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.