1
00:00:00,500 --> 00:00:01,920
GILBERT STRANG: OK.
2
00:00:01,920 --> 00:00:07,520
So this is the next step for
a first-order differential
3
00:00:07,520 --> 00:00:09,060
equation.
4
00:00:09,060 --> 00:00:13,560
We take-- instead
of an exponential,
5
00:00:13,560 --> 00:00:16,000
now we have an oscillating.
6
00:00:16,000 --> 00:00:20,370
Exponentials, the previous
lecture, grew or decayed,
7
00:00:20,370 --> 00:00:22,170
now we have an oscillate.
8
00:00:22,170 --> 00:00:28,900
We have AC, alternating
current in this problem,
9
00:00:28,900 --> 00:00:35,180
instead of real exponentials,
we have oscillation, vibration,
10
00:00:35,180 --> 00:00:40,840
all the applications that
involve circular motion,
11
00:00:40,840 --> 00:00:44,370
going around and around instead
of going off exponentially.
12
00:00:44,370 --> 00:00:45,610
OK.
13
00:00:45,610 --> 00:00:48,530
So here's the point.
14
00:00:48,530 --> 00:00:51,700
Again I'm looking for
a particular solution.
15
00:00:51,700 --> 00:00:54,920
The particular solution--
it would be nice
16
00:00:54,920 --> 00:00:57,140
if we could say the
particular solution
17
00:00:57,140 --> 00:01:00,100
was just some multiple
of the cosine.
18
00:01:00,100 --> 00:01:01,550
But that won't work.
19
00:01:01,550 --> 00:01:04,750
So that makes this
problem one step harder
20
00:01:04,750 --> 00:01:06,300
than the exponential.
21
00:01:06,300 --> 00:01:09,400
We need to allow
the signs in there.
22
00:01:09,400 --> 00:01:14,750
Because, if I look for a cosine,
if I tried only this part,
23
00:01:14,750 --> 00:01:16,730
I could match that.
24
00:01:16,730 --> 00:01:20,090
I'd have a times the cosine,
that would be cosine.
25
00:01:20,090 --> 00:01:25,520
But the derivative of a
cosine is a sine function.
26
00:01:25,520 --> 00:01:27,960
So signs are going
to get in there
27
00:01:27,960 --> 00:01:31,580
and we have to allow
them into the solution.
28
00:01:31,580 --> 00:01:32,080
OK.
29
00:01:32,080 --> 00:01:34,680
So that's the right
thing to assume.
30
00:01:34,680 --> 00:01:37,300
Actually there will
be, you'll see,
31
00:01:37,300 --> 00:01:42,280
three different ways to write
the answer to this problem.
32
00:01:42,280 --> 00:01:45,740
And this is the first sort
of most straightforward,
33
00:01:45,740 --> 00:01:48,730
but not the best
in the long run.
34
00:01:48,730 --> 00:01:49,230
OK.
35
00:01:49,230 --> 00:01:52,590
Straightforward one, I'm
going to substitute that
36
00:01:52,590 --> 00:01:57,560
into the equation and find
M and N. That's my job.
37
00:01:57,560 --> 00:01:59,510
Find these numbers.
38
00:01:59,510 --> 00:02:01,560
So put that into the equation.
39
00:02:01,560 --> 00:02:04,270
On the left side I
want the derivative,
40
00:02:04,270 --> 00:02:09,870
so that we'll be omega--
well derivative of the cosine
41
00:02:09,870 --> 00:02:17,218
is minus omega m sine omega t.
42
00:02:17,218 --> 00:02:20,390
The derivative brought
out this factor omega.
43
00:02:20,390 --> 00:02:23,020
The derivative of
cosine was sine.
44
00:02:23,020 --> 00:02:25,240
Now the derivative
of this brings out
45
00:02:25,240 --> 00:02:31,700
a factor of omega--
omega N cosine omega t.
46
00:02:31,700 --> 00:02:36,320
And that should equal
a times y-- there's y,
47
00:02:36,320 --> 00:02:46,350
so I just multiply by
a-- a M cosine omega t,
48
00:02:46,350 --> 00:02:50,950
and a N sine omega t.
49
00:02:50,950 --> 00:02:52,690
That's the ay part.
50
00:02:52,690 --> 00:02:59,110
And now I have the source
term plus cosine omega t.
51
00:02:59,110 --> 00:03:04,300
And that has to be
true for all time.
52
00:03:04,300 --> 00:03:07,970
And now I need the equation.
53
00:03:07,970 --> 00:03:09,570
What do I do with this?
54
00:03:09,570 --> 00:03:11,700
I'm looking for two
things, M and N.
55
00:03:11,700 --> 00:03:13,680
I'm looking for two equations.
56
00:03:13,680 --> 00:03:16,990
So I match the cosine terms.
57
00:03:16,990 --> 00:03:22,880
I match that term, that cosine
term, and the source term.
58
00:03:22,880 --> 00:03:27,760
So they all multiply
cosine omega t.
59
00:03:27,760 --> 00:03:33,090
So I want omega
n-- so I bring this
60
00:03:33,090 --> 00:03:38,200
over on the other side--
minus a m plus omega n
61
00:03:38,200 --> 00:03:43,230
equals-- here I just have
one cosine-- equals 1.
62
00:03:43,230 --> 00:03:46,390
Minus a M plus omega N equals 1.
63
00:03:46,390 --> 00:03:49,010
And now I'll match
the sine terms.
64
00:03:49,010 --> 00:03:56,310
So in the sine terms, I have
a minus omega M, sine omega t.
65
00:03:56,310 --> 00:03:58,580
And I have to bring
this on the other side,
66
00:03:58,580 --> 00:04:03,090
so that'll be a minus
a N sine omega t.
67
00:04:03,090 --> 00:04:07,060
And there's no sine
omega t in the source.
68
00:04:07,060 --> 00:04:09,340
There's my two equations.
69
00:04:09,340 --> 00:04:12,050
Those are my two
equations for M and N.
70
00:04:12,050 --> 00:04:14,310
So I just solve
those two equations
71
00:04:14,310 --> 00:04:18,600
and I've got the particular
solution that I look for.
72
00:04:18,600 --> 00:04:21,300
So, it's two equations,
two unknowns.
73
00:04:21,300 --> 00:04:24,160
It's the basic problem
of linear algebra.
74
00:04:24,160 --> 00:04:26,470
I'm inclined to just write
down the answer, which
75
00:04:26,470 --> 00:04:30,000
I prepared in advance.
76
00:04:30,000 --> 00:04:34,460
And it turns out
to be minus a over
77
00:04:34,460 --> 00:04:37,720
omega squared plus a squared.
78
00:04:37,720 --> 00:04:42,550
And N turns out to have
that same omega squared
79
00:04:42,550 --> 00:04:46,900
plus a squared, and
above it goes omega.
80
00:04:51,000 --> 00:04:58,870
If you check this equation,
for example, omega times the M
81
00:04:58,870 --> 00:05:03,700
will give me a a omega
with a minus with a minus.
82
00:05:03,700 --> 00:05:08,310
And then a times N will
also have an a omega.
83
00:05:08,310 --> 00:05:10,650
And the same omega
squared plus a
84
00:05:10,650 --> 00:05:15,120
squared, they cancel to give 0.
85
00:05:15,120 --> 00:05:17,240
And this equation
is also solved.
86
00:05:17,240 --> 00:05:23,060
So one more important
problem solved.
87
00:05:23,060 --> 00:05:27,440
Well, we found the
particular solution.
88
00:05:27,440 --> 00:05:32,210
I haven't added in-- I haven't
match the initial condition.
89
00:05:32,210 --> 00:05:35,980
Now in many, many cases,
it's this particular solution
90
00:05:35,980 --> 00:05:37,410
that's of interest.
91
00:05:37,410 --> 00:05:42,510
This here-- let me put a
box around our solution--
92
00:05:42,510 --> 00:05:46,600
and we substituted that in
the differential equation.
93
00:05:46,600 --> 00:05:49,670
We discovered M. We
discovered N. We've
94
00:05:49,670 --> 00:05:52,090
got this particular solution.
95
00:05:52,090 --> 00:05:56,310
And that's the oscillation
that keeps going.
96
00:05:56,310 --> 00:06:01,490
That if we're listening
to radio or if we
97
00:06:01,490 --> 00:06:06,230
have alternating current,
this is what we see,
98
00:06:06,230 --> 00:06:09,070
the null solution.
99
00:06:09,070 --> 00:06:13,720
The thing that's coming
with no source term.
100
00:06:13,720 --> 00:06:17,270
Usually a is negative
and that disappears.
101
00:06:17,270 --> 00:06:19,670
That's called the
transient term.
102
00:06:19,670 --> 00:06:26,410
So the null solution would be
have an ae to the at as always.
103
00:06:26,410 --> 00:06:31,570
But I'm not so interested in
that because it disappears.
104
00:06:31,570 --> 00:06:33,810
You don't hear it
after a minute.
105
00:06:33,810 --> 00:06:38,010
And this is the solution that
you're-- this is what your ear
106
00:06:38,010 --> 00:06:39,380
is hearing.
107
00:06:39,380 --> 00:06:41,170
OK.
108
00:06:41,170 --> 00:06:43,810
So we've got one
form of the answer.
109
00:06:43,810 --> 00:06:48,140
Now, that's a pretty nice
form, but it's not perfect.
110
00:06:48,140 --> 00:06:53,660
I can't see exactly--
this can be simplified
111
00:06:53,660 --> 00:06:55,640
in a really nice way.
112
00:06:55,640 --> 00:06:58,340
So when we work with
sines and cosines,
113
00:06:58,340 --> 00:07:00,900
it's this next step
that's important.
114
00:07:03,610 --> 00:07:07,200
I believe that that
same yp of t can
115
00:07:07,200 --> 00:07:13,330
be written in a different
way as what also--
116
00:07:13,330 --> 00:07:20,810
another form, a different--
well I should say,
117
00:07:20,810 --> 00:07:26,880
another form for
the same y of t.
118
00:07:26,880 --> 00:07:35,830
Another form will
be the same y of t.
119
00:07:35,830 --> 00:07:40,610
You see what I don't like is
having a cosine and a sine
120
00:07:40,610 --> 00:07:45,500
because those are out of
phase, and they're combining it
121
00:07:45,500 --> 00:07:47,390
into something, and
I want to find out
122
00:07:47,390 --> 00:07:48,810
what their combining into.
123
00:07:48,810 --> 00:07:50,210
And it's really nice.
124
00:07:50,210 --> 00:08:00,420
They're combining into a single
cosine, but not just omega t,
125
00:08:00,420 --> 00:08:02,770
there's a lag, a phase shift.
126
00:08:02,770 --> 00:08:06,120
The angle involved is
often called the phase.
127
00:08:06,120 --> 00:08:11,230
So the two, sine
and cosine, combine
128
00:08:11,230 --> 00:08:14,680
to give a phase shift
with some amplitude,
129
00:08:14,680 --> 00:08:18,450
maybe I'll call it G, the gain.
130
00:08:18,450 --> 00:08:20,930
Or often it would be
called capital R just
131
00:08:20,930 --> 00:08:25,860
for-- because it's-- sort of
what you're seeing here is
132
00:08:25,860 --> 00:08:27,260
polar coordinates.
133
00:08:27,260 --> 00:08:31,596
So I want to match
this, which has the G
134
00:08:31,596 --> 00:08:36,799
and the alpha-- polar
coordinates is really
135
00:08:36,799 --> 00:08:38,950
the right way to think of this.
136
00:08:38,950 --> 00:08:44,270
G and an alpha, a
magnitude and an angle.
137
00:08:44,270 --> 00:08:48,950
I want to match that with
the form I already had.
138
00:08:48,950 --> 00:08:52,950
So I'll use a little
trigonometry here
139
00:08:52,950 --> 00:08:57,010
to remember that
this is equal to-- I
140
00:08:57,010 --> 00:09:00,820
have a G. Do you
remember a formula
141
00:09:00,820 --> 00:09:03,220
for the cosine of a minus b?
142
00:09:03,220 --> 00:09:09,640
The cosine of a difference
is cosine of omega t,
143
00:09:09,640 --> 00:09:13,610
cosine of alpha,
plus-- it's a plus
144
00:09:13,610 --> 00:09:15,720
here because it's
a minus there--
145
00:09:15,720 --> 00:09:18,920
sine omega t sine alpha.
146
00:09:21,900 --> 00:09:26,680
So I have just written this
out in the two-term form,
147
00:09:26,680 --> 00:09:29,820
and I did that so that I could
match the two-term form I
148
00:09:29,820 --> 00:09:30,500
already had.
149
00:09:30,500 --> 00:09:33,120
So can I just do that matching?
150
00:09:33,120 --> 00:09:39,260
The cosine omega t, the
M must be G cosine alpha.
151
00:09:42,640 --> 00:09:48,565
And they N must be G sine alpha.
152
00:09:54,180 --> 00:09:58,220
So I now have two equations.
153
00:09:58,220 --> 00:10:02,550
The M and the N, I still
remember what those are.
154
00:10:02,550 --> 00:10:03,940
I figured those out.
155
00:10:03,940 --> 00:10:09,060
But now I want to convert the
M N form to the G alpha form,
156
00:10:09,060 --> 00:10:10,750
and this is what I have to do.
157
00:10:10,750 --> 00:10:14,200
And it's the usual thing
with polar coordinates.
158
00:10:14,200 --> 00:10:18,740
How can I get-- how do I
discover what G is there,
159
00:10:18,740 --> 00:10:22,260
and what alpha is?
160
00:10:22,260 --> 00:10:27,820
The trick is-- the one
fundamental identity
161
00:10:27,820 --> 00:10:31,520
when you see cosines and sines--
is to remember that cosine
162
00:10:31,520 --> 00:10:33,140
squared plus sine squared is 1.
163
00:10:33,140 --> 00:10:35,240
I'm going to use
that, have to use it.
164
00:10:35,240 --> 00:10:37,610
So I'll square both sides.
165
00:10:37,610 --> 00:10:41,260
I'll have M squared,
and I'll add.
166
00:10:41,260 --> 00:10:47,050
So I'll have M squared
plus N squared is
167
00:10:47,050 --> 00:10:52,640
G squared cosine squared alpha.
168
00:10:52,640 --> 00:10:57,850
G squared times
cosine squared alpha--
169
00:10:57,850 --> 00:11:03,010
when I square that one--
and sine squared alpha
170
00:11:03,010 --> 00:11:04,240
when I square that one.
171
00:11:04,240 --> 00:11:06,770
And again, the
point is that's one.
172
00:11:06,770 --> 00:11:09,840
So that's just G squared.
173
00:11:09,840 --> 00:11:11,430
So what do I learned?
174
00:11:11,430 --> 00:11:13,820
G is the square root of this.
175
00:11:13,820 --> 00:11:19,380
G is the square root of
M squared plus N squared.
176
00:11:19,380 --> 00:11:24,360
And I'm always freedom plug-in
the M and the N that I found.
177
00:11:24,360 --> 00:11:28,310
OK that's-- ah,
what about alpha?
178
00:11:28,310 --> 00:11:29,890
That's the angle.
179
00:11:29,890 --> 00:11:33,890
So I have to-- again,
I'm thinking trig here.
180
00:11:33,890 --> 00:11:37,220
How am I going to
get alpha here?
181
00:11:37,220 --> 00:11:39,250
I want to get G
out of this formula
182
00:11:39,250 --> 00:11:42,150
now, and just
focus on the alpha.
183
00:11:42,150 --> 00:11:45,404
Previously I got alpha
out of it and got the G.
184
00:11:45,404 --> 00:11:49,540
Now the way to do
is take the ratio.
185
00:11:49,540 --> 00:11:51,930
If I take the ratio
of that to that,
186
00:11:51,930 --> 00:11:55,940
divide one by the other,
the G's will cancel.
187
00:11:55,940 --> 00:11:58,550
So I'll take the
ratio of that to that
188
00:11:58,550 --> 00:12:14,220
to get G sine alpha divided by
g cosine alpha is then N over M.
189
00:12:14,220 --> 00:12:17,240
And the G's cancel as I wanted.
190
00:12:17,240 --> 00:12:19,540
So now I have an
equation for alpha.
191
00:12:19,540 --> 00:12:23,910
Or more exactly, I have an
equation for tangent of alpha.
192
00:12:23,910 --> 00:12:33,260
Sine over cosine is tangent
of alpha is N over M.
193
00:12:33,260 --> 00:12:36,630
So this is called
the-- you could call it
194
00:12:36,630 --> 00:12:38,910
the sinusoidal identity.
195
00:12:38,910 --> 00:12:41,210
What is that word sinusoid?
196
00:12:41,210 --> 00:12:47,020
Sinusoid is a word for any
mixture of sines and cosines,
197
00:12:47,020 --> 00:12:52,990
any mixture of sines and
cosines of the same omega t.
198
00:12:52,990 --> 00:12:57,560
So the sinusoidal
identity says that I
199
00:12:57,560 --> 00:13:03,170
can rewrite that solution
into this solution.
200
00:13:08,040 --> 00:13:11,480
And I really see the key
number in the whole thing
201
00:13:11,480 --> 00:13:14,940
is the gain, the magnitude.
202
00:13:14,940 --> 00:13:18,360
It's how loud the
station comes through
203
00:13:18,360 --> 00:13:20,300
if we're tuning a radio.
204
00:13:20,300 --> 00:13:23,720
So and again, this
is the response
205
00:13:23,720 --> 00:13:29,410
that's keeps going because
the cosine oscillates forever.
206
00:13:29,410 --> 00:13:32,390
There will be also
something coming
207
00:13:32,390 --> 00:13:36,660
from the initial condition
that we expect to die out.
208
00:13:36,660 --> 00:13:38,750
So I mentioned at the
very start that there
209
00:13:38,750 --> 00:13:46,110
were three forms of the
answer to this cosine input,
210
00:13:46,110 --> 00:13:48,240
and I've given you two.
211
00:13:48,240 --> 00:13:51,280
I've given you the M and N form.
212
00:13:51,280 --> 00:13:54,270
You could say rectangular
coordinates-- cosines
213
00:13:54,270 --> 00:13:55,380
and sines.
214
00:13:55,380 --> 00:13:59,500
I've given you the
polar form, which
215
00:13:59,500 --> 00:14:04,350
is a gain, a
magnitude, and a phase.
216
00:14:04,350 --> 00:14:09,280
And the third one
involves complex numbers.
217
00:14:09,280 --> 00:14:14,570
I have to make that a separate
lecture, maybe even two.
218
00:14:14,570 --> 00:14:18,910
So complex numbers,
where do they come in?
219
00:14:18,910 --> 00:14:22,030
It's a totally real equation.
220
00:14:22,030 --> 00:14:25,030
If I think about all
this that I've done,
221
00:14:25,030 --> 00:14:34,710
it was all totally
real, but there's
222
00:14:34,710 --> 00:14:38,510
a link-- the key fact about
complex numbers, Euler's
223
00:14:38,510 --> 00:14:43,620
great formula will give me a
connection between cosine omega
224
00:14:43,620 --> 00:14:50,580
t and sine omega t with
e to the I omega t.
225
00:14:50,580 --> 00:14:54,970
So at the price of introducing
that complex number,
226
00:14:54,970 --> 00:15:04,250
imaginary number i, or j
for electrical engineers,
227
00:15:04,250 --> 00:15:06,570
we're back to exponentials.
228
00:15:06,570 --> 00:15:08,220
We're back to exponentials.
229
00:15:08,220 --> 00:15:10,170
So that'll come in
the next lecture.
230
00:15:10,170 --> 00:15:17,720
This is one more example
of a nice source function.
231
00:15:17,720 --> 00:15:21,545
Maybe I could just say, what
are the nicest source functions?
232
00:15:24,270 --> 00:15:28,040
So this is the source
function here and it was nice.
233
00:15:28,040 --> 00:15:30,080
Exponential was even nicer.
234
00:15:30,080 --> 00:15:32,170
Constant was best of all.
235
00:15:32,170 --> 00:15:35,970
And I want to-- another
one I want to introduce
236
00:15:35,970 --> 00:15:38,410
is a delta function.
237
00:15:38,410 --> 00:15:41,310
So that's-- a delta
function is an impulse,
238
00:15:41,310 --> 00:15:43,390
something that
happens in an instant.
239
00:15:43,390 --> 00:15:46,870
And that's an interesting, very
interesting and very important
240
00:15:46,870 --> 00:15:47,740
possibility.
241
00:15:47,740 --> 00:15:48,240
OK.
242
00:15:48,240 --> 00:15:50,070
Thank you.