1
00:00:02,200 --> 00:00:06,700
GILBERT STRONG: This is the
second video on second order

2
00:00:06,700 --> 00:00:09,870
differential equations,
constant coefficients,

3
00:00:09,870 --> 00:00:13,410
but now we have a
right hand side.

4
00:00:13,410 --> 00:00:17,740
And the first one was free
harmonic motion with a zero,

5
00:00:17,740 --> 00:00:24,790
but now I'm making this motion,
I'm pushing this motion,

6
00:00:24,790 --> 00:00:30,020
but at a frequency omega.

7
00:00:30,020 --> 00:00:32,580
This is my forcing term.

8
00:00:32,580 --> 00:00:36,370
So I think I'm having a
forcing frequency, omega,

9
00:00:36,370 --> 00:00:41,150
and remember that for this
one, for the no solution,

10
00:00:41,150 --> 00:00:45,030
there was a natural
frequency omega n.

11
00:00:45,030 --> 00:00:48,400
It's very important
are those close,

12
00:00:48,400 --> 00:00:51,260
are those well separated?

13
00:00:51,260 --> 00:00:54,280
That governs whether the
bridge that you're walking over

14
00:00:54,280 --> 00:00:57,460
oscillates too much
and eventually falls.

15
00:00:57,460 --> 00:01:00,110
Or in the extreme
case, are they equal?

16
00:01:00,110 --> 00:01:04,260
If omega n is equal to omega
that's called resonance.

17
00:01:04,260 --> 00:01:06,083
Let me put that word in.

18
00:01:06,083 --> 00:01:06,583
Resonance.

19
00:01:09,430 --> 00:01:12,950
When omega equals omega n.

20
00:01:12,950 --> 00:01:16,680
And we're not going
to deal with today,

21
00:01:16,680 --> 00:01:21,960
but you should know that
always the formula has an omega

22
00:01:21,960 --> 00:01:25,440
minus omega n dividing by that.

23
00:01:25,440 --> 00:01:28,620
So if that is 0, if
omega equals omega n

24
00:01:28,620 --> 00:01:31,240
our formula has to change.

25
00:01:31,240 --> 00:01:34,215
Today, this won't happen.

26
00:01:34,215 --> 00:01:34,714
No.

27
00:01:38,050 --> 00:01:40,190
So what's the formula?

28
00:01:40,190 --> 00:01:42,360
What is yp?

29
00:01:42,360 --> 00:01:44,235
I'm looking for a
particular solution.

30
00:01:47,700 --> 00:01:53,220
That's a nice function and
also important in practice.

31
00:01:53,220 --> 00:01:58,190
So I would like to hope that
the particular solution could

32
00:01:58,190 --> 00:02:04,659
be some multiple of
that cosine omega t.

33
00:02:04,659 --> 00:02:08,410
And in this problem
that's possible.

34
00:02:08,410 --> 00:02:13,550
Because if I have a cosine, I've
got a cosine on the right hand

35
00:02:13,550 --> 00:02:18,500
side, and if that cosine comes
here, it's on the left side,

36
00:02:18,500 --> 00:02:22,620
and the second derivative of
the cosine is, again, a cosine,

37
00:02:22,620 --> 00:02:26,070
I'm going to have a match
of cosine omega t terms.

38
00:02:26,070 --> 00:02:31,350
And then I'll just choose
the right number capital Y.

39
00:02:31,350 --> 00:02:33,710
I won't be able to
do that when there's

40
00:02:33,710 --> 00:02:36,510
a first derivative in there,
because the first derivative

41
00:02:36,510 --> 00:02:39,990
of cosine will bring in signs.

42
00:02:39,990 --> 00:02:42,800
I'll have a mixture
of cosines and sines

43
00:02:42,800 --> 00:02:45,910
and then I better
allow for that mixture.

44
00:02:45,910 --> 00:02:47,910
But here I don't have to.

45
00:02:47,910 --> 00:02:50,850
There's the forcing function.

46
00:02:50,850 --> 00:02:54,143
Response, this is
the forced response.

47
00:03:01,700 --> 00:03:04,660
I'd like to get used
to that word, response,

48
00:03:04,660 --> 00:03:07,370
for the solution.

49
00:03:07,370 --> 00:03:10,840
Here's the input, the
response is the output.

50
00:03:10,840 --> 00:03:14,060
So let me just plug
that into the equation

51
00:03:14,060 --> 00:03:16,380
and find capital Y.

52
00:03:16,380 --> 00:03:24,175
So here I have m, second
derivative is going to be a Y,

53
00:03:24,175 --> 00:03:27,340
and second derivative
will bring out a minus

54
00:03:27,340 --> 00:03:30,835
omega squared times the cosine.

55
00:03:34,160 --> 00:03:42,930
And here I have kY is Y times
the cosine equal the cosine.

56
00:03:48,830 --> 00:03:52,070
I could have a constant
there, but the whole thing

57
00:03:52,070 --> 00:03:59,650
would be no more interesting,
no more difficult than with a 1.

58
00:03:59,650 --> 00:04:01,790
So what do I do?

59
00:04:01,790 --> 00:04:04,750
The nice thing is here
I have all cosines,

60
00:04:04,750 --> 00:04:09,870
so I'm just going to have
minus omega squared m and a k.

61
00:04:09,870 --> 00:04:12,800
So it's k minus m omega squared.

62
00:04:12,800 --> 00:04:14,820
Can I write it that way?

63
00:04:14,820 --> 00:04:20,160
Times Y. I'm going to
cancel the cosines.

64
00:04:20,160 --> 00:04:21,680
That's just a 1.

65
00:04:21,680 --> 00:04:24,410
On the side is a 1.

66
00:04:24,410 --> 00:04:28,130
I've canceled the
cosine, so I've kept kY.

67
00:04:28,130 --> 00:04:34,810
I've kept the 1, and I've
kept a minus omega squared mY.

68
00:04:34,810 --> 00:04:37,270
So that tells me Y right away.

69
00:04:37,270 --> 00:04:41,380
It's just like plugging in
an exponential and canceling

70
00:04:41,380 --> 00:04:43,020
exponentials all the way along.

71
00:04:43,020 --> 00:04:47,740
Here, I'm canceling cosines
all the way because every term

72
00:04:47,740 --> 00:04:49,080
was a cosine.

73
00:04:49,080 --> 00:04:51,285
So I know Y. So I
know the answer.

74
00:04:54,560 --> 00:05:01,320
So the final answer
is Y(t) is Yn.

75
00:05:01,320 --> 00:05:05,910
Well, let me put Y
particular first plus Yn.

76
00:05:05,910 --> 00:05:08,990
So I've just found Y particular.

77
00:05:08,990 --> 00:05:14,790
Y particular is this
capital Y cosine omega t.

78
00:05:14,790 --> 00:05:22,770
So it's cosine omega t times
Y and Y is 1 over this.

79
00:05:22,770 --> 00:05:29,220
Here's goes Y. Down below I
have k minus m omega squared.

80
00:05:32,910 --> 00:05:33,850
Right?

81
00:05:33,850 --> 00:05:36,980
That's what we just found,
that particular solution.

82
00:05:36,980 --> 00:05:40,340
The capital Y, the
multiplying constant,

83
00:05:40,340 --> 00:05:43,800
was 1 over that constant.

84
00:05:43,800 --> 00:05:52,470
And now comes the C1
cosine of omega nt

85
00:05:52,470 --> 00:05:56,395
and the C2 sine of omega nt.

86
00:06:00,180 --> 00:06:04,930
Remember, omega n is
different from omega.

87
00:06:04,930 --> 00:06:08,930
Actually, this is
pretty nice here.

88
00:06:08,930 --> 00:06:12,380
I could write that
another way so you

89
00:06:12,380 --> 00:06:14,440
would see the important here.

90
00:06:14,440 --> 00:06:17,170
So remember, what
is omega n squared?

91
00:06:17,170 --> 00:06:24,610
Can I just remember that
omega n squared is k over m.

92
00:06:24,610 --> 00:06:25,930
Right?

93
00:06:25,930 --> 00:06:26,430
Yup.

94
00:06:30,580 --> 00:06:35,570
k is the same as-- I'm going
to put that m up here--

95
00:06:35,570 --> 00:06:39,995
k is the same as
m omega n squared.

96
00:06:43,900 --> 00:06:46,800
k is the same as
m omega n squared

97
00:06:46,800 --> 00:06:49,536
and here I'm subtracting
m omega squared.

98
00:06:52,740 --> 00:06:57,980
You'll see the whole point of
resonance or near resonance

99
00:06:57,980 --> 00:07:05,880
when the bridge is getting
forced buy a frequency close

100
00:07:05,880 --> 00:07:08,170
to its resonant frequency.

101
00:07:08,170 --> 00:07:11,360
This difference, omega
squared, the difference

102
00:07:11,360 --> 00:07:14,825
between the two frequencies
squared is in the denominator

103
00:07:14,825 --> 00:07:19,920
and will be small and
then the effect is large.

104
00:07:19,920 --> 00:07:25,240
And if we get those too close,
the effect is too large.

105
00:07:25,240 --> 00:07:30,350
So we'll see this cosine omega
t over this is, I would call,

106
00:07:30,350 --> 00:07:44,790
the frequency response
is this factor.

107
00:07:44,790 --> 00:07:50,680
1 over m omega n squared
minus omega squared.

108
00:07:54,310 --> 00:07:59,150
That's the key
multiplier for when

109
00:07:59,150 --> 00:08:04,280
the forcing term is a pure
frequency, that frequency

110
00:08:04,280 --> 00:08:05,590
gets exploded.

111
00:08:05,590 --> 00:08:09,620
And now, of course, what are
capital C1 and capital C2?

112
00:08:09,620 --> 00:08:12,870
We find those from
the initial condition.

113
00:08:12,870 --> 00:08:16,440
At t equals 0, we
put in t equals 0,

114
00:08:16,440 --> 00:08:20,600
and that tells us
what C1 has to be.

115
00:08:20,600 --> 00:08:28,070
And we put in t equals 0 again
to match the velocity Y prime

116
00:08:28,070 --> 00:08:31,010
at 0, and that tells us C2.

117
00:08:31,010 --> 00:08:33,370
Are you OK with that?

118
00:08:33,370 --> 00:08:38,890
Just look at the beauty
of that solution.

119
00:08:38,890 --> 00:08:42,110
This is null part.

120
00:08:42,110 --> 00:08:44,840
This is the forced part,
the particular part,

121
00:08:44,840 --> 00:08:49,820
the cosine divided
by that constant.

122
00:08:49,820 --> 00:08:52,510
There's one more
equation, one more

123
00:08:52,510 --> 00:08:59,580
forcing term I'd like often
and always and now to discuss.

124
00:08:59,580 --> 00:09:03,880
And that is a delta
function, an impulse.

125
00:09:03,880 --> 00:09:08,920
So I'm going to add
one more example.

126
00:09:08,920 --> 00:09:18,060
my double prime plus ky
equal the delta function.

127
00:09:18,060 --> 00:09:18,920
Delta function.

128
00:09:23,410 --> 00:09:24,685
It's called an impulse.

129
00:09:30,620 --> 00:09:32,680
So I'd like to solve
that equation also.

130
00:09:32,680 --> 00:09:36,790
When the forcing term just
happens at one second,

131
00:09:36,790 --> 00:09:38,050
at the initial second.

132
00:09:38,050 --> 00:09:40,510
At t equals 0, the
delta function,

133
00:09:40,510 --> 00:09:43,490
I'm hitting the spring.

134
00:09:43,490 --> 00:09:48,620
So the spring is sitting or
the pendulum is sitting there.

135
00:09:48,620 --> 00:09:51,170
Actually, let's set it at rest.

136
00:09:51,170 --> 00:09:52,100
Here's my pendulum.

137
00:09:52,100 --> 00:09:53,290
I'll try to draw a pendulum.

138
00:09:57,510 --> 00:09:58,070
I don't know.

139
00:09:58,070 --> 00:09:59,460
That's not much of a pendulum.

140
00:09:59,460 --> 00:10:03,210
But it's good enough.

141
00:10:03,210 --> 00:10:07,910
This equation says what
happens if I hit it

142
00:10:07,910 --> 00:10:12,480
with a point source?

143
00:10:12,480 --> 00:10:16,990
At t equals 0, I hit it but
I give it a finite velocity.

144
00:10:20,500 --> 00:10:24,260
It doesn't move in
that instant second.

145
00:10:24,260 --> 00:10:26,190
This is where delta
functions come in

146
00:10:26,190 --> 00:10:29,730
so let me give you the
result of what happens

147
00:10:29,730 --> 00:10:31,870
and then we'll see them again.

148
00:10:31,870 --> 00:10:33,480
So what am I doing?

149
00:10:33,480 --> 00:10:37,050
I want to solve this equation
when the forcing function is

150
00:10:37,050 --> 00:10:38,820
a delta function.

151
00:10:38,820 --> 00:10:42,890
So I'm going to call y
the impulse response.

152
00:10:42,890 --> 00:10:47,620
It's the solution that comes
when the forcing function is

153
00:10:47,620 --> 00:10:48,370
an impulse.

154
00:10:48,370 --> 00:10:50,140
So y is the impulse response.

155
00:10:50,140 --> 00:10:52,370
In fact, it's so
important, I'm going

156
00:10:52,370 --> 00:10:55,560
to give it its own letter. g.

157
00:10:55,560 --> 00:10:59,080
Now, can I turn that y into a g?

158
00:10:59,080 --> 00:11:10,415
So that g is g of t is
the impulse response.

159
00:11:18,670 --> 00:11:20,810
If I can solve that equation.

160
00:11:20,810 --> 00:11:23,430
You might say, not so easy.

161
00:11:23,430 --> 00:11:27,140
With a delta function, it's
not even a genuine function.

162
00:11:27,140 --> 00:11:28,140
It's a little bit crazy.

163
00:11:28,140 --> 00:11:31,300
It all happens in one second.

164
00:11:31,300 --> 00:11:33,840
I'm sorry, in one instant.

165
00:11:33,840 --> 00:11:38,050
Not over one second,
but one moment.

166
00:11:38,050 --> 00:11:40,460
But I can solve it.

167
00:11:40,460 --> 00:11:43,770
I can solve it for this reason.

168
00:11:43,770 --> 00:11:46,600
I can think of it
as an impulse here

169
00:11:46,600 --> 00:11:51,860
or I have an option,
another way which clearly I

170
00:11:51,860 --> 00:11:59,600
can think of it as solving it
with no force mg double prime.

171
00:11:59,600 --> 00:12:06,010
Same problem, same
solution is 0.

172
00:12:06,010 --> 00:12:13,140
But I start from rest.

173
00:12:13,140 --> 00:12:14,120
Nothing's happening.

174
00:12:14,120 --> 00:12:17,020
y of 0 is 0.

175
00:12:17,020 --> 00:12:23,580
And it starts from an initial
velocity, y prime of 0.

176
00:12:23,580 --> 00:12:27,530
The impulse starts it
out like a golf ball.

177
00:12:27,530 --> 00:12:30,750
Just go.

178
00:12:30,750 --> 00:12:33,470
And there's a 1 over m there.

179
00:12:33,470 --> 00:12:37,580
I'll discuss that another time.

180
00:12:37,580 --> 00:12:43,320
What I want to see now is that
I have either this somewhat

181
00:12:43,320 --> 00:12:48,730
mysterious equation or this
totally normal equation,

182
00:12:48,730 --> 00:12:53,060
even a no equation starting
from y of 0 equals 0.

183
00:12:53,060 --> 00:12:55,890
But with an initial
velocity that the impulse

184
00:12:55,890 --> 00:12:57,520
gave to the system.

185
00:12:57,520 --> 00:13:00,860
And I should be calling this g.

186
00:13:00,860 --> 00:13:02,402
This is the g.

187
00:13:02,402 --> 00:13:04,760
We'll see impulse
responses again,

188
00:13:04,760 --> 00:13:07,950
but let's see it this time
by solving this equation.

189
00:13:07,950 --> 00:13:11,430
So I plan to solve that
equation and actually we

190
00:13:11,430 --> 00:13:14,010
solved it last time.

191
00:13:14,010 --> 00:13:16,860
You remember the
solution to this one?

192
00:13:16,860 --> 00:13:21,680
When it starts from
0, there's no cosine.

193
00:13:21,680 --> 00:13:26,060
But when the initial velocity
is 1 over m, there is a sign.

194
00:13:26,060 --> 00:13:33,350
So I'm going to just
write down the g of t,

195
00:13:33,350 --> 00:13:38,170
which is just sine of omega nt.

196
00:13:38,170 --> 00:13:40,820
And why is it the
natural frequency?

197
00:13:40,820 --> 00:13:46,280
Because I'm solving the no.

198
00:13:46,280 --> 00:13:48,030
I'm looking for a no solution.

199
00:13:48,030 --> 00:13:53,310
But the previous video on
no solutions gets me this.

200
00:13:53,310 --> 00:14:02,880
Only I have to divide by, get 1
over m as the initial velocity.

201
00:14:02,880 --> 00:14:07,340
You'll see that that will
solve the no equation.

202
00:14:07,340 --> 00:14:17,160
This is what happens to the
pendulum or the golf ball.

203
00:14:17,160 --> 00:14:18,900
Well, pendulum much better.

204
00:14:18,900 --> 00:14:21,370
Actually, golf ball
is poor example.

205
00:14:21,370 --> 00:14:22,430
Sorry about that.

206
00:14:22,430 --> 00:14:24,990
Golf balls don't
swing back and forth.

207
00:14:24,990 --> 00:14:26,680
They tend to go.

208
00:14:26,680 --> 00:14:30,690
I'm looking at pendulums,
springs going up and down.

209
00:14:30,690 --> 00:14:36,750
So the spring starts out, has
an initial velocity of 1 over m

210
00:14:36,750 --> 00:14:39,100
and then after that
nothing happens.

211
00:14:39,100 --> 00:14:42,286
So that is the impulse response.

212
00:14:48,590 --> 00:14:50,390
The response to an impulse.

213
00:14:50,390 --> 00:14:51,700
And why do I like that?

214
00:14:51,700 --> 00:14:53,850
First of all, its beautiful.

215
00:14:53,850 --> 00:14:55,290
Simple answer.

216
00:14:55,290 --> 00:14:59,010
Secondly, every
forcing function,

217
00:14:59,010 --> 00:15:03,170
and the output
comes from this one.

218
00:15:03,170 --> 00:15:05,020
We'll see that point.

219
00:15:05,020 --> 00:15:08,950
So we've introduced forcing
functions, cos omega

220
00:15:08,950 --> 00:15:15,290
t, where the particular solution
was a multiple of cos omega t.

221
00:15:15,290 --> 00:15:19,260
And now, we've introduced
a forcing function delta,

222
00:15:19,260 --> 00:15:25,970
the delta function where the
response is a sine function.

223
00:15:25,970 --> 00:15:27,770
Thank you.