1
00:00:00 --> 00:00:06

2

3

4
00:00:07 --> 00:00:13
OK, let's get started.
I'm assuming that,

5
00:00:10 --> 00:00:16
A, you went recitation
yesterday, B,

6
00:00:13 --> 00:00:19
that even if you didn't,
you know how to separate

7
00:00:17 --> 00:00:23
variables, and you know how to
construct simple models,

8
00:00:21 --> 00:00:27
solve physical problems with
differential equations,

9
00:00:25 --> 00:00:31
and possibly even solve them.
So, you should have learned

10
00:00:31 --> 00:00:37
that either in high school,
or 18.01 here,

11
00:00:35 --> 00:00:41
or, yeah.
So, I'm going to start from

12
00:00:38 --> 00:00:44
that point, assume you know
that.

13
00:00:42 --> 00:00:48
I'm not going to tell you what
differential equations are,

14
00:00:47 --> 00:00:53
or what modeling is.
If you still are uncertain

15
00:00:51 --> 00:00:57
about those things,
the book has a very long and

16
00:00:56 --> 00:01:02
good explanation of it.
Just read that stuff.

17
00:01:00 --> 00:01:06
So, we are talking about first
order ODEs.

18
00:01:06 --> 00:01:12
ODE: I'll only use two
acronyms.

19
00:01:08 --> 00:01:14
ODE is ordinary differential
equations.

20
00:01:12 --> 00:01:18
I think all of MIT knows that,
whether they've been taking the

21
00:01:17 --> 00:01:23
course or not.
So, we are talking about

22
00:01:21 --> 00:01:27
first-order ODEs,
which in standard form,

23
00:01:25 --> 00:01:31
are written,
you isolate the derivative of y

24
00:01:29 --> 00:01:35
with respect to,
x, let's say,

25
00:01:31 --> 00:01:37
on the left-hand side,
and on the right-hand side you

26
00:01:36 --> 00:01:42
write everything else.
You can't always do this very

27
00:01:42 --> 00:01:48
well, but for today,
I'm going to assume that it has

28
00:01:47 --> 00:01:53
been done and it's doable.
So, for example,

29
00:01:50 --> 00:01:56
some of the ones that will be
considered either today or in

30
00:01:56 --> 00:02:02
the problem set are things like
y prime equals x over y.

31
00:02:01 --> 00:02:07
That's pretty simple.

32
00:02:05 --> 00:02:11
The problem set has y prime
equals, let's see,

33
00:02:11 --> 00:02:17
x minus y squared.

34
00:02:15 --> 00:02:21
And, it also has y prime equals
y minus x squared.

35
00:02:22 --> 00:02:28
There are others,

36
00:02:25 --> 00:02:31
too.
Now, when you look at this,

37
00:02:29 --> 00:02:35
this, of course,
you can solve by separating

38
00:02:35 --> 00:02:41
variables.
So, this is solvable.

39
00:02:39 --> 00:02:45
This one is-- and neither of
these can you separate

40
00:02:43 --> 00:02:49
variables.
And they look extremely

41
00:02:46 --> 00:02:52
similar.
But they are extremely

42
00:02:48 --> 00:02:54
dissimilar.
The most dissimilar about them

43
00:02:52 --> 00:02:58
is that this one is easily
solvable.

44
00:02:54 --> 00:03:00
And you will learn,
if you don't know already,

45
00:02:58 --> 00:03:04
next time next Friday how to
solve this one.

46
00:03:03 --> 00:03:09
This one, which looks almost
the same, is unsolvable in a

47
00:03:06 --> 00:03:12
certain sense.
Namely, there are no elementary

48
00:03:09 --> 00:03:15
functions which you can write
down, which will give a solution

49
00:03:13 --> 00:03:19
of that differential equation.
So, right away,

50
00:03:16 --> 00:03:22
one confronts the most
significant fact that even for

51
00:03:19 --> 00:03:25
the simplest possible
differential equations,

52
00:03:22 --> 00:03:28
those which only involve the
first derivative,

53
00:03:25 --> 00:03:31
it's possible to write down
extremely looking simple guys.

54
00:03:30 --> 00:03:36
I'll put this one up in blue to
indicate that it's bad.

55
00:03:35 --> 00:03:41
Whoops, sorry,
I mean, not really bad,

56
00:03:38 --> 00:03:44
but recalcitrant.
It's not solvable in the

57
00:03:42 --> 00:03:48
ordinary sense in which you
think of an equation is

58
00:03:46 --> 00:03:52
solvable.
And, since those equations are

59
00:03:50 --> 00:03:56
the rule rather than the
exception, I'm going about this

60
00:03:55 --> 00:04:01
first day to not solving a
single differential equation,

61
00:04:00 --> 00:04:06
but indicating to you what you
do when you meet a blue equation

62
00:04:06 --> 00:04:12
like that.
What do you do with it?

63
00:04:11 --> 00:04:17
So, this first day is going to
be devoted to geometric ways of

64
00:04:17 --> 00:04:23
looking at differential
equations and numerical.

65
00:04:21 --> 00:04:27
At the very end,
I'll talk a little bit about

66
00:04:25 --> 00:04:31
numerical ways.
And you'll work on both of

67
00:04:29 --> 00:04:35
those for the first problem set.
So, what's our geometric view

68
00:04:35 --> 00:04:41
of differential equations?
Well, it's something that's

69
00:04:41 --> 00:04:47
contrasted with the usual
procedures, by which you solve

70
00:04:45 --> 00:04:51
things and find elementary
functions which solve them.

71
00:04:49 --> 00:04:55
I'll call that the analytic
method.

72
00:04:52 --> 00:04:58
So, on the one hand,
we have the analytic ideas,

73
00:04:56 --> 00:05:02
in which you write down
explicitly the equation,

74
00:04:59 --> 00:05:05
y prime equals f of x,y.

75
00:05:04 --> 00:05:10
And, you look for certain
functions, which are called its

76
00:05:07 --> 00:05:13
solutions.
Now, so there's the ODE.

77
00:05:09 --> 00:05:15
And, y1 of x,
notice I don't use a separate

78
00:05:12 --> 00:05:18
letter.
I don't use g or h or something

79
00:05:14 --> 00:05:20
like that for the solution
because the letters multiply so

80
00:05:18 --> 00:05:24
quickly, that is,
multiply in the sense of

81
00:05:20 --> 00:05:26
rabbits, that after a while,
if you keep using different

82
00:05:24 --> 00:05:30
letters for each new idea,
you can't figure out what

83
00:05:27 --> 00:05:33
you're talking about.
So, I'll use y1 means,

84
00:05:32 --> 00:05:38
it's a solution of this
differential equation.

85
00:05:37 --> 00:05:43
Of course, the differential
equation has many solutions

86
00:05:43 --> 00:05:49
containing an arbitrary
constant.

87
00:05:46 --> 00:05:52
So, we'll call this the
solution.

88
00:05:50 --> 00:05:56
Now, the geometric view,
the geometric guy that

89
00:05:54 --> 00:06:00
corresponds to this version of
writing the equation,

90
00:06:00 --> 00:06:06
is something called a direction
field.

91
00:06:06 --> 00:06:12
And, the solution is,
from the geometric point of

92
00:06:09 --> 00:06:15
view, something called an
integral curve.

93
00:06:12 --> 00:06:18
So, let me explain if you don't
know what the direction field

94
00:06:16 --> 00:06:22
is.
I know for some of you,

95
00:06:18 --> 00:06:24
I'm reviewing what you learned
in high school.

96
00:06:21 --> 00:06:27
Those of you who had the BC
syllabus in high school should

97
00:06:25 --> 00:06:31
know these things.
But, it never hurts to get a

98
00:06:28 --> 00:06:34
little more practice.
And, in any event,

99
00:06:31 --> 00:06:37
I think the computer stuff that
you will be doing on the problem

100
00:06:36 --> 00:06:42
set, a certain amount of it
should be novel to you.

101
00:06:41 --> 00:06:47
It was novel to me,
so why not to you?

102
00:06:43 --> 00:06:49
So, what's a direction field?
Well, the direction field is,

103
00:06:47 --> 00:06:53
you take the plane,
and in each point of the

104
00:06:51 --> 00:06:57
plane-- of course,
that's an impossibility.

105
00:06:54 --> 00:07:00
But, you pick some points of
the plane.

106
00:06:56 --> 00:07:02
You draw what's called a little
line element.

107
00:07:01 --> 00:07:07
So, there is a point.
It's a little line,

108
00:07:04 --> 00:07:10
and the only thing which
distinguishes it outside of its

109
00:07:08 --> 00:07:14
position in the plane,
so here's the point,

110
00:07:11 --> 00:07:17
(x,y), at which we are drawing
this line element,

111
00:07:15 --> 00:07:21
is its slope.
And, what is its slope?

112
00:07:18 --> 00:07:24
Its slope is to be f of x,y.

113
00:07:21 --> 00:07:27
And now, You fill up the plane
with these things until you're

114
00:07:26 --> 00:07:32
tired of putting then in.
So, I'm going to get tired

115
00:07:30 --> 00:07:36
pretty quickly.
So, I don't know,

116
00:07:34 --> 00:07:40
let's not make them all go the
same way.

117
00:07:36 --> 00:07:42
That sort of seems cheating.
How about here?

118
00:07:40 --> 00:07:46
Here's a few randomly chosen
line elements that I put in,

119
00:07:44 --> 00:07:50
and I putted the slopes at
random since I didn't have any

120
00:07:48 --> 00:07:54
particular differential equation
in mind.

121
00:07:50 --> 00:07:56
Now, the integral curve,
so those are the line elements.

122
00:07:54 --> 00:08:00
The integral curve is a curve,
which goes through the plane,

123
00:07:58 --> 00:08:04
and at every point is tangent
to the line element there.

124
00:08:04 --> 00:08:10
So, this is the integral curve.
Hey, wait a minute,

125
00:08:07 --> 00:08:13
I thought tangents were the
line element there didn't even

126
00:08:12 --> 00:08:18
touch it.
Well, I can't fill up the plane

127
00:08:15 --> 00:08:21
with line elements.
Here, at this point,

128
00:08:17 --> 00:08:23
there was a line element,
which I didn't bother drawing

129
00:08:22 --> 00:08:28
in.
And, it was tangent to that.

130
00:08:24 --> 00:08:30
Same thing over here:
if I drew the line element

131
00:08:27 --> 00:08:33
here, I would find that the
curve had exactly the right

132
00:08:31 --> 00:08:37
slope there.
So, the point is the integral,

133
00:08:37 --> 00:08:43
what distinguishes the integral
curve is that everywhere it has

134
00:08:43 --> 00:08:49
the direction,
that's the way I'll indicate

135
00:08:47 --> 00:08:53
that it's tangent,
has the direction of the field

136
00:08:52 --> 00:08:58
everywhere at all points on the
curve, of course,

137
00:08:57 --> 00:09:03
where it doesn't go.
It doesn't have any mission to

138
00:09:02 --> 00:09:08
fulfill.
Now, I say that this integral

139
00:09:04 --> 00:09:10
curve is the graph of the
solution to the differential

140
00:09:08 --> 00:09:14
equation.
In other words,

141
00:09:10 --> 00:09:16
writing down analytically the
differential equation is the

142
00:09:14 --> 00:09:20
same geometrically as drawing
this direction field,

143
00:09:18 --> 00:09:24
and solving analytically for a
solution of the differential

144
00:09:22 --> 00:09:28
equation is the same thing as
geometrically drawing an

145
00:09:26 --> 00:09:32
integral curve.
So, what am I saying?

146
00:09:30 --> 00:09:36
I say that an integral curve,
all right, let me write it this

147
00:09:39 --> 00:09:45
way.
I'll make a little theorem out

148
00:09:44 --> 00:09:50
of it, that y1 of x is
a solution to the differential

149
00:09:53 --> 00:09:59
equation if, and only if,
the graph, the curve associated

150
00:10:01 --> 00:10:07
with this, the graph of y1 of x
is an integral curve.

151
00:10:11 --> 00:10:17
Integral curve of what?
Well, of the direction field

152
00:10:14 --> 00:10:20
associated with that equation.
But there isn't quite enough

153
00:10:18 --> 00:10:24
room to write that on the board.
But, you could put it in your

154
00:10:22 --> 00:10:28
notes, if you take notes.
So, this is the relation

155
00:10:25 --> 00:10:31
between the two,
the integral curves of the

156
00:10:28 --> 00:10:34
graphs or solutions.
Now, why is that so?

157
00:10:31 --> 00:10:37
Well, in fact,
all I have to do to prove this,

158
00:10:34 --> 00:10:40
if you can call it a proof at
all, is simply to translate what

159
00:10:38 --> 00:10:44
each side really means.
What does it really mean to say

160
00:10:42 --> 00:10:48
that a given function is a
solution to the differential

161
00:10:45 --> 00:10:51
equation?
Well, it means that if you plug

162
00:10:48 --> 00:10:54
it into the differential
equation, it satisfies it.

163
00:10:52 --> 00:10:58
Okay, what is that?
So, how do I plug it into the

164
00:10:55 --> 00:11:01
differential equation and check
that it satisfies it?

165
00:11:00 --> 00:11:06
Well, doing it in the abstract,
I first calculate its

166
00:11:04 --> 00:11:10
derivative.
And then, how will it look

167
00:11:07 --> 00:11:13
after I plugged it into the
differential equation?

168
00:11:12 --> 00:11:18
Well, I don't do anything to
the x, but wherever I see y,

169
00:11:17 --> 00:11:23
I plug in this particular
function.

170
00:11:20 --> 00:11:26
So, in notation,
that would be written this way.

171
00:11:24 --> 00:11:30
So, for this to be a solution
means this, that that equation

172
00:11:29 --> 00:11:35
is satisfied.
Okay, what does it mean for the

173
00:11:35 --> 00:11:41
graph to be an integral curve?
Well, it means that at each

174
00:11:42 --> 00:11:48
point, the slope of this curve,
it means that the slope of y1

175
00:11:49 --> 00:11:55
of x should be,
at each point, x1 y1.

176
00:11:52 --> 00:11:58
It should be equal to the slope

177
00:11:58 --> 00:12:04
of the direction field at that
point.

178
00:12:04 --> 00:12:10
And then, what is the slope of
the direction field at that

179
00:12:08 --> 00:12:14
point?
Well, it is f of that

180
00:12:10 --> 00:12:16
particular, well,
at the point,

181
00:12:12 --> 00:12:18
x, y1 of x.
If you like,

182
00:12:15 --> 00:12:21
you can put a subscript,
one, on there,

183
00:12:18 --> 00:12:24
send a one here or a zero
there, to indicate that you mean

184
00:12:22 --> 00:12:28
a particular point.
But, it looks better if you

185
00:12:26 --> 00:12:32
don't.
But, there's some possibility

186
00:12:28 --> 00:12:34
of confusion.
I admit to that.

187
00:12:32 --> 00:12:38
So, the slope of the direction
field, what is that slope?

188
00:12:35 --> 00:12:41
Well, by the way,
I calculated the direction

189
00:12:38 --> 00:12:44
field.
Its slope at the point was to

190
00:12:41 --> 00:12:47
be x, whatever the value of x
was, and whatever the value of

191
00:12:45 --> 00:12:51
y1 of x was,
substituted into the right-hand

192
00:12:49 --> 00:12:55
side of the equation.
So, what the slope of this

193
00:12:52 --> 00:12:58
function of that curve of the
graph should be equal to the

194
00:12:56 --> 00:13:02
slope of the direction field.
Now, what does this say?

195
00:13:01 --> 00:13:07
Well, what's the slope of y1 of
x?

196
00:13:03 --> 00:13:09
That's y1 prime of x.

197
00:13:05 --> 00:13:11
That's from the first day of
18.01, calculus.

198
00:13:08 --> 00:13:14
What's the slope of the
direction field?

199
00:13:11 --> 00:13:17
This?
Well, it's this.

200
00:13:12 --> 00:13:18
And, that's with the right hand
side.

201
00:13:14 --> 00:13:20
So, saying these two guys are
the same or equal,

202
00:13:17 --> 00:13:23
is exactly, analytically,
the same as saying these two

203
00:13:21 --> 00:13:27
guys are equal.
So, in other words,

204
00:13:23 --> 00:13:29
the proof consists of,
what does this really mean?

205
00:13:26 --> 00:13:32
What does this really mean?
And after you see what both

206
00:13:29 --> 00:13:35
really mean, you say,
yeah, they're the same.

207
00:13:34 --> 00:13:40
So, I don't how to write that.
It's okay: same,

208
00:13:39 --> 00:13:45
same, how's that?
This is the same as that.

209
00:13:44 --> 00:13:50
Okay, well, this leaves us the
interesting question of how do

210
00:13:52 --> 00:13:58
you draw a direction from the,
well, this being 2003,

211
00:13:58 --> 00:14:04
mostly computers draw them for
you.

212
00:14:04 --> 00:14:10
Nonetheless,
you do have to know a certain

213
00:14:07 --> 00:14:13
amount.
I've given you a couple of

214
00:14:09 --> 00:14:15
exercises where you have to draw
the direction field yourself.

215
00:14:14 --> 00:14:20
This is so you get a feeling
for it, and also because humans

216
00:14:19 --> 00:14:25
don't draw direction fields the
same way computers do.

217
00:14:23 --> 00:14:29
So, let's first of all,
how did computers do it?

218
00:14:27 --> 00:14:33
They are very stupid.
There's no problem.

219
00:14:32 --> 00:14:38
Since they go very fast and
have unlimited amounts of energy

220
00:14:37 --> 00:14:43
to waste, the computer method is
the naive one.

221
00:14:42 --> 00:14:48
You pick the point.
You pick a point,

222
00:14:45 --> 00:14:51
and generally,
they are usually equally

223
00:14:49 --> 00:14:55
spaced.
You determine some spacing,

224
00:14:52 --> 00:14:58
that one: blah,
blah, blah, blah,

225
00:14:55 --> 00:15:01
blah, blah, blah,
equally spaced.

226
00:15:00 --> 00:15:06
And, at each point,
it computes f of x,

227
00:15:04 --> 00:15:10
y at the point,
finds, meets,

228
00:15:08 --> 00:15:14
and computes the value of f of
(x, y), that function,

229
00:15:14 --> 00:15:20
and the next thing is,
on the screen,

230
00:15:17 --> 00:15:23
it draws, at (x,
y), the little line element

231
00:15:22 --> 00:15:28
having slope f of x,y.

232
00:15:26 --> 00:15:32
In other words,
it does what the differential

233
00:15:30 --> 00:15:36
equation tells it to do.
And the only thing that it does

234
00:15:36 --> 00:15:42
is you can, if you are telling
the thing to draw the direction

235
00:15:40 --> 00:15:46
field, about the only option you
have is telling what the spacing

236
00:15:43 --> 00:15:49
should be, and sometimes people
don't like to see a whole line.

237
00:15:46 --> 00:15:52
They only like to see a little
bit of a half line.

238
00:15:49 --> 00:15:55
And, you can sometimes tell,
according to the program,

239
00:15:52 --> 00:15:58
tell the computer how long you
want that line to be,

240
00:15:55 --> 00:16:01
if you want it teeny or a
little bigger.

241
00:15:57 --> 00:16:03
Once in awhile you want you
want it narrower on it,

242
00:16:00 --> 00:16:06
but not right now.
Okay, that's what a computer

243
00:16:04 --> 00:16:10
does.
What does a human do?

244
00:16:05 --> 00:16:11
This is what it means to be
human.

245
00:16:08 --> 00:16:14
You use your intelligence.
From a human point of view,

246
00:16:12 --> 00:16:18
this stuff has been done in the
wrong order.

247
00:16:15 --> 00:16:21
And the reason it's been done
in the wrong order:

248
00:16:18 --> 00:16:24
because for each new point,
it requires a recalculation of

249
00:16:22 --> 00:16:28
f of (x, y).

250
00:16:24 --> 00:16:30
That is horrible.
The computer doesn't mind,

251
00:16:27 --> 00:16:33
but a human does.
So, for a human,

252
00:16:31 --> 00:16:37
the way to do it is not to
begin by picking the point,

253
00:16:35 --> 00:16:41
but to begin by picking the
slope that you would like to

254
00:16:40 --> 00:16:46
see.
So, you begin by taking the

255
00:16:42 --> 00:16:48
slope.
Let's call it the value of the

256
00:16:45 --> 00:16:51
slope, C.
So, you pick a number.

257
00:16:48 --> 00:16:54
C is two.
I want to see where are all the

258
00:16:51 --> 00:16:57
points in the plane where the
slope of that line element would

259
00:16:56 --> 00:17:02
be two?
Well, they will satisfy an

260
00:16:58 --> 00:17:04
equation.
The equation is f of (x,

261
00:17:02 --> 00:17:08
y) equals, in general,
it will be C.

262
00:17:07 --> 00:17:13
So, what you do is plot this,
plot the equation,

263
00:17:10 --> 00:17:16
plot this equation.
Notice, it's not the

264
00:17:14 --> 00:17:20
differential equation.
You can't exactly plot a

265
00:17:17 --> 00:17:23
differential equation.
It's a curve,

266
00:17:20 --> 00:17:26
an ordinary curve.
But which curve will depend;

267
00:17:24 --> 00:17:30
it's, in fact,
from the 18.02 point of view,

268
00:17:28 --> 00:17:34
the level curve of C,
sorry, it's a level curve of f

269
00:17:32 --> 00:17:38
of (x, y), the function f of x
and y corresponding to the level

270
00:17:37 --> 00:17:43
of value C.
But we are not going to call it

271
00:17:42 --> 00:17:48
that because this is not 18.02.
Instead, we're going to call it

272
00:17:48 --> 00:17:54
an isocline.
And then, you plot,

273
00:17:51 --> 00:17:57
well, you've done it.
So, you've got this isocline,

274
00:17:56 --> 00:18:02
except I'm going to use a
solution curve,

275
00:18:00 --> 00:18:06
solid lines,
only for integral curves.

276
00:18:03 --> 00:18:09
When we do plot isoclines,
to indicate that they are not

277
00:18:09 --> 00:18:15
solutions, we'll use dashed
lines for doing them.

278
00:18:15 --> 00:18:21
One of the computer things does
and the other one doesn't.

279
00:18:18 --> 00:18:24
But they use different colors,
also.

280
00:18:20 --> 00:18:26
There are different ways of
telling you what's an isocline

281
00:18:23 --> 00:18:29
and what's the solution curve.
So, and what do you do?

282
00:18:26 --> 00:18:32
So, these are all the points
where the slope is going to be

283
00:18:29 --> 00:18:35
C.
And now, what you do is draw in

284
00:18:32 --> 00:18:38
as many as you want of line
elements having slope C.

285
00:18:35 --> 00:18:41
Notice how efficient that is.
If you want 50 million of them

286
00:18:39 --> 00:18:45
and have the time,
draw in 50 million.

287
00:18:41 --> 00:18:47
If two or three are enough,
draw in two or three.

288
00:18:45 --> 00:18:51
You will be looking at the
picture.

289
00:18:47 --> 00:18:53
You will see what the curve
looks like, and that will give

290
00:18:51 --> 00:18:57
you your judgment as to how you
are to do that.

291
00:18:54 --> 00:19:00
So, in general,
a picture drawn that way,

292
00:18:57 --> 00:19:03
so let's say,
an isocline corresponding to C

293
00:18:59 --> 00:19:05
equals zero.
The line elements,

294
00:19:03 --> 00:19:09
and I think for an isocline,
for the purposes of this

295
00:19:07 --> 00:19:13
lecture, it would be a good idea
to put isoclines.

296
00:19:10 --> 00:19:16
Okay, so I'm going to put
solution curves in pink,

297
00:19:14 --> 00:19:20
or whatever this color is,
and isoclines are going to be

298
00:19:18 --> 00:19:24
in orange, I guess.
So, isocline,

299
00:19:21 --> 00:19:27
represented by a dashed line,
and now you will put in the

300
00:19:25 --> 00:19:31
line elements of,
we'll need lots of chalk for

301
00:19:28 --> 00:19:34
that.
So, I'll use white chalk.

302
00:19:32 --> 00:19:38
Y horizontal?
Because according to this the

303
00:19:34 --> 00:19:40
slope is supposed to be zero
there.

304
00:19:37 --> 00:19:43
And at the same way,
how about an isocline where the

305
00:19:40 --> 00:19:46
slope is negative one?
Let's suppose here C is equal

306
00:19:44 --> 00:19:50
to negative one.
Okay, then it will look like

307
00:19:47 --> 00:19:53
this.
These are supposed to be lines

308
00:19:49 --> 00:19:55
of slope negative one.
Don't shoot me if they are not.

309
00:19:53 --> 00:19:59
So, that's the principle.
So, this is how you will fill

310
00:19:56 --> 00:20:02
up the plane to draw a direction
field: by plotting the isoclines

311
00:20:01 --> 00:20:07
first.
And then, once you have the

312
00:20:04 --> 00:20:10
isoclines there,
you will have line elements.

313
00:20:07 --> 00:20:13
And you can draw a direction
field.

314
00:20:09 --> 00:20:15
Okay, so, for the next few
minutes, I'd like to work a

315
00:20:12 --> 00:20:18
couple of examples for you to
show how this works out in

316
00:20:15 --> 00:20:21
practice.

317
00:20:17 --> 00:20:23

318

319

320
00:20:34 --> 00:20:40
So, the first equation is going
to be y prime equals minus x

321
00:20:45 --> 00:20:51
over y.
Okay, first thing,

322
00:20:53 --> 00:20:59
what are the isoclines?
Well, the isoclines are going

323
00:21:03 --> 00:21:09
to be y.
Well, negative x over y is

324
00:21:08 --> 00:21:14
equal to C.
Maybe I better make two steps

325
00:21:12 --> 00:21:18
out of this.
Minus x over y is equal to C.

326
00:21:16 --> 00:21:22
But, of course,
nobody draws a curve in that

327
00:21:19 --> 00:21:25
form.
You'll want it in the form y

328
00:21:22 --> 00:21:28
equals minus one over 
C times x.

329
00:21:26 --> 00:21:32
So, there's our isocline.
Why don't I put that up in

330
00:21:32 --> 00:21:38
orange since it's going to be,
that's the color I'll draw it

331
00:21:36 --> 00:21:42
in.
In other words,

332
00:21:38 --> 00:21:44
for different values of C,
now this thing is aligned.

333
00:21:42 --> 00:21:48
It's aligned,
in fact, through the origin.

334
00:21:45 --> 00:21:51
This looks pretty simple.
Okay, so here's our plane.

335
00:21:50 --> 00:21:56
The isoclines are going to be
lines through the origin.

336
00:21:54 --> 00:22:00
And now, let's put them in,
suppose, for example,

337
00:21:58 --> 00:22:04
C is equal to one.
Well, if C is equal to one,

338
00:22:06 --> 00:22:12
then it's the line,
y equals minus x.

339
00:22:14 --> 00:22:20
So, this is the isocline.
I'll put, down here,

340
00:22:23 --> 00:22:29
C equals minus one.
And, along it,

341
00:22:30 --> 00:22:36
no, something's wrong.
I'm sorry?

342
00:22:38 --> 00:22:44
C is one, not negative one,
right, thanks.

343
00:22:42 --> 00:22:48
Thanks.
So, C equals one.

344
00:22:44 --> 00:22:50
So, it should be little line
segments of slope one will be

345
00:22:50 --> 00:22:56
the line elements,
things of slope one.

346
00:22:54 --> 00:23:00
OK, now how about C equals
negative one?

347
00:23:00 --> 00:23:06
If C equals negative one,
then it's the line,

348
00:23:03 --> 00:23:09
y equals x.
And so, that's the isocline.

349
00:23:07 --> 00:23:13
Notice, still dash because
these are isoclines.

350
00:23:11 --> 00:23:17
Here, C is negative one.
And so, the slope elements look

351
00:23:15 --> 00:23:21
like this.
Notice, they are perpendicular.

352
00:23:19 --> 00:23:25
Now, notice that they are
always going to be perpendicular

353
00:23:23 --> 00:23:29
to the line because the slope of
this line is minus one over C.

354
00:23:30 --> 00:23:36
But, the slope of the line
element is going to be C.

355
00:23:33 --> 00:23:39
Those numbers,
minus one over C and C,

356
00:23:36 --> 00:23:42
are negative reciprocals.
And, you know that two lines

357
00:23:40 --> 00:23:46
whose slopes are negative
reciprocals are perpendicular.

358
00:23:44 --> 00:23:50
So, the line elements are going
to be perpendicular to these.

359
00:23:49 --> 00:23:55
And therefore,
I hardly even have to bother

360
00:23:52 --> 00:23:58
calculating, doing any more
calculation.

361
00:23:55 --> 00:24:01
Here's going to be a,
well, how about this one?

362
00:24:00 --> 00:24:06
Here's a controversial
isocline.

363
00:24:02 --> 00:24:08
Is that an isocline?
Well, wait a minute.

364
00:24:05 --> 00:24:11
That doesn't correspond to
anything looking like this.

365
00:24:10 --> 00:24:16
Ah-ha, but it would if I put C
multiplied through by C.

366
00:24:14 --> 00:24:20
And then, it would correspond
to C being zero.

367
00:24:18 --> 00:24:24
In other words,
don't write it like this.

368
00:24:21 --> 00:24:27
Multiply through by C.
It will read C y equals

369
00:24:25 --> 00:24:31
negative x.
And then, when C is zero,

370
00:24:29 --> 00:24:35
I have x equals zero,
which is exactly the y-axis.

371
00:24:35 --> 00:24:41
So, that really is included.
How about the x-axis?

372
00:24:38 --> 00:24:44
Well, the x-axis is not
included.

373
00:24:40 --> 00:24:46
However, most people include it
anyway.

374
00:24:43 --> 00:24:49
This is very common to be a
sort of sloppy and bending the

375
00:24:47 --> 00:24:53
edges of corners a little bit,
and hoping nobody will notice.

376
00:24:51 --> 00:24:57
We'll say that corresponds to C
equals infinity.

377
00:24:55 --> 00:25:01
I hope nobody wants to fight
about that.

378
00:24:58 --> 00:25:04
If you do, go fight with
somebody else.

379
00:25:02 --> 00:25:08
So, if C is infinity,
that means the little line

380
00:25:05 --> 00:25:11
segment should have infinite
slope, and by common consent,

381
00:25:10 --> 00:25:16
that means it should be
vertical.

382
00:25:12 --> 00:25:18
And so, we can even count this
as sort of an isocline.

383
00:25:17 --> 00:25:23
And, I'll make the dashes
smaller, indicate it has a lower

384
00:25:21 --> 00:25:27
status than the others.
And, I'll put this in,

385
00:25:25 --> 00:25:31
do this weaselly thing of
putting it in quotation marks to

386
00:25:29 --> 00:25:35
indicate that I'm not
responsible for it.

387
00:25:34 --> 00:25:40
Okay, now, we now have to put
it the integral curves.

388
00:25:39 --> 00:25:45
Well, nothing could be easier.
I'm looking for curves which

389
00:25:45 --> 00:25:51
are everywhere perpendicular to
these rays.

390
00:25:50 --> 00:25:56
Well, you know from geometry
that those are circles.

391
00:25:55 --> 00:26:01
So, the integral curves are
circles.

392
00:26:00 --> 00:26:06
And, it's an elementary
exercise, which I would not

393
00:26:04 --> 00:26:10
deprive you of the pleasure of.
Solve the ODE by separation of

394
00:26:08 --> 00:26:14
variables.
In other words,

395
00:26:10 --> 00:26:16
we've gotten the,
so the circles are ones with a

396
00:26:14 --> 00:26:20
center at the origin,
of course, equal some constant.

397
00:26:18 --> 00:26:24
I'll call it C1,
so it's not confused with this

398
00:26:22 --> 00:26:28
C.
They look like that,

399
00:26:24 --> 00:26:30
and now you should solve this
by separating variables,

400
00:26:28 --> 00:26:34
and just confirm that the
solutions are,

401
00:26:31 --> 00:26:37
in fact, those circles.
One interesting thing,

402
00:26:36 --> 00:26:42
and so I confirm this,
I won't do it because I want to

403
00:26:40 --> 00:26:46
do geometric and numerical
things today.

404
00:26:42 --> 00:26:48
So, if you solve it by
separating variables,

405
00:26:45 --> 00:26:51
one interesting thing to note
is that if I write the solution

406
00:26:49 --> 00:26:55
as y equals y1 of x, well,

407
00:26:52 --> 00:26:58
it'll look something like the
square root of C1 minus,

408
00:26:56 --> 00:27:02
let's make this squared because
that's the way people usually

409
00:27:00 --> 00:27:06
put the radius,
minus x squared.

410
00:27:03 --> 00:27:09
And so, a solution,

411
00:27:06 --> 00:27:12
a typical solution looks like
this.

412
00:27:09 --> 00:27:15
Well, what's the solution over
here?

413
00:27:11 --> 00:27:17
Well, that one solution will be
goes from here to here.

414
00:27:15 --> 00:27:21
If you like,
it has a negative side to it.

415
00:27:18 --> 00:27:24
So, I'll make,
let's say, plus.

416
00:27:21 --> 00:27:27
There's another solution,
which has a negative value.

417
00:27:25 --> 00:27:31
But let's use the one with the
positive value of the square

418
00:27:29 --> 00:27:35
root.
My point is this,

419
00:27:32 --> 00:27:38
that that solution,
the domain of that solution,

420
00:27:35 --> 00:27:41
really only goes from here to
here.

421
00:27:38 --> 00:27:44
It's not the whole x-axis.
It's just a limited piece of

422
00:27:42 --> 00:27:48
the x-axis where that solution
is defined.

423
00:27:45 --> 00:27:51
There's no way of extending it
further.

424
00:27:48 --> 00:27:54
And, there's no way of
predicting, by looking at the

425
00:27:52 --> 00:27:58
differential equation,
that a typical solution was

426
00:27:56 --> 00:28:02
going to have a limited domain
like that.

427
00:28:01 --> 00:28:07
In other words,
you could find a solution,

428
00:28:04 --> 00:28:10
but how far out is it going to
go?

429
00:28:07 --> 00:28:13
Sometimes, it's impossible to
tell, except by either finding

430
00:28:12 --> 00:28:18
it explicitly,
or by asking a computer to draw

431
00:28:16 --> 00:28:22
a picture of it,
and seeing if that gives you

432
00:28:19 --> 00:28:25
some insight.
It's one of the many

433
00:28:22 --> 00:28:28
difficulties in handling
differential equations.

434
00:28:26 --> 00:28:32
You don't know what the domain
of a solution is going to be

435
00:28:31 --> 00:28:37
until you've actually calculated
it.

436
00:28:36 --> 00:28:42
Now, a slightly more
complicated example is going to

437
00:28:40 --> 00:28:46
be, let's see, y prime 
equals one plus x minus y.

438
00:28:43 --> 00:28:49
It's not a lot more

439
00:28:46 --> 00:28:52
complicated, and as a computer
exercise, you will work with,

440
00:28:51 --> 00:28:57
still, more complicated ones.
But here, the isoclines would

441
00:28:56 --> 00:29:02
be what?
Well, I set that equal to C.

442
00:29:00 --> 00:29:06
Can you do the algebra in your
head?

443
00:29:02 --> 00:29:08
An isocline will have the
equation: this equals C.

444
00:29:07 --> 00:29:13
So, I'm going to put the y on
the right hand side,

445
00:29:11 --> 00:29:17
and that C on the left hand
side.

446
00:29:13 --> 00:29:19
So, it will have the equation y
equals one plus x minus C,

447
00:29:19 --> 00:29:25
or a nicer way to
write it would be x plus one

448
00:29:23 --> 00:29:29
minus C.
I guess it really doesn't

449
00:29:28 --> 00:29:34
matter.
So there's the equation of the

450
00:29:31 --> 00:29:37
isocline.
Let's quickly draw the

451
00:29:34 --> 00:29:40
direction field.
And notice, by the way,

452
00:29:36 --> 00:29:42
it's a simple equation,
but you cannot separate

453
00:29:39 --> 00:29:45
variables.
So, I will not,

454
00:29:41 --> 00:29:47
today at any rate,
be able to check the answer.

455
00:29:44 --> 00:29:50
I will not be able to get an
analytic answer.

456
00:29:47 --> 00:29:53
All we'll be able to do now is
get a geometric answer.

457
00:29:50 --> 00:29:56
But notice how quickly,
relatively quickly,

458
00:29:53 --> 00:29:59
one can get it.
So, I'm feeling for how the

459
00:29:56 --> 00:30:02
solutions behave to this
equation.

460
00:30:00 --> 00:30:06
All right, let's see,
what should we plot first?

461
00:30:05 --> 00:30:11
I like C equals one,
no, don't do C equals one.

462
00:30:10 --> 00:30:16
Let's do C equals zero,
first.

463
00:30:13 --> 00:30:19
C equals zero.
That's the line.

464
00:30:16 --> 00:30:22
y equals x plus 1.

465
00:30:19 --> 00:30:25
Okay, let me run and get that
chalk.

466
00:30:23 --> 00:30:29
So, I'll isoclines are in
orange.

467
00:30:27 --> 00:30:33
If so, when C equals zero,
y equals x plus one.

468
00:30:32 --> 00:30:38
So, let's say it's this curve.
C equals zero.

469
00:30:38 --> 00:30:44
How about C equals negative
one?

470
00:30:42 --> 00:30:48
Then it's y equals x plus two.

471
00:30:47 --> 00:30:53
It's this curve.
Well, let's label it down here.

472
00:30:53 --> 00:30:59
So, this is C equals negative
one.

473
00:30:57 --> 00:31:03
C equals negative two would be
y equals x, no,

474
00:31:02 --> 00:31:08
what am I doing?
C equals negative one is y

475
00:31:08 --> 00:31:14
equals x plus two.
That's right.

476
00:31:12 --> 00:31:18
Well, how about the other side?
If C equals plus one,

477
00:31:16 --> 00:31:22
well, then it's going to go
through the origin.

478
00:31:20 --> 00:31:26
It looks like a little more
room down here.

479
00:31:24 --> 00:31:30
How about, so if this is going
to be C equals one,

480
00:31:28 --> 00:31:34
then I sort of get the idea.
C equals two will look like

481
00:31:34 --> 00:31:40
this.
They're all going to be

482
00:31:37 --> 00:31:43
parallel lines because all
that's changing is the

483
00:31:42 --> 00:31:48
y-intercept, as I do this thing.
So, here, it's C equals two.

484
00:31:47 --> 00:31:53
That's probably enough.
All right, let's put it in the

485
00:31:53 --> 00:31:59
line elements.
All right, C equals negative

486
00:31:57 --> 00:32:03
one.
These will be perpendicular.

487
00:32:00 --> 00:32:06
C equals zero,
like this.

488
00:32:04 --> 00:32:10
C equals one.
Oh, this is interesting.

489
00:32:06 --> 00:32:12
I can't even draw in the line
elements because they seem to

490
00:32:10 --> 00:32:16
coincide with the curve itself,
with the line itself.

491
00:32:14 --> 00:32:20
They write y along the line,
and that makes it hard to draw

492
00:32:18 --> 00:32:24
them in.
How about C equals two?

493
00:32:20 --> 00:32:26
Well, here, the line elements
will be slanty.

494
00:32:23 --> 00:32:29
They'll have slope two,
so a pretty slanty up.

495
00:32:26 --> 00:32:32
And, I can see if a C equals
three in the same way.

496
00:32:31 --> 00:32:37
There are going to be even more
slantier up.

497
00:32:34 --> 00:32:40
And here, they're going to be
even more slanty down.

498
00:32:37 --> 00:32:43
This is not very scientific
terminology or mathematical,

499
00:32:41 --> 00:32:47
but you get the idea.
Okay, so there's our quick

500
00:32:45 --> 00:32:51
version of the direction field.
All we have to do is put in

501
00:32:49 --> 00:32:55
some integral curves now.
Well, it looks like it's doing

502
00:32:53 --> 00:32:59
this.
It gets less slanty here.

503
00:32:55 --> 00:33:01
It levels out,
has slope zero.

504
00:32:59 --> 00:33:05
And now, in this part of the
plain, the slope seems to be

505
00:33:03 --> 00:33:09
rising.
So, it must do something like

506
00:33:06 --> 00:33:12
that.
This guy must do something like

507
00:33:08 --> 00:33:14
this.
I'm a little doubtful of what I

508
00:33:11 --> 00:33:17
should be doing here.
Or, how about going from the

509
00:33:15 --> 00:33:21
other side?
Well, it rises,

510
00:33:17 --> 00:33:23
gets a little,
should it cross this?

511
00:33:20 --> 00:33:26
What should I do?
Well, there's one integral

512
00:33:23 --> 00:33:29
curve, which is easy to see.
It's this one.

513
00:33:26 --> 00:33:32
This line is both an isocline
and an integral curve.

514
00:33:32 --> 00:33:38
It's everything,
except drawable,

515
00:33:35 --> 00:33:41
[LAUGHTER] so,
you understand this is the same

516
00:33:41 --> 00:33:47
line.
It's both orange and pink at

517
00:33:45 --> 00:33:51
the same time.
But I don't know what

518
00:33:49 --> 00:33:55
combination color that would
make.

519
00:33:53 --> 00:33:59
It doesn't look like a line,
but be sympathetic.

520
00:34:00 --> 00:34:06
Now, the question is,
what's happening in this

521
00:34:04 --> 00:34:10
corridor?
In the corridor,

522
00:34:06 --> 00:34:12
that's not a mathematical word
either, between the isoclines

523
00:34:12 --> 00:34:18
for, well, what are they?
They are the isoclines for C

524
00:34:18 --> 00:34:24
equals two, and C equals zero.
How does that corridor look?

525
00:34:23 --> 00:34:29
Well: something like this.
Over here, the lines all look

526
00:34:29 --> 00:34:35
like that.
And here, they all look like

527
00:34:33 --> 00:34:39
this.
The slope is two.

528
00:34:36 --> 00:34:42
And, a hapless solution gets in
there.

529
00:34:39 --> 00:34:45
What's it to do?
Well, do you see that if a

530
00:34:43 --> 00:34:49
solution gets in that corridor,
an integral curve gets in that

531
00:34:49 --> 00:34:55
corridor, no escape is possible.
It's like a lobster trap.

532
00:34:54 --> 00:35:00
The lobster can walk in.
But it cannot walk out because

533
00:34:58 --> 00:35:04
things are always going in.
How could it escape?

534
00:35:03 --> 00:35:09
Well, it would have to double
back, somehow,

535
00:35:06 --> 00:35:12
and remember,
to escape, it has to be,

536
00:35:10 --> 00:35:16
to escape on the left side,
it must be going horizontally.

537
00:35:17 --> 00:35:23
But, how could it do that
without doubling back first and

538
00:35:20 --> 00:35:26
having the wrong slope?
The slope of everything in this

539
00:35:24 --> 00:35:30
corridor is positive,
and to double back and escape,

540
00:35:28 --> 00:35:34
it would at some point have to
have negative slope.

541
00:35:32 --> 00:35:38
It can't do that.
Well, could it escape on the

542
00:35:35 --> 00:35:41
right-hand side?
No, because at the moment when

543
00:35:39 --> 00:35:45
it wants to cross,
it will have to have a slope

544
00:35:42 --> 00:35:48
less than this line.
But all these spiky guys are

545
00:35:46 --> 00:35:52
pointing; it can't escape that
way either.

546
00:35:50 --> 00:35:56
So, no escape is possible.
It has to continue on,

547
00:35:53 --> 00:35:59
there.
But, more than that is true.

548
00:35:56 --> 00:36:02
So, a solution can't escape.
Once it's in there,

549
00:36:01 --> 00:36:07
it can't escape.
It's like, what do they call

550
00:36:04 --> 00:36:10
those plants,
I forget, pitcher plants.

551
00:36:07 --> 00:36:13
All they hear is they are going
down.

552
00:36:10 --> 00:36:16
So, it looks like that.
And so, the poor little insect

553
00:36:14 --> 00:36:20
falls in.
They could climb up the walls

554
00:36:17 --> 00:36:23
except that all the hairs are
going the wrong direction,

555
00:36:22 --> 00:36:28
and it can't get over them.
Well, let's think of it that

556
00:36:26 --> 00:36:32
way: this poor trap solution.
So, it does what it has to do.

557
00:36:32 --> 00:36:38
Now, there's more to it than
that.

558
00:36:35 --> 00:36:41
Because there are two
principles involved here that

559
00:36:39 --> 00:36:45
you should know,
that help a lot in drawing

560
00:36:43 --> 00:36:49
these pictures.
Principle number one is that

561
00:36:46 --> 00:36:52
two integral curves cannot cross
at an angle.

562
00:36:50 --> 00:36:56
Two integral curves can't
cross, I mean,

563
00:36:53 --> 00:36:59
by crossing at an angle like
that.

564
00:36:56 --> 00:37:02
I'll indicate what I mean by a
picture like that.

565
00:37:02 --> 00:37:08
Now, why not?
This is an important principle.

566
00:37:05 --> 00:37:11
Let's put that up in the white
box.

567
00:37:08 --> 00:37:14
They can't cross because if two
integral curves,

568
00:37:12 --> 00:37:18
are trying to cross,
well, one will look like this.

569
00:37:16 --> 00:37:22
It's an integral curve because
it has this slope.

570
00:37:20 --> 00:37:26
And, the other integral curve
has this slope.

571
00:37:24 --> 00:37:30
And now, they fight with each
other.

572
00:37:27 --> 00:37:33
What is the true slope at that
point?

573
00:37:32 --> 00:37:38
Well, the direction field only
allows you to have one slope.

574
00:37:36 --> 00:37:42
If there's a line element at
that point, it has a definite

575
00:37:40 --> 00:37:46
slope.
And therefore,

576
00:37:41 --> 00:37:47
it cannot have both the slope
and that one.

577
00:37:44 --> 00:37:50
It's as simple as that.
So, the reason is you can't

578
00:37:48 --> 00:37:54
have two slopes.
The direction field doesn't

579
00:37:51 --> 00:37:57
allow it.
Well, that's a big,

580
00:37:53 --> 00:37:59
big help because if I know,
here's an integral curve,

581
00:37:57 --> 00:38:03
and if I know that none of
these other pink integral curves

582
00:38:01 --> 00:38:07
are allowed to cross it,
how else can I do it?

583
00:38:06 --> 00:38:12
Well, they can't escape.
They can't cross.

584
00:38:09 --> 00:38:15
It's sort of clear that they
must get closer and closer to

585
00:38:13 --> 00:38:19
it.
You know, I'd have to work a

586
00:38:16 --> 00:38:22
little to justify that.
But I think that nobody would

587
00:38:20 --> 00:38:26
have any doubt of it who did a
little experimentation.

588
00:38:24 --> 00:38:30
In other words,
all these curves joined that

589
00:38:28 --> 00:38:34
little tube and get closer and
closer to this line,

590
00:38:32 --> 00:38:38
y equals x.
And there, without solving the

591
00:38:37 --> 00:38:43
differential equation,
it's clear that all of these

592
00:38:42 --> 00:38:48
solutions, how do they behave?
As x goes to infinity,

593
00:38:47 --> 00:38:53
they become asymptotic to,
they become closer and closer

594
00:38:52 --> 00:38:58
to the solution,
x.

595
00:38:54 --> 00:39:00
Is x a solution?
Yeah, because y equals x is an

596
00:38:58 --> 00:39:04
integral curve.
Is x a solution?

597
00:39:02 --> 00:39:08
Yeah, because if I plug in y
equals x, I get what?

598
00:39:07 --> 00:39:13
On the right-hand side,
I get one.

599
00:39:10 --> 00:39:16
And on the left-hand side,
I get one.

600
00:39:14 --> 00:39:20
One equals one.
So, this is a solution.

601
00:39:18 --> 00:39:24
Let's indicate that it's a
solution.

602
00:39:21 --> 00:39:27
So, analytically,
we've discovered an analytic

603
00:39:26 --> 00:39:32
solution to the differential
equation, namely,

604
00:39:31 --> 00:39:37
Y equals X, just by this
geometric process.

605
00:39:37 --> 00:39:43
Now, there's one more principle
like that, which is less

606
00:39:41 --> 00:39:47
obvious.
But you do have to know it.

607
00:39:44 --> 00:39:50
So, you are not allowed to
cross.

608
00:39:46 --> 00:39:52
That's clear.
But it's much,

609
00:39:49 --> 00:39:55
much, much, much,
much less obvious that two

610
00:39:52 --> 00:39:58
integral curves cannot touch.
That is, they cannot even be

611
00:39:57 --> 00:40:03
tangent.
Two integral curves cannot be

612
00:40:00 --> 00:40:06
tangent.

613
00:40:02 --> 00:40:08

614

615

616
00:40:10 --> 00:40:16
I'll indicate that by the word
touch, which is what a lot of

617
00:40:19 --> 00:40:25
people say.
In other words,

618
00:40:23 --> 00:40:29
if this is illegal,
so is this.

619
00:40:28 --> 00:40:34
It can't happen.
You know, without that,

620
00:40:33 --> 00:40:39
for example,
it might be,

621
00:40:35 --> 00:40:41
I might feel that there would
be nothing in this to prevent

622
00:40:39 --> 00:40:45
those curves from joining.
Why couldn't these pink curves

623
00:40:43 --> 00:40:49
join the line,
y equals x?

624
00:40:45 --> 00:40:51
You know, it's a solution.
They just pitch a ride,

625
00:40:49 --> 00:40:55
as it were.
The answer is they cannot do

626
00:40:52 --> 00:40:58
that because they have to just
get asymptotic to it,

627
00:40:55 --> 00:41:01
ever, ever closer.
They can't join y equals x

628
00:40:59 --> 00:41:05
because at the point where they
join, you have that situation.

629
00:41:05 --> 00:41:11
Now, why can't you to have
this?

630
00:41:09 --> 00:41:15
That's much more sophisticated
than this, and the reason is

631
00:41:17 --> 00:41:23
because of something called the
Existence and Uniqueness

632
00:41:24 --> 00:41:30
Theorem, which says that there
is through a point,

633
00:41:31 --> 00:41:37
x zero y zero,
that y prime equals f of

634
00:41:38 --> 00:41:44
(x, y) has only one,

635
00:41:43 --> 00:41:49
and only one solution.
One has one solution.

636
00:41:49 --> 00:41:55
In mathematics speak,
that means at least one

637
00:41:53 --> 00:41:59
solution.
It doesn't mean it has just one

638
00:41:56 --> 00:42:02
solution.
That's mathematical convention.

639
00:41:59 --> 00:42:05
It has one solution,
at least one solution.

640
00:42:02 --> 00:42:08
But, the killer is,
only one solution.

641
00:42:06 --> 00:42:12
That's what you have to say in
mathematics if you want just

642
00:42:10 --> 00:42:16
one, one, and only one solution
through the point

643
00:42:15 --> 00:42:21
 x zero y zero.
So, the fact that it has one,

644
00:42:18 --> 00:42:24
that is the existence part.
The fact that it has only one

645
00:42:23 --> 00:42:29
is the uniqueness part of the
theorem.

646
00:42:26 --> 00:42:32
Now, like all good mathematical
theorems, this one does have

647
00:42:31 --> 00:42:37
hypotheses.
So, this is not going to be a

648
00:42:35 --> 00:42:41
course, I warn you,
those of you who are

649
00:42:39 --> 00:42:45
theoretically inclined,
very rich in hypotheses.

650
00:42:44 --> 00:42:50
But, hypotheses for those one
or that f of (x,

651
00:42:48 --> 00:42:54
y) should be a
continuous function.

652
00:42:52 --> 00:42:58
Now, like polynomial,
signs, should be continuous

653
00:42:57 --> 00:43:03
near, in the vicinity of that
point.

654
00:43:02 --> 00:43:08
That guarantees existence,
and what guarantees uniqueness

655
00:43:08 --> 00:43:14
is the hypothesis that you would
not guess by yourself.

656
00:43:14 --> 00:43:20
Neither would I.
What guarantees the uniqueness

657
00:43:19 --> 00:43:25
is that also,
it's partial derivative with

658
00:43:24 --> 00:43:30
respect to y should be
continuous, should be continuous

659
00:43:30 --> 00:43:36
near x zero y zero.

660
00:43:35 --> 00:43:41
Well, I have to make a
decision.

661
00:43:38 --> 00:43:44
I don't have time to talk about
Euler's method.

662
00:43:43 --> 00:43:49
I'll refer you to the,
there's one page of notes,

663
00:43:49 --> 00:43:55
and I couldn't do any more than
just repeat what's on those

664
00:43:55 --> 00:44:01
notes.
So, I'll trust you to read

665
00:43:59 --> 00:44:05
that.
And instead,

666
00:44:02 --> 00:44:08
let me give you an example
which will solidify these things

667
00:44:09 --> 00:44:15
in your mind a little bit.
I think that's a better course.

668
00:44:17 --> 00:44:23
The example is not in your
notes, and therefore,

669
00:44:22 --> 00:44:28
remember, you heard it here
first.

670
00:44:27 --> 00:44:33
Okay, so what's the example?
So, there is that differential

671
00:44:34 --> 00:44:40
equation.
Now, let's just solve it by

672
00:44:38 --> 00:44:44
separating variables.
Can you do it in your head?

673
00:44:42 --> 00:44:48
dy over dx, put all the y's on
the left.

674
00:44:44 --> 00:44:50
It will look like dy over one
minus y.

675
00:44:48 --> 00:44:54
Put all the dx's on the left.
So, the dx here goes on the

676
00:44:52 --> 00:44:58
right, rather.
That will be dx.

677
00:44:54 --> 00:45:00
And then, the x goes down into
the denominator.

678
00:44:57 --> 00:45:03
So now, it looks like that.
And, if I integrate both sides,

679
00:45:03 --> 00:45:09
I get the log of one minus y,
I guess, maybe with a,

680
00:45:08 --> 00:45:14
I never bothered with that,
but you can.

681
00:45:12 --> 00:45:18
It should be absolute values.
All right, put an absolute

682
00:45:17 --> 00:45:23
value, plus a constant.
And now, if I exponentiate both

683
00:45:23 --> 00:45:29
sides, the constant is positive.
So, this is going to look like

684
00:45:29 --> 00:45:35
y.
One minus y equals x

685
00:45:33 --> 00:45:39
And, the constant will be e to

686
00:45:36 --> 00:45:42
the C1.
And, I'll just make that a new

687
00:45:39 --> 00:45:45
constant, Cx.
And now, by letting C be

688
00:45:42 --> 00:45:48
negative, that's why you can get
rid of the absolute values,

689
00:45:45 --> 00:45:51
if you allow C to have negative
values as well as positive

690
00:45:49 --> 00:45:55
values.
Let's write this in a more

691
00:45:51 --> 00:45:57
human form.
So, y is equal to one minus Cx.

692
00:45:53 --> 00:45:59
Good, all right,

693
00:45:55 --> 00:46:01
let's just plot those.
So, these are the solutions.

694
00:46:00 --> 00:46:06
It's a pretty easy equation,
pretty easy solution method,

695
00:46:05 --> 00:46:11
just separation of variables.
What do they look like?

696
00:46:11 --> 00:46:17
Well, these are all lines whose
intercept is at one.

697
00:46:16 --> 00:46:22
And, they have any slope
whatsoever.

698
00:46:19 --> 00:46:25
So, these are the lines that
look like that.

699
00:46:24 --> 00:46:30
Okay, now let me ask,
existence and uniqueness.

700
00:46:29 --> 00:46:35
Existence: through which points
of the plane does the solution

701
00:46:35 --> 00:46:41
go?
Answer: through every point of

702
00:46:39 --> 00:46:45
the plane, through any point
here, I can find one and only

703
00:46:44 --> 00:46:50
one of those lines,
except for these stupid guys

704
00:46:48 --> 00:46:54
here on the stalk of the flower.
Here, for each of these points,

705
00:46:53 --> 00:46:59
there is no existence.
There is no solution to this

706
00:46:57 --> 00:47:03
differential equation,
which goes through any of these

707
00:47:02 --> 00:47:08
wiggly points on the y-axis,
with one exception.

708
00:47:07 --> 00:47:13
This point is oversupplied.
At this point,

709
00:47:10 --> 00:47:16
it's not existence that fails.
It's uniqueness that fails:

710
00:47:14 --> 00:47:20
no uniqueness.
There are lots of things which

711
00:47:18 --> 00:47:24
go through here.
Now, is that a violation of the

712
00:47:21 --> 00:47:27
existence and uniqueness
theorem?

713
00:47:24 --> 00:47:30
It cannot be a violation
because the theorem has no

714
00:47:28 --> 00:47:34
exceptions.
Otherwise, it wouldn't be a

715
00:47:31 --> 00:47:37
theorem.
So, let's take a look.

716
00:47:34 --> 00:47:40
What's wrong?
We thought we solved it modulo,

717
00:47:37 --> 00:47:43
putting the absolute value
signs on the log.

718
00:47:40 --> 00:47:46
What's wrong?
The answer: what's wrong is to

719
00:47:43 --> 00:47:49
use the theorem you must write
the differential equation in

720
00:47:48 --> 00:47:54
standard form,
in the green form I gave you.

721
00:47:51 --> 00:47:57
Let's write the differential
equation the way we were

722
00:47:54 --> 00:48:00
supposed to.
It says dy / dx equals one

723
00:47:57 --> 00:48:03
minus y divided by x.

724
00:48:02 --> 00:48:08
And now, I see,
the right-hand side is not

725
00:48:05 --> 00:48:11
continuous, in fact,
not even defined when x equals

726
00:48:09 --> 00:48:15
zero, when along the y-axis.
And therefore,

727
00:48:12 --> 00:48:18
the existence and uniqueness is
not guaranteed along the line,

728
00:48:16 --> 00:48:22
x equals zero of the y-axis.
And, in fact,

729
00:48:20 --> 00:48:26
we see that it failed.
Now, as a practical matter,

730
00:48:23 --> 00:48:29
it's the way existence and
uniqueness fails in all ordinary

731
00:48:28 --> 00:48:34
life work with differential
equations is not through

732
00:48:32 --> 00:48:38
sophisticated examples that
mathematicians can construct.

733
00:48:38 --> 00:48:44
But normally,
because f of (x,

734
00:48:40 --> 00:48:46
y) will fail to be
defined somewhere,

735
00:48:43 --> 00:48:49
and those will be the bad
points.

736
00:48:46 --> 00:48:52
Thanks.