistic equation up there,

1082
01:07:13,710 --> 01:07:16,550
with the S curve.

1083
01:07:16,550 --> 01:07:19,880
That took half the lecture.

1084
01:07:19,880 --> 01:07:22,170
The second half
of the lecture has

1085
01:07:22,170 --> 01:07:27,000
started with things
real with sinusoids

1086
01:07:27,000 --> 01:07:29,585
that are combinations
of cosine and sine

1087
01:07:29,585 --> 01:07:35,560
and has written them
in a one term way.

1088
01:07:35,560 --> 01:07:40,340
And now I want to
get the same one term

1089
01:07:40,340 --> 01:07:46,080
picture from using
complex numbers.

1090
01:07:46,080 --> 01:07:48,520
OK.

1091
01:07:48,520 --> 01:07:50,030
OK.

1092
01:07:50,030 --> 01:07:52,440
And everything I
do would be based

1093
01:07:52,440 --> 01:08:02,870
on this great fact from Euler
that e to the i omega t.

1094
01:08:02,870 --> 01:08:06,110
The real part is cosine omega t.

1095
01:08:06,110 --> 01:08:09,246
And the imaginary
part is sine omega t.

1096
01:08:23,029 --> 01:08:26,479
That's a central formula.

1097
01:08:26,479 --> 01:08:29,319
Let me draw it rather
than proving it.

1098
01:08:29,319 --> 01:08:30,650
Let me draw what that means.

1099
01:08:37,760 --> 01:08:39,340
I'm in the complex plane again.

1100
01:08:39,340 --> 01:08:45,310
Real part is the cosine.

1101
01:08:45,310 --> 01:08:48,765
The imaginary part is the sine.

1102
01:08:53,319 --> 01:08:56,920
That number there
is e to the i omega

1103
01:08:56,920 --> 01:09:02,939
t because it's got that real
part and that imaginary part.

1104
01:09:02,939 --> 01:09:04,074
And what's its magnitude?

1105
01:09:06,649 --> 01:09:13,140
What's the R, the polar
distance for cos omega t plus,

1106
01:09:13,140 --> 01:09:17,990
for this number, which
is for this number?

1107
01:09:17,990 --> 01:09:19,899
What's the hypotenuse here?

1108
01:09:19,899 --> 01:09:22,147
Everybody knows.

1109
01:09:22,147 --> 01:09:22,979
AUDIENCE: 1.

1110
01:09:22,979 --> 01:09:23,920
PROFESSOR: 1.

1111
01:09:23,920 --> 01:09:25,470
Hypotenuse is 1.

1112
01:09:25,470 --> 01:09:28,380
Cos squared plus
sine squared is 1.

1113
01:09:28,380 --> 01:09:34,150
So e to the i omega t is
on a circle of radius 1.

1114
01:09:36,979 --> 01:09:41,850
That's the most important
circle in the complex world,

1115
01:09:41,850 --> 01:09:43,810
the circle of radius 1.

1116
01:09:43,810 --> 01:09:46,540
And all these points are on it.

1117
01:09:46,540 --> 01:09:50,765
And their angles are omega t.

1118
01:09:50,765 --> 01:09:53,770
And as t increases,
the angle increases,

1119
01:09:53,770 --> 01:09:55,360
and you go around the circle.

1120
01:09:55,360 --> 01:09:56,030
You've seen it.

1121
01:09:58,780 --> 01:10:05,877
Physics couldn't live
without this model.

1122
01:10:05,877 --> 01:10:06,377
OK.

1123
01:10:10,130 --> 01:10:14,220
So that's basically
what we have to know.

1124
01:10:14,220 --> 01:10:15,650
And now, how do we use it?

1125
01:10:18,300 --> 01:10:23,030
Well, the idea is to
deal with the equation.

1126
01:10:23,030 --> 01:10:26,160
Like, the equation I
had last time was dy,

1127
01:10:26,160 --> 01:10:33,430
dt equals y plus cos t.

1128
01:10:33,430 --> 01:10:39,200
That gave us some trouble
because the solution didn't

1129
01:10:39,200 --> 01:10:42,990
just involve cosines,
it also involved sines.

1130
01:10:42,990 --> 01:10:43,490
Yeah.

1131
01:10:46,700 --> 01:10:54,240
So I want to write that equation
differently, in complex form.

1132
01:10:54,240 --> 01:10:58,130
And this is the key point here.

1133
01:10:58,130 --> 01:11:02,290
So I'm going to look
at the equation dz,

1134
01:11:02,290 --> 01:11:09,410
dt equals z plus e to the i t.

1135
01:11:09,410 --> 01:11:13,380
Well, I'll make that
cos omega t just

1136
01:11:13,380 --> 01:11:24,220
to have a little more, the
units are better, everything's

1137
01:11:24,220 --> 01:11:26,910
better if I have
a frequency there.

1138
01:11:26,910 --> 01:11:30,600
Units of this are seconds
and the units of this

1139
01:11:30,600 --> 01:11:31,639
are 1 over seconds.

1140
01:11:36,630 --> 01:11:38,870
Now, question.

1141
01:11:38,870 --> 01:11:41,270
What's the relation
between the solution

1142
01:11:41,270 --> 01:11:44,820
z to that complex equation
and the solution y

1143
01:11:44,820 --> 01:11:46,026
to that equation?

1144
01:11:50,283 --> 01:11:52,000
Of course, they
have to be related,

1145
01:11:52,000 --> 01:11:56,740
otherwise it was stupid to
move to this complex one.

1146
01:11:56,740 --> 01:12:01,120
My claim is that complex
equation is easy to solve.

1147
01:12:01,120 --> 01:12:05,270
And it gives us the answer
to the real equation.

1148
01:12:05,270 --> 01:12:09,698
And what's the connection
between y and z?

1149
01:12:09,698 --> 01:12:11,180
AUDIENCE: So y's the real part.

1150
01:12:11,180 --> 01:12:13,269
PROFESSOR: Y is,
exactly, say it again.

1151
01:12:13,269 --> 01:12:14,310
AUDIENCE: The real part--

1152
01:12:14,310 --> 01:12:15,710
PROFESSOR: Of z.

1153
01:12:15,710 --> 01:12:17,210
Y is the real part of z.

1154
01:12:20,750 --> 01:12:23,290
So that gives us an idea.

1155
01:12:23,290 --> 01:12:26,856
Solve this equation and
take it's real part.

1156
01:12:30,730 --> 01:12:34,910
If I can solve this equation
without getting into cosine

1157
01:12:34,910 --> 01:12:40,060
and sine separately and
matching, I can stay real.

1158
01:12:40,060 --> 01:12:44,210
I solve the equation by
totally real methods up to now.

1159
01:12:44,210 --> 01:12:48,220
Now I'm going to say,
here's another approach.

1160
01:12:48,220 --> 01:12:51,820
Look at the complex
equation, solve it,

1161
01:12:51,820 --> 01:12:52,940
and take the real part.

1162
01:12:56,180 --> 01:12:57,550
You may prefer one method.

1163
01:12:57,550 --> 01:12:59,020
You may like to stay real.

1164
01:13:01,780 --> 01:13:04,500
In a way, it's a little
more straightforward.

1165
01:13:04,500 --> 01:13:08,690
But the complex one is
the one that will show us,

1166
01:13:08,690 --> 01:13:12,160
it brings out this R,
it brings out the gain,

1167
01:13:12,160 --> 01:13:15,030
it brings out the
important-- engineering

1168
01:13:15,030 --> 01:13:19,310
quantities are important,
if I do it this way.

1169
01:13:19,310 --> 01:13:24,150
Now, I believe that the
solution to that is easy.

1170
01:13:24,150 --> 01:13:27,170
Actually, it is included
in what I did last time.

1171
01:13:27,170 --> 01:13:30,170
It's a linear equation
with a forcing term

1172
01:13:30,170 --> 01:13:33,260
that's a pure exponential.

1173
01:13:33,260 --> 01:13:36,740
And what kind of
solution do I look for?

1174
01:13:36,740 --> 01:13:38,635
I'm looking for a
particular solution.

1175
01:13:41,460 --> 01:13:47,350
If I see an exponential
forcing term, I say, great.

1176
01:13:47,350 --> 01:13:50,300
The solution will
be an exponential.

1177
01:13:50,300 --> 01:13:52,960
So the solution will be sum.

1178
01:13:52,960 --> 01:13:56,780
Z is sum capital Z
e to the i omega t.

1179
01:14:00,960 --> 01:14:01,650
Plug it in.

1180
01:14:05,170 --> 01:14:09,650
What happens if I plug
that in to find capital Z,

1181
01:14:09,650 --> 01:14:12,861
which is just a number?

1182
01:14:12,861 --> 01:14:13,360
Right.

1183
01:14:13,360 --> 01:14:14,900
This is my method.

1184
01:14:14,900 --> 01:14:18,140
This is a linear equation, with
one of those cool right hand

1185
01:14:18,140 --> 01:14:22,285
sides, where the solution has
the same form with a constant,

1186
01:14:22,285 --> 01:14:24,670
and I just have to
find that constant.

1187
01:14:24,670 --> 01:14:26,200
So I plug it in.

1188
01:14:26,200 --> 01:14:27,360
Dz, dt.

1189
01:14:27,360 --> 01:14:32,225
Take the derivative of this,
z i omega will come down.

1190
01:14:32,225 --> 01:14:35,790
E to the i omega t.

1191
01:14:35,790 --> 01:14:39,810
Z is just this, z
to the i omega t.

1192
01:14:39,810 --> 01:14:44,500
And this is just 1
e to the i omega t.

1193
01:14:44,500 --> 01:14:48,450
So I plugged it in, hoping
things would be good.

1194
01:14:48,450 --> 01:14:51,000
And they are because
I can cancel e

1195
01:14:51,000 --> 01:14:55,060
to the i omega t, that's the
beauty of exponentials, leaving

1196
01:14:55,060 --> 01:14:56,420
just a 1 there.

1197
01:14:56,420 --> 01:14:59,090
So what's capital Z?

1198
01:14:59,090 --> 01:15:00,475
What's capital Z then?

1199
01:15:03,630 --> 01:15:05,200
I've got a z here.

1200
01:15:05,200 --> 01:15:07,010
I better bring it over here.

1201
01:15:07,010 --> 01:15:08,790
And I've got the 1 there.

1202
01:15:08,790 --> 01:15:12,050
I think the z is 1 over.

1203
01:15:12,050 --> 01:15:17,190
When I bring that z over here,
do you see what I'm getting?

1204
01:15:17,190 --> 01:15:20,460
It's all multiplying
this e the i omega t.

1205
01:15:20,460 --> 01:15:21,560
It's a number there.

1206
01:15:24,950 --> 01:15:29,370
Z times i omega and
comes over as a minus z.

1207
01:15:29,370 --> 01:15:31,300
What do I have
multiplying z here?

1208
01:15:35,350 --> 01:15:38,480
I see the i omega.

1209
01:15:38,480 --> 01:15:41,490
And what else have I
got multiplying the z?

1210
01:15:41,490 --> 01:15:42,750
AUDIENCE: Minus.

1211
01:15:42,750 --> 01:15:46,018
PROFESSOR: A negative 1 because
it came over with a minus sign.

1212
01:15:49,340 --> 01:15:50,744
Done.

1213
01:15:50,744 --> 01:15:51,410
Equation solved.

1214
01:15:55,040 --> 01:15:56,980
Equation solved.

1215
01:15:56,980 --> 01:15:59,430
Complex equation solved.

1216
01:15:59,430 --> 01:16:03,350
So the point is, the complex
equation was a cinch.

1217
01:16:03,350 --> 01:16:06,190
We just assumed the right
form, plugged it in,

1218
01:16:06,190 --> 01:16:07,712
found the number, we're done.

1219
01:16:10,660 --> 01:16:12,830
But there's one more
step, which is what?

1220
01:16:15,350 --> 01:16:17,590
Take the real part.

1221
01:16:17,590 --> 01:16:19,550
So I have to take the
real part of this.

1222
01:16:19,550 --> 01:16:25,900
So the correct answer is y is
the real part of that number, 1

1223
01:16:25,900 --> 01:16:30,166
over i omega minus 1
times e to the i omega t.

1224
01:16:40,290 --> 01:16:47,850
I'm tempted to stop there, but
just with a little comment.

1225
01:16:47,850 --> 01:16:49,700
How am I going to
find that real part?

1226
01:16:54,660 --> 01:16:57,410
And what form will it have?

1227
01:16:57,410 --> 01:16:59,160
What form will that
real part have?

1228
01:16:59,160 --> 01:17:02,990
Yeah, maybe just to say
what form will it have?

1229
01:17:02,990 --> 01:17:09,389
The real part, it's
going to be a sinusoid.

1230
01:17:13,520 --> 01:17:17,110
But I have a complex number
multiplying this guy.

1231
01:17:17,110 --> 01:17:20,537
The real part is going to
be exactly of the form we--

1232
01:17:20,537 --> 01:17:22,120
well, of course, it
had to be the form

1233
01:17:22,120 --> 01:17:24,470
because that was another
way to solve the equation.

1234
01:17:24,470 --> 01:17:26,480
It's going to be some number.

1235
01:17:26,480 --> 01:17:33,780
And I'll call it g, for
gain, times the real part.

1236
01:17:33,780 --> 01:17:36,650
And so the real part
will be a cosine.

1237
01:17:36,650 --> 01:17:38,990
Yeah, it's just perfect.

1238
01:17:38,990 --> 01:17:41,650
A cosine of omega t.

1239
01:17:41,650 --> 01:17:45,345
And there'll be a phase.

1240
01:17:45,345 --> 01:17:45,845
Yeah.

1241
01:17:49,585 --> 01:17:54,460
i haven't taken that step fully.

1242
01:17:54,460 --> 01:17:57,310
I got to that fully.

1243
01:17:57,310 --> 01:18:02,980
And then I said that that, if
I use some complex arithmetic,

1244
01:18:02,980 --> 01:18:05,190
will come out to be this.

1245
01:18:05,190 --> 01:18:08,100
And you see the beauty
of that answer, which

1246
01:18:08,100 --> 01:18:12,450
was way better than a
sum of sines and cosines.

1247
01:18:12,450 --> 01:18:13,620
We see the gain.

1248
01:18:13,620 --> 01:18:15,080
We see the amplitude.

1249
01:18:15,080 --> 01:18:17,040
And we see the phase shift.

1250
01:18:17,040 --> 01:18:17,960
Yeah.

1251
01:18:17,960 --> 01:18:20,640
So I don't know, that
would be a good exercise

1252
01:18:20,640 --> 01:18:23,320
in complex numbers.

1253
01:18:23,320 --> 01:18:30,460
Find g and find phi, in taking
the real part of this thing.

1254
01:18:30,460 --> 01:18:31,020
Yeah.

1255
01:18:31,020 --> 01:18:34,640
It's a pure exercise in
using complex numbers.

1256
01:18:34,640 --> 01:18:36,430
I don't feel like
doing it today.

1257
01:18:39,940 --> 01:18:43,440
If we do it, you just
see a lot of formulas.

1258
01:18:43,440 --> 01:18:45,360
Here, you see the point.

1259
01:18:45,360 --> 01:18:48,560
The point was that
the complex equation

1260
01:18:48,560 --> 01:18:51,730
could be solved in one line.

1261
01:18:51,730 --> 01:18:53,750
We just did it.

1262
01:18:53,750 --> 01:18:57,380
But that left us the problem
of taking the real part.

1263
01:18:57,380 --> 01:19:00,170
That was the e to
the i omega t there.

1264
01:19:00,170 --> 01:19:02,130
Left us the problem of
taking the real part.

1265
01:19:02,130 --> 01:19:05,360
And that's a practice
with complex arithmetic.

1266
01:19:05,360 --> 01:19:07,190
So you've got the choice.

1267
01:19:07,190 --> 01:19:12,030
Either stay real--
sign plus cosine.

1268
01:19:12,030 --> 01:19:18,830
And then use the sinusoidal
identity, polar form.

1269
01:19:18,830 --> 01:19:22,140
Or get the polar form from here.

1270
01:19:22,140 --> 01:19:24,500
Same answer both ways.