1
00:00:40,000 --> 00:00:46,000
This is also written in the
form, it's the k that's on the

2
00:00:45,000 --> 00:00:51,000
right hand side.
Actually, I found that source

3
00:00:49,000 --> 00:00:55,000
is of considerable difficulty.
And, in general,

4
00:00:53,000 --> 00:00:59,000
it is.
For these, the temperature

5
00:00:56,000 --> 00:01:02,000
concentration model,
it's natural to have the k on

6
00:01:01,000 --> 00:01:07,000
the right-hand side,
and to separate out the (q)e as

7
00:01:05,000 --> 00:01:11,000
part of it.
Another model for which that's

8
00:01:10,000 --> 00:01:16,000
true is mixing,
as I think I will show you on

9
00:01:14,000 --> 00:01:20,000
Monday.
On the other hand,

10
00:01:16,000 --> 00:01:22,000
there are some common
first-order models for which

11
00:01:21,000 --> 00:01:27,000
it's not a natural way to
separate things out.

12
00:01:25,000 --> 00:01:31,000
Examples would be the RC
circuit, radioactive decay,

13
00:01:29,000 --> 00:01:35,000
stuff like that.
So, this is not a universal

14
00:01:34,000 --> 00:01:40,000
utility.
But I thought that that form of

15
00:01:38,000 --> 00:01:44,000
writing it was a sufficient
utility to make a special case,

16
00:01:43,000 --> 00:01:49,000
and I emphasize it very heavily
in the nodes.

17
00:01:47,000 --> 00:01:53,000
Let's look at the equation.
And, this form will be good

18
00:01:52,000 --> 00:01:58,000
enough, the y prime.
When you solve it,

19
00:01:56,000 --> 00:02:02,000
let me remind you how the
solutions look,

20
00:02:00,000 --> 00:02:06,000
because that explains the
terminology.

21
00:02:05,000 --> 00:02:11,000
The solution looks like,
after you have done the

22
00:02:08,000 --> 00:02:14,000
integrating factor and
multiplied through,

23
00:02:11,000 --> 00:02:17,000
and integrated both sides,
in short, what you're supposed

24
00:02:16,000 --> 00:02:22,000
to do, the solution looks like y
equals, there's the term e to

25
00:02:20,000 --> 00:02:26,000
the negative k out
front times an integral which

26
00:02:25,000 --> 00:02:31,000
you can either make definite or
indefinite, according to your

27
00:02:30,000 --> 00:02:36,000
preference.
q of t times e to the kt inside

28
00:02:34,000 --> 00:02:40,000
dt,
it will help you to remember

29
00:02:37,000 --> 00:02:43,000
the opposite signs if you think
that when q is a constant,

30
00:02:41,000 --> 00:02:47,000
one, for example,
you want these two guys to

31
00:02:44,000 --> 00:02:50,000
cancel out and produce a
constant solution.

32
00:02:47,000 --> 00:02:53,000
That's a good way to remember
that the signs have to be

33
00:02:50,000 --> 00:02:56,000
opposite.
But, I don't encourage you to

34
00:02:53,000 --> 00:02:59,000
remember the formula at all.
It's just a convenient thing

35
00:02:56,000 --> 00:03:02,000
for me to be able to use right
now.

36
00:03:00,000 --> 00:03:06,000
And then, there's the other
term, which comes by putting up

37
00:03:04,000 --> 00:03:10,000
the arbitrary constant
explicitly, c e to the negative

38
00:03:09,000 --> 00:03:15,000
kt.
So, you could either write it

39
00:03:12,000 --> 00:03:18,000
this way, where this is somewhat
vague, or you could make it

40
00:03:17,000 --> 00:03:23,000
definite by putting a zero here
and a t there,

41
00:03:21,000 --> 00:03:27,000
and change the dummy variable
inside according to the way the

42
00:03:26,000 --> 00:03:32,000
notes tell you to do it.
Now, when you do this,

43
00:03:31,000 --> 00:03:37,000
and if k is positive,
that's absolutely essential,

44
00:03:35,000 --> 00:03:41,000
only when that is so,
then this term,

45
00:03:39,000 --> 00:03:45,000
as I told you a week or so ago,
this term goes to zero because

46
00:03:45,000 --> 00:03:51,000
k is positive as t goes to
infinity.

47
00:03:48,000 --> 00:03:54,000
So, this goes to zero as t
goes, and it doesn't matter what

48
00:03:53,000 --> 00:03:59,000
c is, as t goes to infinity.
This term stays some sort of

49
00:03:59,000 --> 00:04:05,000
function.
And so, this term is called the

50
00:04:03,000 --> 00:04:09,000
steady-state or long-term
solution, or it's called both,

51
00:04:08,000 --> 00:04:14,000
a long-term solution.
And this, which disappears,

52
00:04:14,000 --> 00:04:20,000
gets smaller and smaller as
time goes on,

53
00:04:17,000 --> 00:04:23,000
is therefore called the
transient because it disappears

54
00:04:21,000 --> 00:04:27,000
at the time increases to
infinity.

55
00:04:24,000 --> 00:04:30,000
So, this part uses the initial
condition, uses the initial

56
00:04:28,000 --> 00:04:34,000
value.
Let's call it y of zero,

57
00:04:31,000 --> 00:04:37,000
assuming that you
started the initial value,

58
00:04:35,000 --> 00:04:41,000
t, when t was equal to zero,
which is a common thing to do,

59
00:04:40,000 --> 00:04:46,000
although of course not
necessary.

60
00:04:44,000 --> 00:04:50,000
The starting value appears in
this term.

61
00:04:47,000 --> 00:04:53,000
This one is just some function.
Now, the general picture or the

62
00:04:53,000 --> 00:04:59,000
way that looks is,
the steady-state solution will

63
00:04:57,000 --> 00:05:03,000
be some solution like,
I don't know,

64
00:05:01,000 --> 00:05:07,000
like that, let's say.
So, that's a steady-state

65
00:05:05,000 --> 00:05:11,000
solution, the SSS.
Well, what do the other guys

66
00:05:09,000 --> 00:05:15,000
look like?
Well, the steady-state solution

67
00:05:12,000 --> 00:05:18,000
has this starting point.
Other solutions can have any of

68
00:05:16,000 --> 00:05:22,000
these other starting points.
So, in the beginning,

69
00:05:20,000 --> 00:05:26,000
they won't look like the
steady-state solution.

70
00:05:23,000 --> 00:05:29,000
But, we know that as time goes
on, they must approach it

71
00:05:27,000 --> 00:05:33,000
because this term represents the
difference between the solution

72
00:05:31,000 --> 00:05:37,000
and the steady-state solution.
So, this term is going to zero.

73
00:05:37,000 --> 00:05:43,000
And therefore,
whatever these guys do to start

74
00:05:41,000 --> 00:05:47,000
out with, after a while they
must follow the steady-state

75
00:05:45,000 --> 00:05:51,000
solution more and more closely.
They must, in short,

76
00:05:49,000 --> 00:05:55,000
be asymptotic to it.
So, the solutions to any

77
00:05:53,000 --> 00:05:59,000
equation of that form will look
like this.

78
00:05:56,000 --> 00:06:02,000
Up here, maybe it started at
127.

79
00:05:58,000 --> 00:06:04,000
That's okay.
After a while,

80
00:06:00,000 --> 00:06:06,000
it's going to start approaching
that green curve.

81
00:06:06,000 --> 00:06:12,000
Of course, they won't cross
each other.

82
00:06:09,000 --> 00:06:15,000
That's the rock star,
and these are the groupies

83
00:06:13,000 --> 00:06:19,000
trying to get close to it.
Now, but something follows from

84
00:06:18,000 --> 00:06:24,000
that picture.
Which is the steady-state

85
00:06:22,000 --> 00:06:28,000
solution?
What, in short,

86
00:06:24,000 --> 00:06:30,000
is so special about this green
curve?

87
00:06:27,000 --> 00:06:33,000
All these other white solution
curves have that same property,

88
00:06:33,000 --> 00:06:39,000
the same property that all the
other white curves and the green

89
00:06:38,000 --> 00:06:44,000
curve, too, are trying to get
close to them.

90
00:06:44,000 --> 00:06:50,000
In other words,
there is nothing special about

91
00:06:47,000 --> 00:06:53,000
the green curve.
It's just that they all want to

92
00:06:51,000 --> 00:06:57,000
get close to each other.
And therefore,

93
00:06:54,000 --> 00:07:00,000
though you can write a formula
like this, there isn't one

94
00:06:58,000 --> 00:07:04,000
steady-state solution.
There are many.

95
00:07:01,000 --> 00:07:07,000
Now, this produces vagueness.
You talk about the steady-state

96
00:07:07,000 --> 00:07:13,000
solution; which one are you
talking about?

97
00:07:09,000 --> 00:07:15,000
I have no answer to that;
the usual answer is whichever

98
00:07:13,000 --> 00:07:19,000
one looks simplest.
Normally, the one that will

99
00:07:16,000 --> 00:07:22,000
look simplest is the one where c
is zero.

100
00:07:19,000 --> 00:07:25,000
But, if this is a peculiar
function, it might be that for

101
00:07:23,000 --> 00:07:29,000
some other value of c,
you get an even simpler

102
00:07:26,000 --> 00:07:32,000
expression.
So, the steady-state solution:

103
00:07:29,000 --> 00:07:35,000
about the best I can see,
either you integrate that,

104
00:07:32,000 --> 00:07:38,000
don't use an arbitrary
constant, and use what you get,

105
00:07:36,000 --> 00:07:42,000
or pick the simplest.
Pick the value of c,

106
00:07:41,000 --> 00:07:47,000
which gives you the simplest
answer.

107
00:07:46,000 --> 00:07:52,000
Pick the simplest function,
and that's what usually called

108
00:07:53,000 --> 00:07:59,000
the steady-state solution.
Now, from that point of view,

109
00:08:00,000 --> 00:08:06,000
what I'm calling the input in
this input response point of

110
00:08:05,000 --> 00:08:11,000
view, which we are going to be
using, by the way,

111
00:08:08,000 --> 00:08:14,000
constantly, well,
pretty much all term long,

112
00:08:12,000 --> 00:08:18,000
but certainly for the next
month or so, I'm constantly

113
00:08:16,000 --> 00:08:22,000
going to be coming back to it.
The input is the q of t.

114
00:08:21,000 --> 00:08:27,000
In other words,

115
00:08:23,000 --> 00:08:29,000
it seems rather peculiar.
But the input is the right-hand

116
00:08:27,000 --> 00:08:33,000
side of the equation of the
differential equation.

117
00:08:31,000 --> 00:08:37,000
And the reason is because I'm
always thinking of the

118
00:08:35,000 --> 00:08:41,000
temperature model.
The external water bath at

119
00:08:41,000 --> 00:08:47,000
temperature T external,
the internal thing here,

120
00:08:44,000 --> 00:08:50,000
the problem is,
given this function,

121
00:08:47,000 --> 00:08:53,000
the external water bath
temperature is driving,

122
00:08:51,000 --> 00:08:57,000
so to speak,
the temperature of the inside.

123
00:08:54,000 --> 00:09:00,000
And therefore,
the input is the temperature of

124
00:08:57,000 --> 00:09:03,000
the water bath.
I don't like the word output,

125
00:09:01,000 --> 00:09:07,000
although it would be the
natural thing because this

126
00:09:05,000 --> 00:09:11,000
temperature doesn't look like an
output.

127
00:09:07,000 --> 00:09:13,000
Anyone might be willing to say,
yeah, you are inputting the

128
00:09:11,000 --> 00:09:17,000
value of the temperature here.
This, it's more likely,

129
00:09:15,000 --> 00:09:21,000
the normal term is response.
This thing, this plus the water

130
00:09:19,000 --> 00:09:25,000
bath, is a little system.
And the response of the system,

131
00:09:22,000 --> 00:09:28,000
i.e.
the change in the internal

132
00:09:24,000 --> 00:09:30,000
temperature is governed by the
driving external temperature.

133
00:09:28,000 --> 00:09:34,000
So, the input is q of t,
and the response of

134
00:09:32,000 --> 00:09:38,000
the system is the solution to
the differential equation.

135
00:09:45,000 --> 00:09:51,000
Now, if the thing is special,
as it's going to be for most of

136
00:09:49,000 --> 00:09:55,000
this period, it has that special
form, then I'm going to,

137
00:09:54,000 --> 00:10:00,000
I really want to call q sub e
the input.

138
00:09:58,000 --> 00:10:04,000
I want to call q sub e the
input, and there is no standard

139
00:10:02,000 --> 00:10:08,000
way of doing that,
although there's a most common

140
00:10:06,000 --> 00:10:12,000
way.
So, I'm just calling it the

141
00:10:10,000 --> 00:10:16,000
physical input,
in other words,

142
00:10:12,000 --> 00:10:18,000
the temperature input,
or the concentration input.

143
00:10:16,000 --> 00:10:22,000
And, that will be my (q)e of t,
and by the

144
00:10:21,000 --> 00:10:27,000
subscript e, you will understand
that I'm writing it in that form

145
00:10:26,000 --> 00:10:32,000
and thinking of this model,
or concentration model,

146
00:10:30,000 --> 00:10:36,000
or mixing model as I will show
you on Monday.

147
00:10:35,000 --> 00:10:41,000
By the way, this is often
handled, I mean,

148
00:10:37,000 --> 00:10:43,000
how would you handle this to
get rid of a k?

149
00:10:41,000 --> 00:10:47,000
Well, divide through by k.
So, this equation is often,

150
00:10:45,000 --> 00:10:51,000
in the literature,
written this way:

151
00:10:47,000 --> 00:10:53,000
one over k times y prime plus y
is equal to, well,

152
00:10:51,000 --> 00:10:57,000
now they call it q of t,
not (q)e of t because they've

153
00:10:55,000 --> 00:11:01,000
gotten rid of this funny factor.
But

154
00:10:59,000 --> 00:11:05,000
I will continue to call it (q)e.
So, in other words,

155
00:11:04,000 --> 00:11:10,000
and this part this is just,
frankly, called the input.

156
00:11:09,000 --> 00:11:15,000
It doesn't say physical or
anything.

157
00:11:11,000 --> 00:11:17,000
And, this is the solution,
it's then the response,

158
00:11:16,000 --> 00:11:22,000
and this funny coefficient of y
prime,

159
00:11:19,000 --> 00:11:25,000
that's not in standard linear
form, is it, anymore?

160
00:11:23,000 --> 00:11:29,000
But, it's a standard form if
you want to do this input

161
00:11:28,000 --> 00:11:34,000
response analysis.
So, this is also a way of

162
00:11:31,000 --> 00:11:37,000
writing the equation.
I'm not going to use it because

163
00:11:38,000 --> 00:11:44,000
how many standard forms could
this poor little course absorb?

164
00:11:43,000 --> 00:11:49,000
I'll stick to that one.
Okay, you have,

165
00:11:47,000 --> 00:11:53,000
then, the superposition
principle, which I don't think

166
00:11:52,000 --> 00:11:58,000
I'm going to-- the solution,
which solution?

167
00:11:57,000 --> 00:12:03,000
Well, normally it means any
solution, or in other words,

168
00:12:02,000 --> 00:12:08,000
the steady-state solution.
Now, notice that terminology

169
00:12:07,000 --> 00:12:13,000
only makes sense if k is
positive.

170
00:12:10,000 --> 00:12:16,000
And, in fact,
there is nothing like the

171
00:12:13,000 --> 00:12:19,000
picture, the picture doesn't
look at all like this if k is

172
00:12:17,000 --> 00:12:23,000
negative, and therefore,
the terms would steady state,

173
00:12:20,000 --> 00:12:26,000
transient would be totally
inappropriate if k were

174
00:12:24,000 --> 00:12:30,000
negative.
So, this assumes definitely

175
00:12:26,000 --> 00:12:32,000
that k has to be greater than
zero.

176
00:12:30,000 --> 00:12:36,000
Otherwise, no.
So, I'll call this the physical

177
00:12:33,000 --> 00:12:39,000
input.
And then, you have the

178
00:12:35,000 --> 00:12:41,000
superposition principle,
which I really can't improve

179
00:12:40,000 --> 00:12:46,000
upon what's written in the
notes, this superposition of

180
00:12:44,000 --> 00:12:50,000
inputs.
Whether they are physical

181
00:12:47,000 --> 00:12:53,000
inputs or nonphysical inputs,
if the input q of t produces

182
00:12:51,000 --> 00:12:57,000
the response, y of t,

183
00:12:54,000 --> 00:13:00,000
and q two of t produces the
response, y two of t,

184
00:13:02,000 --> 00:13:08,000
-- then a simple calculation
with the differential equation

185
00:13:07,000 --> 00:13:13,000
shows you that by,
so to speak,

186
00:13:10,000 --> 00:13:16,000
adding, that the sum of these
two, I stated it very generally

187
00:13:15,000 --> 00:13:21,000
in the notes but it corresponds,
we will have as the response

188
00:13:21,000 --> 00:13:27,000
y1, the steady-state response y1
plus y2,

189
00:13:25,000 --> 00:13:31,000
and a constant times y1.
That's an expression,

190
00:13:30,000 --> 00:13:36,000
essentially,
of the linear,

191
00:13:32,000 --> 00:13:38,000
it uses the fact that the
special form of the equation,

192
00:13:35,000 --> 00:13:41,000
and we will have a very
efficient and elegant way of

193
00:13:38,000 --> 00:13:44,000
seeing this when we study higher
order equations.

194
00:13:41,000 --> 00:13:47,000
For now, I will just,
the little calculation that's

195
00:13:45,000 --> 00:13:51,000
done in the notes will suffice
for first-order equations.

196
00:13:48,000 --> 00:13:54,000
If you don't have a complicated
equation, there's no point in

197
00:13:52,000 --> 00:13:58,000
making a fuss over proofs using
it.

198
00:13:54,000 --> 00:14:00,000
But essentially,
it uses the fact that the

199
00:13:57,000 --> 00:14:03,000
equation is linear.
Or, that's bad,

200
00:14:01,000 --> 00:14:07,000
so linearity of the ODE.
In other words,

201
00:14:04,000 --> 00:14:10,000
it's a consequence of the fact
that the equation looks the way

202
00:14:09,000 --> 00:14:15,000
it does.
And, something like this would

203
00:14:12,000 --> 00:14:18,000
not, in any sense,
be true if the equation,

204
00:14:15,000 --> 00:14:21,000
for example,
had here a y squared

205
00:14:18,000 --> 00:14:24,000
instead of t.
Everything I'm saying this

206
00:14:21,000 --> 00:14:27,000
period would be total nonsense
and totally inapplicable.

207
00:14:27,000 --> 00:14:33,000
Now, today, what I wanted to
discuss was, what's in the notes

208
00:14:32,000 --> 00:14:38,000
that I gave you today,
which is, what happens when the

209
00:14:36,000 --> 00:14:42,000
physical input is trigonometric?
For certain reasons,

210
00:14:41,000 --> 00:14:47,000
that's the most important case
there is.

211
00:14:44,000 --> 00:14:50,000
It's because of the existence
of what are called Fourier

212
00:14:49,000 --> 00:14:55,000
series, and there are a couple
of words about them.

213
00:14:53,000 --> 00:14:59,000
That's something we will be
studying in about three weeks or

214
00:14:58,000 --> 00:15:04,000
so.
What's going on,

215
00:15:01,000 --> 00:15:07,000
roughly, is that,
so I'm going to take the

216
00:15:06,000 --> 00:15:12,000
equation in the form y prime
plus ky equals k times

217
00:15:12,000 --> 00:15:18,000
(q)e of t,
and the input that I'm

218
00:15:17,000 --> 00:15:23,000
interested in is when this is a
simple one that you use on the

219
00:15:23,000 --> 00:15:29,000
visual that you did about two
points worth of work for handing

220
00:15:30,000 --> 00:15:36,000
in today, cosine omega t.

221
00:15:36,000 --> 00:15:42,000
So, if you like,
k here.

222
00:15:37,000 --> 00:15:43,000
So, the (q)e is cosine omega t.
That was the physical input.

223
00:15:41,000 --> 00:15:47,000
And, omega, as you know,
is, you have to be careful when

224
00:15:44,000 --> 00:15:50,000
you use the word frequency.
I assume you got this from

225
00:15:48,000 --> 00:15:54,000
physics class all last semester.
But anyway, just to remind you,

226
00:15:52,000 --> 00:15:58,000
there's a whole yoga of five or
six terms that go whenever

227
00:15:56,000 --> 00:16:02,000
you're talking about
trigonometric functions.

228
00:16:00,000 --> 00:16:06,000
Instead of giving a long
explanation, the end of the

229
00:16:03,000 --> 00:16:09,000
second page of the notes just
gives you a reference list of

230
00:16:08,000 --> 00:16:14,000
what you are expected to know
for 18.03 and physics as well,

231
00:16:12,000 --> 00:16:18,000
with a brief one or two line
description of what each of

232
00:16:17,000 --> 00:16:23,000
those means.
So, think of it as something to

233
00:16:20,000 --> 00:16:26,000
refer back to if you have
forgotten.

234
00:16:23,000 --> 00:16:29,000
But, omega is what's called the
angular frequency or the

235
00:16:27,000 --> 00:16:33,000
circular frequency.
It's somewhat misleading to

236
00:16:31,000 --> 00:16:37,000
call it the frequency,
although I probably will.

237
00:16:36,000 --> 00:16:42,000
It's the angular frequency.
It's, in other words,

238
00:16:39,000 --> 00:16:45,000
it's the number of complete
oscillations.

239
00:16:42,000 --> 00:16:48,000
This cosine omega t
is going up and down right?

240
00:16:47,000 --> 00:16:53,000
So, a complete oscillation as
it goes down and then returns to

241
00:16:51,000 --> 00:16:57,000
where it started.
That's a complete oscillation.

242
00:16:55,000 --> 00:17:01,000
This is only half an
oscillation because you didn't

243
00:16:58,000 --> 00:17:04,000
give it a chance to get back.
Okay, so the number of complete

244
00:17:03,000 --> 00:17:09,000
oscillations in how much time,
well, in two pi,

245
00:17:06,000 --> 00:17:12,000
in the distance,
two pi on the t-axis in the

246
00:17:09,000 --> 00:17:15,000
interval of length two pi
because, for example,

247
00:17:13,000 --> 00:17:19,000
if omega is one,
cosine t takes two

248
00:17:16,000 --> 00:17:22,000
pi to repeat itself,
right?

249
00:17:20,000 --> 00:17:26,000
If omega were two,
it would repeat itself.

250
00:17:23,000 --> 00:17:29,000
It would make two complete
oscillations in the interval,

251
00:17:28,000 --> 00:17:34,000
two pi.
So, it's what happens to the

252
00:17:31,000 --> 00:17:37,000
interval, two pi,
not what happens in the time

253
00:17:35,000 --> 00:17:41,000
interval, one,
which is the natural meaning of

254
00:17:39,000 --> 00:17:45,000
the word frequency.
There's always this factor of

255
00:17:43,000 --> 00:17:49,000
two pi that floats around to
make all of your formulas and

256
00:17:48,000 --> 00:17:54,000
solutions incorrect.
Okay, now, so,

257
00:17:51,000 --> 00:17:57,000
what I'm out to do is,
the problem is for the physical

258
00:17:55,000 --> 00:18:01,000
input, (q)e cosine omega t,

259
00:17:59,000 --> 00:18:05,000
find the response.
In other words,

260
00:18:02,000 --> 00:18:08,000
solve the differential
equation.

261
00:18:07,000 --> 00:18:13,000
In short, for the visual that
you looked at,

262
00:18:11,000 --> 00:18:17,000
I think I've forgot the colors
now.

263
00:18:14,000 --> 00:18:20,000
The input was in green,
maybe, but I do remember that

264
00:18:19,000 --> 00:18:25,000
the response was in yellow.
I think I remember that.

265
00:18:24,000 --> 00:18:30,000
So, find the response,
yellow, and the input was,

266
00:18:28,000 --> 00:18:34,000
what color was it,
green?

267
00:18:30,000 --> 00:18:36,000
Blue, blue.
Light blue.

268
00:18:34,000 --> 00:18:40,000
Okay, so we've got to solve the
differential equation.

269
00:18:39,000 --> 00:18:45,000
Now, it's a question of how I'm
going to solve the differential

270
00:18:46,000 --> 00:18:52,000
equation.
I'm going to use complex

271
00:18:49,000 --> 00:18:55,000
numbers throughout,
A because that's the way it's

272
00:18:54,000 --> 00:19:00,000
usually done.
B, to give you practice using

273
00:18:59,000 --> 00:19:05,000
complex numbers,
and I don't think I need any

274
00:19:04,000 --> 00:19:10,000
other reasons.
So, I'm going to use complex

275
00:19:09,000 --> 00:19:15,000
numbers.
I'm going to complexify.

276
00:19:13,000 --> 00:19:19,000
To use complex numbers,
what you do is complexification

277
00:19:18,000 --> 00:19:24,000
of the problem.
So, I'm going to complexify the

278
00:19:23,000 --> 00:19:29,000
problem, turn it into the domain
of complex numbers.

279
00:19:29,000 --> 00:19:35,000
So, take the differential
equation, turn it into a

280
00:19:33,000 --> 00:19:39,000
differential equation involving
complex numbers,

281
00:19:37,000 --> 00:19:43,000
solve that, and then go back to
the real domain to get the

282
00:19:42,000 --> 00:19:48,000
answer, since it's easier to
integrate exponentials.

283
00:19:46,000 --> 00:19:52,000
And therefore,
try to introduce,

284
00:19:49,000 --> 00:19:55,000
try to change the trigonometric
functions into complex

285
00:19:53,000 --> 00:19:59,000
exponentials,
simply because the work will be

286
00:19:57,000 --> 00:20:03,000
easier to do.
All right, so let's do it.

287
00:20:02,000 --> 00:20:08,000
To change this differential
equation, remember,

288
00:20:05,000 --> 00:20:11,000
I've got cosine omega t here.

289
00:20:09,000 --> 00:20:15,000
I'm going to use the fact that
e to the i omega t,

290
00:20:14,000 --> 00:20:20,000
Euler's formula,
that the real part of it is

291
00:20:17,000 --> 00:20:23,000
cosine omega t.
So, I'm going to view this as

292
00:20:22,000 --> 00:20:28,000
the real part of this complex
function.

293
00:20:25,000 --> 00:20:31,000
But, I will throw at the
imaginary part,

294
00:20:28,000 --> 00:20:34,000
too, since at one point we will
need it.

295
00:20:31,000 --> 00:20:37,000
Now, what is the equation,
then, that it's going to turn

296
00:20:36,000 --> 00:20:42,000
into?
The complexified equation is

297
00:20:41,000 --> 00:20:47,000
going to be y prime plus ky
equals, and now,

298
00:20:46,000 --> 00:20:52,000
instead of the right hand side,
k times cosine omega t,

299
00:20:53,000 --> 00:20:59,000
I'll use the whole complex
exponential, e i omega t.

300
00:21:00,000 --> 00:21:06,000
Now, I have a problem because

301
00:21:06,000 --> 00:21:12,000
y, here, in this equation,
y means the real function which

302
00:21:09,000 --> 00:21:15,000
solves that problem.
I therefore cannot continue to

303
00:21:13,000 --> 00:21:19,000
call this y because I want y to
be a real function.

304
00:21:16,000 --> 00:21:22,000
I have to change its name.
Since this is complex function

305
00:21:20,000 --> 00:21:26,000
on the right-hand side,
I will have to expect a complex

306
00:21:24,000 --> 00:21:30,000
solution to the differential
equation.

307
00:21:28,000 --> 00:21:34,000
I'm going to call that complex
solution y tilda.

308
00:21:32,000 --> 00:21:38,000
Now, that's what I would also
use as the designation for the

309
00:21:38,000 --> 00:21:44,000
variable.
So, y tilda is the complex

310
00:21:42,000 --> 00:21:48,000
solution.
And, it's going to have the

311
00:21:46,000 --> 00:21:52,000
form y1 plus i times y2.

312
00:21:49,000 --> 00:21:55,000
It's going to be the complex
solution.

313
00:21:53,000 --> 00:21:59,000
And now, what I say is,
so, solve it.

314
00:21:57,000 --> 00:22:03,000
Find this complex solution.
So, find the program is to find

315
00:22:03,000 --> 00:22:09,000
y tilde, --
-- that's the complex solution.

316
00:22:08,000 --> 00:22:14,000
And then I say,
all you have to do is take the

317
00:22:12,000 --> 00:22:18,000
real part of that,
and that will answer the

318
00:22:16,000 --> 00:22:22,000
original problem.
Then, y1, that's the real part

319
00:22:20,000 --> 00:22:26,000
of it, right?
It's a function,

320
00:22:23,000 --> 00:22:29,000
you know, like this is cosine
plus sine, as it was over here,

321
00:22:28,000 --> 00:22:34,000
it will naturally be something
different.

322
00:22:31,000 --> 00:22:37,000
It will be something different,
but that part of it,

323
00:22:36,000 --> 00:22:42,000
the real part will solve the
original problem,

324
00:22:40,000 --> 00:22:46,000
the original,
real, ODE.

325
00:22:44,000 --> 00:22:50,000
Now, you will say,
you expect us to believe that?

326
00:22:47,000 --> 00:22:53,000
Well, yes, in fact.
I think we've got a lot to do,

327
00:22:51,000 --> 00:22:57,000
so since the argument for this
is given in the nodes,

328
00:22:54,000 --> 00:23:00,000
so, read this in the notes.
It only takes a line or two of

329
00:22:58,000 --> 00:23:04,000
standard work with
differentiation.

330
00:23:02,000 --> 00:23:08,000
So, read in the notes the
argument for that,

331
00:23:05,000 --> 00:23:11,000
why that's so.
It just amounts to separating

332
00:23:09,000 --> 00:23:15,000
real and imaginary parts.
Okay, so let's,

333
00:23:13,000 --> 00:23:19,000
now, solve this.
Since that's our program,

334
00:23:17,000 --> 00:23:23,000
all we have to find is the
solution.

335
00:23:20,000 --> 00:23:26,000
Well, just use integrating
factors and just do it.

336
00:23:25,000 --> 00:23:31,000
So, the integrating factor will
be, what, e to the,

337
00:23:29,000 --> 00:23:35,000
I don't want to use that
formula.

338
00:23:34,000 --> 00:23:40,000
So, the integrating factor will
be e to the kt is the

339
00:23:38,000 --> 00:23:44,000
integrating factor.
If I multiply through both

340
00:23:42,000 --> 00:23:48,000
sides by the integrating factor,
then the left-hand side will

341
00:23:46,000 --> 00:23:52,000
become y e to the kt,
the way it always does,

342
00:23:50,000 --> 00:23:56,000
prime, Y tilde, sorry,

343
00:23:53,000 --> 00:23:59,000
and the right-hand side will
be, now I'm going to start

344
00:23:57,000 --> 00:24:03,000
combining exponentials.
It will be k times e to the

345
00:24:03,000 --> 00:24:09,000
power i times omega t plus k.

346
00:24:11,000 --> 00:24:17,000
I'm going to write that k plus
omega t.

347
00:24:31,000 --> 00:24:37,000
i omega t plus k.

348
00:24:36,000 --> 00:24:42,000
Thank you.
i omega t plus k,

349
00:24:40,000 --> 00:24:46,000
or k plus i omega t.

350
00:24:47,000 --> 00:24:53,000
kt?
Sorry.

351
00:24:48,000 --> 00:24:54,000
So, it's k times e to the i
omega t times e to the kt.

352
00:24:57,000 --> 00:25:03,000
So, that's (k plus i omega)

353
00:25:06,000 --> 00:25:12,000
times t. Sorry.

354
00:25:10,000 --> 00:25:16,000
So, y tilda e to the kt
is k divided by,

355
00:25:17,000 --> 00:25:23,000
now I integrate this,
so it essentially reproduces

356
00:25:23,000 --> 00:25:29,000
itself, except you have to put
down on the bottom k plus i

357
00:25:30,000 --> 00:25:36,000
omega.
I'll take the final step.

358
00:25:35,000 --> 00:25:41,000
What's y tilda equals,
see, when you do it this way,

359
00:25:38,000 --> 00:25:44,000
then you don't get a messy
looking formula that you

360
00:25:42,000 --> 00:25:48,000
substitute into and that is
scary looking.

361
00:25:44,000 --> 00:25:50,000
This is never scary.
Now, I'm going to do two things

362
00:25:48,000 --> 00:25:54,000
simultaneously.
First of all,

363
00:25:49,000 --> 00:25:55,000
here, if I multiply,
after I get the answer,

364
00:25:52,000 --> 00:25:58,000
I'm going to want to multiply
it by e to the negative kt,

365
00:25:56,000 --> 00:26:02,000
right,
to solve for y tilda.

366
00:26:00,000 --> 00:26:06,000
If I multiply this by e to the
negative kt, then that just gets

367
00:26:04,000 --> 00:26:10,000
rid of the k that I put in,
and left back with e to the i

368
00:26:08,000 --> 00:26:14,000
omega t.
So, that side is easy.

369
00:26:10,000 --> 00:26:16,000
All that is left is e to the i
omega t.

370
00:26:14,000 --> 00:26:20,000
Now, what's interesting is this
thing out here,

371
00:26:18,000 --> 00:26:24,000
k plus i omega.
I'm going to take a typical

372
00:26:22,000 --> 00:26:28,000
step of scaling it.
And you scale it.

373
00:26:24,000 --> 00:26:30,000
I'm going to divide the top and
bottom by k.

374
00:26:29,000 --> 00:26:35,000
And, what does that produce?
One divided by one plus i times

375
00:26:35,000 --> 00:26:41,000
omega over k.

376
00:26:40,000 --> 00:26:46,000
What I've done is take these
two separate constants,

377
00:26:45,000 --> 00:26:51,000
and shown that the critical
thing is their ratio.

378
00:26:51,000 --> 00:26:57,000
Okay, now, what I have to do
now is take the real part.

379
00:26:57,000 --> 00:27:03,000
Now, there are two ways to do
this.

380
00:27:01,000 --> 00:27:07,000
There are two ways to do this.
Both are instructive.

381
00:27:08,000 --> 00:27:14,000
So, there are two methods.
I have a multiplication.

382
00:27:13,000 --> 00:27:19,000
The problem is,
of course, that these two

383
00:27:17,000 --> 00:27:23,000
things are multiplied together.
And, one of them is,

384
00:27:23,000 --> 00:27:29,000
essentially,
in Cartesian form,

385
00:27:26,000 --> 00:27:32,000
and the other is,
essentially,

386
00:27:29,000 --> 00:27:35,000
in polar form.
You have to make a decision.

387
00:27:35,000 --> 00:27:41,000
Either go polar,
it sounds like go postal,

388
00:27:40,000 --> 00:27:46,000
doesn't it, or worse,
like a bear,

389
00:27:45,000 --> 00:27:51,000
savage, attack it savagely,
which that's a very good,

390
00:27:52,000 --> 00:27:58,000
aggressive attitude to have
when doing a problem,

391
00:27:58,000 --> 00:28:04,000
or we can go Cartesian.
Going polar is a little faster,

392
00:28:05,000 --> 00:28:11,000
and I think it's what's done in
the nodes.

393
00:28:08,000 --> 00:28:14,000
The notes to do both of these.
They just do the first.

394
00:28:11,000 --> 00:28:17,000
On the other hand,
they give you a formula,

395
00:28:14,000 --> 00:28:20,000
which is the critical thing
that you will need to go

396
00:28:18,000 --> 00:28:24,000
Cartesian.
I hope I can do both of them if

397
00:28:21,000 --> 00:28:27,000
we sort of hurry along.
How do I go polar?

398
00:28:24,000 --> 00:28:30,000
To go polar,
what you want to do is express

399
00:28:27,000 --> 00:28:33,000
this thing in polar form.
Now, one of the things I didn't

400
00:28:32,000 --> 00:28:38,000
emphasize enough,
probably, when I talked to you

401
00:28:35,000 --> 00:28:41,000
about complex numbers last time
is, so I will remind you,

402
00:28:40,000 --> 00:28:46,000
which saves my conscience and
doesn't hurt yours,

403
00:28:43,000 --> 00:28:49,000
suppose you have alpha as a
complex number.

404
00:28:47,000 --> 00:28:53,000
See, this complex number is a
reciprocal.

405
00:28:50,000 --> 00:28:56,000
The good number is what's down
below.

406
00:28:52,000 --> 00:28:58,000
Unfortunately,
it's downstairs.

407
00:28:55,000 --> 00:29:01,000
You should know,
like you know the back of your

408
00:28:58,000 --> 00:29:04,000
hand, which nobody knows,
one over alpha.

409
00:29:03,000 --> 00:29:09,000
So that's the form.
The number I'm interested in,

410
00:29:05,000 --> 00:29:11,000
that coefficient,
it is of the form one over

411
00:29:08,000 --> 00:29:14,000
alpha.
One over alpha times alpha is

412
00:29:10,000 --> 00:29:16,000
equal to one.

413
00:29:13,000 --> 00:29:19,000
And, from that,
it follows, first of all,

414
00:29:15,000 --> 00:29:21,000
if I take absolute values,
if the absolute value of one

415
00:29:19,000 --> 00:29:25,000
over alpha times the absolute
value of this is equal to one,

416
00:29:22,000 --> 00:29:28,000
so, this is equal to one over
the absolute value of alpha.

417
00:29:26,000 --> 00:29:32,000
I think you all knew that.
I'm a little less certain you

418
00:29:29,000 --> 00:29:35,000
knew how to take care of the
angles.

419
00:29:33,000 --> 00:29:39,000
How about the argument?
Well, the argument of the

420
00:29:36,000 --> 00:29:42,000
angle, in other words,
the angle of one over alpha

421
00:29:40,000 --> 00:29:46,000
plus, because when you multiply,
angles add.

422
00:29:44,000 --> 00:29:50,000
Remember that.
Plus, the angle associated with

423
00:29:48,000 --> 00:29:54,000
alpha has to be the angle
associated with one.

424
00:29:51,000 --> 00:29:57,000
But what's that?
One is out here.

425
00:29:54,000 --> 00:30:00,000
What's the angle of one?
Zero.

426
00:30:06,000 --> 00:30:12,000
Therefore, the argument,
the absolute value of this

427
00:30:10,000 --> 00:30:16,000
thing is want over the absolute
value.

428
00:30:14,000 --> 00:30:20,000
That's easy.
And, you should know that the

429
00:30:18,000 --> 00:30:24,000
argument of want over alpha is
equal to minus the argument of

430
00:30:23,000 --> 00:30:29,000
alpha.
So, when you take reciprocal,

431
00:30:27,000 --> 00:30:33,000
the angle turns into its
negative.

432
00:30:30,000 --> 00:30:36,000
Okay, I'm going to use that
now, because my aim is to turn

433
00:30:35,000 --> 00:30:41,000
this into polar form.
So, let's do that someplace,

434
00:30:40,000 --> 00:30:46,000
I guess here.
So, I want the polar form for

435
00:30:48,000 --> 00:30:54,000
one over one plus i times omega
over k.

436
00:31:00,000 --> 00:31:06,000
Okay, I will draw a picture.

437
00:31:04,000 --> 00:31:10,000
Here's one.
Here is omega over k.

438
00:31:09,000 --> 00:31:15,000
Let's call this angle phi.

439
00:31:12,000 --> 00:31:18,000
It's a natural thing to call
it.

440
00:31:15,000 --> 00:31:21,000
It's a right triangle,
of course.

441
00:31:18,000 --> 00:31:24,000
Okay, now, this is going to be
a complex number times e to an

442
00:31:24,000 --> 00:31:30,000
angle.
Now, what's the angle going to

443
00:31:28,000 --> 00:31:34,000
be?
Well, this is a complex number,

444
00:31:32,000 --> 00:31:38,000
the angle for the complex
number.

445
00:31:35,000 --> 00:31:41,000
So, the argument of the complex
number, one plus i times omega

446
00:31:40,000 --> 00:31:46,000
over k is how much?

447
00:31:43,000 --> 00:31:49,000
Well, there's the complex
number one plus i over one plus

448
00:31:48,000 --> 00:31:54,000
i times omega over k.

449
00:31:53,000 --> 00:31:59,000
Its angle is phi.
So, the argument of this is

450
00:31:57,000 --> 00:32:03,000
phi, and therefore,
the argument of its reciprocal

451
00:32:01,000 --> 00:32:07,000
is negative phi.
So, it's e to the minus i phi.

452
00:32:06,000 --> 00:32:12,000
And, what's A?

453
00:32:09,000 --> 00:32:15,000
A is one over the absolute
value of that complex number.

454
00:32:14,000 --> 00:32:20,000
Well, the absolute value of
this complex number is one plus

455
00:32:20,000 --> 00:32:26,000
omega over k squared.

456
00:32:24,000 --> 00:32:30,000
So, the A is going to be one
over that, the square root of

457
00:32:29,000 --> 00:32:35,000
one plus omega over k,
the quantity squared,

458
00:32:33,000 --> 00:32:39,000
times e to the minus i phi.

459
00:32:39,000 --> 00:32:45,000
See, I did that.

460
00:32:43,000 --> 00:32:49,000
That's a critical step.
You must turn that coefficient.

461
00:32:46,000 --> 00:32:52,000
If you want to go polar,
you must turn is that

462
00:32:49,000 --> 00:32:55,000
coefficient, write that
coefficient in the polar form.

463
00:32:52,000 --> 00:32:58,000
And for that,
you need these basic facts

464
00:32:54,000 --> 00:33:00,000
about, draw the complex number,
draw its angle,

465
00:32:57,000 --> 00:33:03,000
and so on and so forth.
And now, what's there for the

466
00:33:02,000 --> 00:33:08,000
solution?
Once you've done that,

467
00:33:06,000 --> 00:33:12,000
the work is over.
What's the complex solution?

468
00:33:10,000 --> 00:33:16,000
The complex solution is this.
I've just found the polar form

469
00:33:16,000 --> 00:33:22,000
for this.
So, I multiply it by e to the i

470
00:33:20,000 --> 00:33:26,000
omega t,
which means these things add.

471
00:33:25,000 --> 00:33:31,000
So, it's equal to A,
this A, times e to the i omega

472
00:33:30,000 --> 00:33:36,000
t minus i times phi.

473
00:33:37,000 --> 00:33:43,000
Or, in other words,
the coefficient is one over,

474
00:33:42,000 --> 00:33:48,000
this is a real number,
now, square root of one plus

475
00:33:47,000 --> 00:33:53,000
omega over k squared.

476
00:33:53,000 --> 00:33:59,000
And, this is e to the,
see if I get it right,

477
00:33:58,000 --> 00:34:04,000
now.
And finally,

478
00:34:00,000 --> 00:34:06,000
now, what's the answer to our
real problem?

479
00:34:05,000 --> 00:34:11,000
y1: the real answer.
I mean: the really real answer.

480
00:34:11,000 --> 00:34:17,000
What is it?
Well, this is a real number.

481
00:34:13,000 --> 00:34:19,000
So, I simply reproduce that as
the coefficient out front.

482
00:34:17,000 --> 00:34:23,000
And for the other part,
I want the real part of that.

483
00:34:20,000 --> 00:34:26,000
But you can write that down
instantly.

484
00:34:23,000 --> 00:34:29,000
So, let's recopy the
coefficient.

485
00:34:25,000 --> 00:34:31,000
And then, I want just the real
part of this.

486
00:34:28,000 --> 00:34:34,000
Well, this is e to the i times
some crazy angle.

487
00:34:32,000 --> 00:34:38,000
So, the real part is the cosine
of that crazy angle.

488
00:34:36,000 --> 00:34:42,000
So, it's the cosine of omega t
minus phi.

489
00:34:40,000 --> 00:34:46,000
And, if somebody says,

490
00:34:42,000 --> 00:34:48,000
yeah, well, okay,
I got the omega k,

491
00:34:45,000 --> 00:34:51,000
I know what that is.
That came from the problem,

492
00:34:49,000 --> 00:34:55,000
the driving frequency,
driving angular frequency.

493
00:34:52,000 --> 00:34:58,000
That was omega,
and k, I guess,

494
00:34:55,000 --> 00:35:01,000
k was the conductivity,
the thing which told you how

495
00:34:59,000 --> 00:35:05,000
quickly the heat that penetrated
the walls of the little inner

496
00:35:03,000 --> 00:35:09,000
chamber.
So, that's okay,

497
00:35:07,000 --> 00:35:13,000
but what's this phi?
Well, the best way to get phi

498
00:35:11,000 --> 00:35:17,000
is just to draw that picture,
but if you want a formula for

499
00:35:15,000 --> 00:35:21,000
phi, phi will be,
well, I guess from the picture,

500
00:35:19,000 --> 00:35:25,000
it's the arc tangent of omega,
k, divided by k,

501
00:35:23,000 --> 00:35:29,000
over one,
which I don't have to put

502
00:35:28,000 --> 00:35:34,000
in.
So, it's this number,

503
00:35:30,000 --> 00:35:36,000
phi, in reference to this
function.

504
00:35:34,000 --> 00:35:40,000
See, if the phi weren't there,
this would be cosine omega t,

505
00:35:39,000 --> 00:35:45,000
and we all know what that looks

506
00:35:44,000 --> 00:35:50,000
like.
The phi is called the phase lag

507
00:35:48,000 --> 00:35:54,000
or phase delay,
something like that,

508
00:35:51,000 --> 00:35:57,000
the phase lag of the function.
What does it represent?

509
00:35:56,000 --> 00:36:02,000
It represents,
let me draw you a picture.

510
00:36:02,000 --> 00:36:08,000
Let's draw the picture like
this.

511
00:36:05,000 --> 00:36:11,000
Here's cosine omega t.

512
00:36:08,000 --> 00:36:14,000
Now, regular cosine would look
sort of like that.

513
00:36:13,000 --> 00:36:19,000
But, I will indicate that the
angular frequency is not one by

514
00:36:18,000 --> 00:36:24,000
making my cosine squinchy up a
little too much.

515
00:36:23,000 --> 00:36:29,000
Everybody can tell that that's
the cosine on a limp axis,

516
00:36:28,000 --> 00:36:34,000
something for Salvador Dali,
okay.

517
00:36:31,000 --> 00:36:37,000
So, there's cosine of
something.

518
00:36:36,000 --> 00:36:42,000
So, what was it?
Blue?

519
00:36:37,000 --> 00:36:43,000
I don't have blue.
Yes, I have blue.

520
00:36:41,000 --> 00:36:47,000
Okay, so now you will know what
I'm talking about because this

521
00:36:46,000 --> 00:36:52,000
looks just like the screen on
your computer when you put in

522
00:36:52,000 --> 00:36:58,000
the visual for this.
Frequency: your response order

523
00:36:56,000 --> 00:37:02,000
one.
So, this is cosine omega t.

524
00:36:59,000 --> 00:37:05,000
Now, how will cosine omega t

525
00:37:04,000 --> 00:37:10,000
minus phi look?

526
00:37:07,000 --> 00:37:13,000
Well, it'll be moved over.
Let's, for example,

527
00:37:10,000 --> 00:37:16,000
suppose phi were pi over two.
Now, where's pi over two on the

528
00:37:15,000 --> 00:37:21,000
picture?
Well, what I do is cosine omega

529
00:37:18,000 --> 00:37:24,000
t minus this.
I move it over by one,

530
00:37:22,000 --> 00:37:28,000
so that this point becomes that
one, and it looks like,

531
00:37:26,000 --> 00:37:32,000
the site will look like this.
In other words,

532
00:37:30,000 --> 00:37:36,000
I shove it over by,
so this is the point where

533
00:37:33,000 --> 00:37:39,000
omega t equals pi over two.

534
00:37:39,000 --> 00:37:45,000
It's not the value of t.
It's not the value of t.

535
00:37:43,000 --> 00:37:49,000
It's the value of omega t.

536
00:37:46,000 --> 00:37:52,000
And, when I do that,
then the blue curve has been

537
00:37:50,000 --> 00:37:56,000
shoved over one quarter of its
total cycle, and that turns it,

538
00:37:55,000 --> 00:38:01,000
of course, into the sine curve,
which I hope I can draw.

539
00:38:01,000 --> 00:38:07,000
So, this goes up to there,
and then, it's got to get back

540
00:38:05,000 --> 00:38:11,000
through.
Let me stop there while I'm

541
00:38:08,000 --> 00:38:14,000
ahead.
So, this is sine omega t,

542
00:38:11,000 --> 00:38:17,000
the yellow thing,

543
00:38:13,000 --> 00:38:19,000
but that's also,
in another life,

544
00:38:16,000 --> 00:38:22,000
cosine of omega t minus pi over
two.

545
00:38:21,000 --> 00:38:27,000
The main thing is you don't
subtract, the pi over two is not

546
00:38:26,000 --> 00:38:32,000
being subtracted from the t.
It's being subtracted from the

547
00:38:32,000 --> 00:38:38,000
whole expression,
and this whole expression

548
00:38:35,000 --> 00:38:41,000
represents an angle,
which tells you where you are

549
00:38:39,000 --> 00:38:45,000
in the travel,
a long cosine to this.

550
00:38:41,000 --> 00:38:47,000
What this quantity gets to be
two pi, you're back where you

551
00:38:46,000 --> 00:38:52,000
started.
That's not the distance on the

552
00:38:49,000 --> 00:38:55,000
t axis.
It's the angle through which

553
00:38:51,000 --> 00:38:57,000
you go through.
In other words,

554
00:38:54,000 --> 00:39:00,000
does number describes where you
are on the cosine cycle.

555
00:38:58,000 --> 00:39:04,000
It doesn't tell you,
it's not aiming at telling you

556
00:39:01,000 --> 00:39:07,000
exactly where you are on the t
axis.

557
00:39:04,000 --> 00:39:10,000
The response function looks
like one over the square root of

558
00:39:09,000 --> 00:39:15,000
one plus omega over k squared
times cosine omega t minus phi.

559
00:39:19,000 --> 00:39:25,000
And, I asked you on the problem
set, if k goes up,

560
00:39:24,000 --> 00:39:30,000
in other words,
if the conductivity rises,

561
00:39:28,000 --> 00:39:34,000
if heat can get more rapidly
from the outside to the inside,

562
00:39:34,000 --> 00:39:40,000
for example,
how does that affect the

563
00:39:38,000 --> 00:39:44,000
amplitude?
This is the amplitude,

564
00:39:42,000 --> 00:39:48,000
A, and the phase lag.
In other words,

565
00:39:47,000 --> 00:39:53,000
how does this affect the
response?

566
00:39:51,000 --> 00:39:57,000
And now, you can see.
If k goes up,

567
00:39:55,000 --> 00:40:01,000
this fraction is becoming
smaller.

568
00:39:59,000 --> 00:40:05,000
That means the denominator is
becoming smaller,

569
00:40:05,000 --> 00:40:11,000
and therefore,
the amplitude is going up.

570
00:40:12,000 --> 00:40:18,000
What's happening to the phase
lag?

571
00:40:14,000 --> 00:40:20,000
Well, the phase lag looks like
this: phi one omega over k.

572
00:40:20,000 --> 00:40:26,000
If k is going up,

573
00:40:23,000 --> 00:40:29,000
then the size of this side is
going down, and the angle is

574
00:40:28,000 --> 00:40:34,000
going down.
Now, that part is intuitive.

575
00:40:32,000 --> 00:40:38,000
I would have expected everybody
to get that.

576
00:40:36,000 --> 00:40:42,000
It's the heat gets in quickly,
more quickly,

577
00:40:40,000 --> 00:40:46,000
then the amplitude will match
more quickly.

578
00:40:44,000 --> 00:40:50,000
This will rise,
and get fairly close to one,

579
00:40:47,000 --> 00:40:53,000
in fact, and there should be
very little lag in the way the

580
00:40:53,000 --> 00:40:59,000
response follows input.
But how about the other one?

581
00:40:57,000 --> 00:41:03,000
Okay, I give you two minutes.
The other one,

582
00:41:01,000 --> 00:41:07,000
you will figure out yourself.