1
00:00:18,000 --> 00:00:24,000
Okay, those are the formulas.
You will get all of those on

2
00:00:23,000 --> 00:00:29,000
the test, plus a couple more
that I will give you today.

3
00:00:29,000 --> 00:00:35,000
Those will be the basic
formulas of the Laplace

4
00:00:33,000 --> 00:00:39,000
transform.
If I think you need anything

5
00:00:37,000 --> 00:00:43,000
else, I'll give you other stuff,
too.

6
00:00:42,000 --> 00:00:48,000
So, I'm going to leave those on
the board all period.

7
00:00:46,000 --> 00:00:52,000
The basic test for today is to
see how Laplace transforms are

8
00:00:51,000 --> 00:00:57,000
used to solve linear
differential equations with

9
00:00:54,000 --> 00:01:00,000
constant coefficients.
Now, to do that,

10
00:00:57,000 --> 00:01:03,000
we're going to have to take the
Laplace transform of a

11
00:01:02,000 --> 00:01:08,000
derivative.
And, in order to make sense of

12
00:01:06,000 --> 00:01:12,000
that procedure,
we're going to have to ask,

13
00:01:09,000 --> 00:01:15,000
I apologize in advance,
but a slightly theoretical

14
00:01:12,000 --> 00:01:18,000
question, namely,
we have to have some guarantee

15
00:01:15,000 --> 00:01:21,000
in advance that the Laplace
transform is going to exist.

16
00:01:19,000 --> 00:01:25,000
Now, how could the Laplace
transform fail to exist?

17
00:01:23,000 --> 00:01:29,000
Can't I always calculate this?
And the answer is,

18
00:01:26,000 --> 00:01:32,000
no, you can't always calculate
it because this is an improper

19
00:01:30,000 --> 00:01:36,000
integral.
I'm integrating all the way up

20
00:01:33,000 --> 00:01:39,000
to infinity, and you know that
improper integrals don't always

21
00:01:38,000 --> 00:01:44,000
converge.
You know, if the integrand for

22
00:01:42,000 --> 00:01:48,000
example just didn't have the
exponential factor there,

23
00:01:46,000 --> 00:01:52,000
were simply t dt,
that it might look like it made

24
00:01:50,000 --> 00:01:56,000
sense, but that integral doesn't
converge.

25
00:01:53,000 --> 00:01:59,000
And, anyway,
it has no value.

26
00:01:56,000 --> 00:02:02,000
So, I need conditions in
advance, which guarantee that

27
00:02:00,000 --> 00:02:06,000
the Laplace transforms will
exist.

28
00:02:04,000 --> 00:02:10,000
Only under those circumstances
will the formulas make any

29
00:02:07,000 --> 00:02:13,000
sense.
Now, there is a standard

30
00:02:09,000 --> 00:02:15,000
condition that's in your book.
But, I didn't get a chance to

31
00:02:13,000 --> 00:02:19,000
talk about it last time.
So, I thought I'd better spent

32
00:02:16,000 --> 00:02:22,000
the first few minutes today
talking about the condition

33
00:02:20,000 --> 00:02:26,000
because it's what we're going to
need in order to be able to

34
00:02:23,000 --> 00:02:29,000
solve differential equations.
The condition that makes the

35
00:02:27,000 --> 00:02:33,000
Laplace transform definitely
exist for a function is that f

36
00:02:31,000 --> 00:02:37,000
of t shouldn't grow too
rapidly.

37
00:02:35,000 --> 00:02:41,000
It can grow rapidly.
It can grow because the e to

38
00:02:38,000 --> 00:02:44,000
the minus s t is
pulling it down,

39
00:02:41,000 --> 00:02:47,000
trying hard to pull it down to
zero to make the integral

40
00:02:45,000 --> 00:02:51,000
converge.
All we have to do is to

41
00:02:47,000 --> 00:02:53,000
guarantee that it doesn't grow
so rapidly that the e to the

42
00:02:52,000 --> 00:02:58,000
minus s t is powerless to pull
it down.

43
00:02:54,000 --> 00:03:00,000
Now, the condition is it's
what's called a growth

44
00:02:58,000 --> 00:03:04,000
condition.
These are very important in

45
00:03:02,000 --> 00:03:08,000
applications,
and unfortunately,

46
00:03:04,000 --> 00:03:10,000
it's always taught in 18.01,
but it's not always taught in

47
00:03:09,000 --> 00:03:15,000
high school calculus.
And, it's a question of how

48
00:03:13,000 --> 00:03:19,000
fast the function is allowed to
grow.

49
00:03:16,000 --> 00:03:22,000
And, the condition is
universally said this way,

50
00:03:20,000 --> 00:03:26,000
should be of exponential type.
So, what I'm defining is the

51
00:03:24,000 --> 00:03:30,000
phrase "exponential type." I'll
put it in quotation marks for

52
00:03:29,000 --> 00:03:35,000
that reason.
What does this mean?

53
00:03:33,000 --> 00:03:39,000
It's a condition,
a growth condition on a

54
00:03:36,000 --> 00:03:42,000
function, says how fast it can
get big.

55
00:03:40,000 --> 00:03:46,000
It says that f of t in
size, since f of t might get

56
00:03:44,000 --> 00:03:50,000
negatively very large,
and that would hurt,

57
00:03:48,000 --> 00:03:54,000
make the integral hard to
converge, not likely to

58
00:03:52,000 --> 00:03:58,000
converge, use the absolute
value.

59
00:03:54,000 --> 00:04:00,000
In other words,
I don't care if f of t is going

60
00:03:58,000 --> 00:04:04,000
up or going down very low.
Whichever way it goes,

61
00:04:02,000 --> 00:04:08,000
its size should not be bigger
than a rapidly growing

62
00:04:06,000 --> 00:04:12,000
exponential.
And, here's a rapidly growing

63
00:04:10,000 --> 00:04:16,000
exponential.
c is some positive constant,

64
00:04:14,000 --> 00:04:20,000
for some positive constant c
and some positive constant k.

65
00:04:18,000 --> 00:04:24,000
And, this should be true for
all values of t.

66
00:04:20,000 --> 00:04:26,000
All t greater than or equal to
zero.

67
00:04:22,000 --> 00:04:28,000
I don't have to worry about
negative values of t because the

68
00:04:26,000 --> 00:04:32,000
integral doesn't care about
them.

69
00:04:28,000 --> 00:04:34,000
I'm only doing the integration
as t runs from zero to infinity.

70
00:04:33,000 --> 00:04:39,000
In other words,
f of t could have been

71
00:04:38,000 --> 00:04:44,000
an extremely wild function,
sewn a lot of oats or whatever

72
00:04:44,000 --> 00:04:50,000
functions do for negative values
of t, and we don't care.

73
00:04:51,000 --> 00:04:57,000
It's only what's happening from
now from time zero onto

74
00:04:57,000 --> 00:05:03,000
infinity.
As long as it behaves now,

75
00:05:01,000 --> 00:05:07,000
from now on,
it's okay.

76
00:05:05,000 --> 00:05:11,000
All right, so,
the way it should behave is by

77
00:05:07,000 --> 00:05:13,000
being an exponential type.
Now, to try to give you some

78
00:05:11,000 --> 00:05:17,000
feeling for what this means,
these functions,

79
00:05:14,000 --> 00:05:20,000
for example, if k is 100,

80
00:05:16,000 --> 00:05:22,000
do you have any idea what the
plot of e to the 100t

81
00:05:20,000 --> 00:05:26,000
looks like?
It goes straight up.

82
00:05:23,000 --> 00:05:29,000
On every computer you try to
plot it on, e to the 100t

83
00:05:26,000 --> 00:05:32,000
goes like that
unless, of course,

84
00:05:29,000 --> 00:05:35,000
you make the scale t equals
zero to, over here,

85
00:05:32,000 --> 00:05:38,000
is one millionth.
Well, even that won't do.

86
00:05:37,000 --> 00:05:43,000
Okay, so these functions really
can grow quite rapidly.

87
00:05:41,000 --> 00:05:47,000
Let's take an example and see
what's of exponential type,

88
00:05:46,000 --> 00:05:52,000
and then perhaps even more
interestingly,

89
00:05:50,000 --> 00:05:56,000
what isn't.
The function sine t,

90
00:05:53,000 --> 00:05:59,000
is that of exponential type?
Well, sure.

91
00:05:57,000 --> 00:06:03,000
Its absolute value is always
less than or equal to one.

92
00:06:01,000 --> 00:06:07,000
So, it's also this paradigm.
If I take c equal to one,

93
00:06:06,000 --> 00:06:12,000
and what should I take k to be?
Zero.

94
00:06:11,000 --> 00:06:17,000
Take k to be zero,
c equals one,

95
00:06:13,000 --> 00:06:19,000
and in fact sine t
plays that condition.

96
00:06:18,000 --> 00:06:24,000
Here's one that's more
interesting, t to the n.

97
00:06:23,000 --> 00:06:29,000
Think of t to the 100th power.

98
00:06:26,000 --> 00:06:32,000
Is that smaller than some
exponential with maybe a

99
00:06:31,000 --> 00:06:37,000
constant out front?
Well, t to the 100th power

100
00:06:35,000 --> 00:06:41,000
goes straight up,
also.

101
00:06:37,000 --> 00:06:43,000
Well, we feel that if we make
the exponential big enough,

102
00:06:41,000 --> 00:06:47,000
maybe it will win out.
In fact, you don't have to make

103
00:06:45,000 --> 00:06:51,000
the exponential big.
k equals one is good

104
00:06:48,000 --> 00:06:54,000
enough.
In other words,

105
00:06:49,000 --> 00:06:55,000
I don't have to put absolute
value signs around the t to the

106
00:06:53,000 --> 00:06:59,000
n because I'm only
thinking about t as being a

107
00:06:57,000 --> 00:07:03,000
positive number,
anyway.

108
00:07:00,000 --> 00:07:06,000
I say that that's less than or
equal to some constant M,

109
00:07:04,000 --> 00:07:10,000
positive constant M times e to
the t will be good

110
00:07:10,000 --> 00:07:16,000
enough for some M and all t.
Now, why is that?

111
00:07:14,000 --> 00:07:20,000
Why is that?
The way to think of that,

112
00:07:17,000 --> 00:07:23,000
so, what this proves is that,
therefore, t to the n

113
00:07:22,000 --> 00:07:28,000
is of exponential type,
which we could have guessed

114
00:07:27,000 --> 00:07:33,000
because after all we were able
to calculate its Laplace

115
00:07:31,000 --> 00:07:37,000
transform.
Now, just because you can

116
00:07:36,000 --> 00:07:42,000
calculate the Laplace transform
doesn't mean it's of exponential

117
00:07:41,000 --> 00:07:47,000
type, but in practical matters,
it almost always does.

118
00:07:46,000 --> 00:07:52,000
So, t to the n is of
exponential type.

119
00:07:50,000 --> 00:07:56,000
How do you prove that?
Well, the weighted secret is to

120
00:07:55,000 --> 00:08:01,000
look at t to the n divided by e
to the t.

121
00:07:59,000 --> 00:08:05,000
In other words,
look at the quotient.

122
00:08:04,000 --> 00:08:10,000
What I'd like to argue is that
this is bounded by some number,

123
00:08:11,000 --> 00:08:17,000
capital M.
That's the question I'm asking.

124
00:08:15,000 --> 00:08:21,000
Now, why is this so?
Well, I think I can convince

125
00:08:21,000 --> 00:08:27,000
you of it without having to work
very hard.

126
00:08:26,000 --> 00:08:32,000
What does the graph of this
function look like?

127
00:08:33,000 --> 00:08:39,000
It starts here,
so I'm graphing this function,

128
00:08:37,000 --> 00:08:43,000
this ratio.
When t is equal to zero,

129
00:08:41,000 --> 00:08:47,000
its value is zero,
right, because of the

130
00:08:45,000 --> 00:08:51,000
numerator.
What happens as t goes to

131
00:08:48,000 --> 00:08:54,000
infinity?
What happens to this?

132
00:08:51,000 --> 00:08:57,000
What does it approach?
Zero.

133
00:08:54,000 --> 00:09:00,000
And, why?
By L'Hop.

134
00:08:56,000 --> 00:09:02,000
By L'Hopital's rule.
Just keep differentiating,

135
00:09:01,000 --> 00:09:07,000
reapply the rule over and over,
keep differentiating it n

136
00:09:06,000 --> 00:09:12,000
times, and finally you'll have
won the numerator down to t to

137
00:09:12,000 --> 00:09:18,000
the zero,
which isn't doing anything

138
00:09:17,000 --> 00:09:23,000
much.
And, the denominator,

139
00:09:21,000 --> 00:09:27,000
no matter how many times you
differentiate it,

140
00:09:25,000 --> 00:09:31,000
it's still t,
to the t all the time.

141
00:09:27,000 --> 00:09:33,000
So, by using Lopital's rule n
times, you change the top to one

142
00:09:32,000 --> 00:09:38,000
or n factorial,
actually; the bottom stays e to

143
00:09:36,000 --> 00:09:42,000
the t,
and the ratio clearly

144
00:09:39,000 --> 00:09:45,000
approaches zero,
and therefore,

145
00:09:41,000 --> 00:09:47,000
it approached zero to start
with.

146
00:09:45,000 --> 00:09:51,000
So, I don't know what this
function's doing in between.

147
00:09:48,000 --> 00:09:54,000
It's a positive function.
It's continuous because the top

148
00:09:51,000 --> 00:09:57,000
and bottom are continuous,
and the bottom is never zero.

149
00:09:54,000 --> 00:10:00,000
So, it's a continuous function
which starts out at zero and is

150
00:09:58,000 --> 00:10:04,000
positive, and as t goes to
infinity, it gets closer and

151
00:10:01,000 --> 00:10:07,000
closer to the t-axis,
again.

152
00:10:03,000 --> 00:10:09,000
Well, what does t to the n
do?

153
00:10:06,000 --> 00:10:12,000
It might wave around.
It doesn't actually.

154
00:10:10,000 --> 00:10:16,000
But, the point is,
because it's continuous,

155
00:10:15,000 --> 00:10:21,000
starts at zero,
ends at zero,

156
00:10:19,000 --> 00:10:25,000
it's bounded.
It has a maximum somewhere.

157
00:10:23,000 --> 00:10:29,000
And, that maximum is M.
So, it has a maximum.

158
00:10:30,000 --> 00:10:36,000
All you have to know is where
it starts, and where it ends up,

159
00:10:34,000 --> 00:10:40,000
and the fact that it's
continuous.

160
00:10:37,000 --> 00:10:43,000
That guarantees that it has a
maximum.

161
00:10:39,000 --> 00:10:45,000
So, it is less than some
maximum, and that shows that

162
00:10:43,000 --> 00:10:49,000
it's of exponential type.
Now, of course,

163
00:10:46,000 --> 00:10:52,000
before you get the idea that
everything's of exponential

164
00:10:50,000 --> 00:10:56,000
type, let's see what isn't.
I'll give you two functions

165
00:10:54,000 --> 00:11:00,000
that are not of exponential
type, for different reasons.

166
00:11:00,000 --> 00:11:06,000
One over t is not of
exponential type.

167
00:11:03,000 --> 00:11:09,000
Well, of course,
it's not defined that t equals

168
00:11:06,000 --> 00:11:12,000
zero. But, you know,

169
00:11:08,000 --> 00:11:14,000
it's okay for an integral not
to be defined at one point

170
00:11:12,000 --> 00:11:18,000
because you're measuring an
area, and when you measure an

171
00:11:16,000 --> 00:11:22,000
area, what happened to one point
doesn't really matter much.

172
00:11:20,000 --> 00:11:26,000
That's not the thing.
What's wrong with one over t is

173
00:11:24,000 --> 00:11:30,000
that the integral doesn't
converge at zero times one over

174
00:11:28,000 --> 00:11:34,000
t dt.
That integral,

175
00:11:32,000 --> 00:11:38,000
when t is near zero,
this is approximately equal to

176
00:11:36,000 --> 00:11:42,000
one, right?
If t is zero,

177
00:11:38,000 --> 00:11:44,000
this is one.
So, it's like the function,

178
00:11:41,000 --> 00:11:47,000
integral from zero to infinity
of one over t,

179
00:11:46,000 --> 00:11:52,000
near zero it's close to,

180
00:11:50,000 --> 00:11:56,000
it's like the integral from
zero to someplace of no

181
00:11:54,000 --> 00:12:00,000
importance, dt over t.

182
00:11:57,000 --> 00:12:03,000
But, this does not converge.
This is like log t,

183
00:12:01,000 --> 00:12:07,000
and log zero is minus infinity.
So, it doesn't converge.

184
00:12:08,000 --> 00:12:14,000
So, one over t is not
of exponential type.

185
00:12:11,000 --> 00:12:17,000
So, what's the Laplace
transform of one over t?

186
00:12:14,000 --> 00:12:20,000
It doesn't have a Laplace

187
00:12:17,000 --> 00:12:23,000
transform.
Well, what if I put t equals

188
00:12:20,000 --> 00:12:26,000
negative n?
What about t to the minus one?

189
00:12:23,000 --> 00:12:29,000
Well, that only works for

190
00:12:26,000 --> 00:12:32,000
positive integers,
not negative integers.

191
00:12:30,000 --> 00:12:36,000
Okay, so it's not of
exponential type.

192
00:12:33,000 --> 00:12:39,000
However, that's because it
never really gets started

193
00:12:37,000 --> 00:12:43,000
properly.
It's more fun to look at a

194
00:12:40,000 --> 00:12:46,000
function which is not of
exponential type because it

195
00:12:45,000 --> 00:12:51,000
grows too fast.
Now, what's a function that

196
00:12:49,000 --> 00:12:55,000
grows faster that it grows so
rapidly that you can't find any

197
00:12:54,000 --> 00:13:00,000
function e to the k t
which bounds it?

198
00:12:58,000 --> 00:13:04,000
A function which grows too
rapidly, a simple one is e to

199
00:13:03,000 --> 00:13:09,000
the t squared,
grows too rapidly to be of

200
00:13:07,000 --> 00:13:13,000
exponential type.
And, the argument is simple.

201
00:13:13,000 --> 00:13:19,000
No matter what you propose,
it's always,

202
00:13:17,000 --> 00:13:23,000
for the K, no matter how big a
number, use Avogadro's number,

203
00:13:22,000 --> 00:13:28,000
use anything you want.
Ultimately, this is going to be

204
00:13:27,000 --> 00:13:33,000
bigger than k t no matter how
big k is, no matter how big k

205
00:13:32,000 --> 00:13:38,000
is.
When is this going to happen?

206
00:13:36,000 --> 00:13:42,000
This will happen if t squared
is bigger than k t.

207
00:13:40,000 --> 00:13:46,000
In other words,
as soon as t becomes bigger

208
00:13:43,000 --> 00:13:49,000
than k, you might have to wait
quite a while for that to

209
00:13:47,000 --> 00:13:53,000
happen, but, as soon as t gets
bigger than 10 to the 10 to the

210
00:13:52,000 --> 00:13:58,000
23,
this e to the t squared will be

211
00:13:55,000 --> 00:14:01,000
bigger than e to the 10 to the
10 to the 23 times t.

212
00:14:00,000 --> 00:14:06,000
So, e to the t squared,

213
00:14:04,000 --> 00:14:10,000
it's a simple
function, a simple elementary

214
00:14:07,000 --> 00:14:13,000
function.
It grows so rapidly it doesn't

215
00:14:10,000 --> 00:14:16,000
have a Laplace transform.
Okay, so how are we going to

216
00:14:13,000 --> 00:14:19,000
solve differential equations if
e to the t squared?

217
00:14:16,000 --> 00:14:22,000
I won't give you any.
And, the reason I won't give

218
00:14:19,000 --> 00:14:25,000
you any: because I never saw one
occur in real life.

219
00:14:22,000 --> 00:14:28,000
Nature, like sines,
cosines, exponentials,

220
00:14:25,000 --> 00:14:31,000
are fine, I've never seen a
physical, you know,

221
00:14:28,000 --> 00:14:34,000
this is just my ignorance.
But, I've never seen a physical

222
00:14:31,000 --> 00:14:37,000
problem that involved a function
growing as rapidly as e to the t

223
00:14:35,000 --> 00:14:41,000
squared.
That may be just my ignorance.

224
00:14:40,000 --> 00:14:46,000
But, I do know the Laplace
transform won't be used to solve

225
00:14:44,000 --> 00:14:50,000
differential equations involving
such a function.

226
00:14:48,000 --> 00:14:54,000
How about e to the minus t
squared?

227
00:14:52,000 --> 00:14:58,000
That's different.
It looks almost the same,

228
00:14:55,000 --> 00:15:01,000
but e to the minus t squared
does this.

229
00:14:58,000 --> 00:15:04,000
It's very well-behaved.
That's the curve,

230
00:15:02,000 --> 00:15:08,000
of course, that you're all
afraid of.

231
00:15:04,000 --> 00:15:10,000
Don't panic.
Okay.

232
00:15:07,000 --> 00:15:13,000
So, I'd like to explain to you
now how differential equations,

233
00:15:13,000 --> 00:15:19,000
maybe I should save-- I'll tell
you what.

234
00:15:17,000 --> 00:15:23,000
We need more formulas.
So, I'll put them,

235
00:15:20,000 --> 00:15:26,000
why don't I save this board,
and instead,

236
00:15:24,000 --> 00:15:30,000
I'll describe to you the basic
way Laplace transforms are used

237
00:15:30,000 --> 00:15:36,000
to solve differential equations,
what are they called,

238
00:15:35,000 --> 00:15:41,000
a paradigm.
I'll show you the paradigm,

239
00:15:39,000 --> 00:15:45,000
and then we'll fill in the
holes so you have some overall

240
00:15:43,000 --> 00:15:49,000
view of how the procedure goes,
and then you'll understand

241
00:15:46,000 --> 00:15:52,000
where the various pieces fit
into it.

242
00:15:48,000 --> 00:15:54,000
I think you'll understand it
better that way.

243
00:15:51,000 --> 00:15:57,000
So, what do we do?
Start with the differential

244
00:15:54,000 --> 00:16:00,000
equation.
But, right away,

245
00:15:55,000 --> 00:16:01,000
there's a fundamental
difference between what the

246
00:15:58,000 --> 00:16:04,000
Laplace transform does,
and what we've been doing up

247
00:16:01,000 --> 00:16:07,000
until now, namely,
what you have to start with is

248
00:16:04,000 --> 00:16:10,000
not merely the differential
equation.

249
00:16:08,000 --> 00:16:14,000
Let's say we have linear with
constant coefficients.

250
00:16:12,000 --> 00:16:18,000
It's almost never used to solve
any other type of problem.

251
00:16:16,000 --> 00:16:22,000
And, let's take second order so
I don't have to do,

252
00:16:20,000 --> 00:16:26,000
because that's the kind we've
been working with all term.

253
00:16:25,000 --> 00:16:31,000
But, it's allowed to be
inhomogeneous,

254
00:16:28,000 --> 00:16:34,000
so, f of t.
Let's call the something else,

255
00:16:32,000 --> 00:16:38,000
another letter,
h of t.

256
00:16:36,000 --> 00:16:42,000
I'll want f of t for
the function I'm taking the

257
00:16:39,000 --> 00:16:45,000
Laplace transform of.
All right, now,

258
00:16:42,000 --> 00:16:48,000
the difference is that up to
now, you know techniques for

259
00:16:45,000 --> 00:16:51,000
solving this just as it stands.
The Laplace transform does not

260
00:16:50,000 --> 00:16:56,000
know how to solve this just
doesn't stands.

261
00:16:52,000 --> 00:16:58,000
The Laplace transform must have
an initial value problem.

262
00:16:56,000 --> 00:17:02,000
In other words,
you must supply from the

263
00:16:59,000 --> 00:17:05,000
beginning the initial conditions
that the y is to satisfy.

264
00:17:04,000 --> 00:17:10,000
Now, I don't want to say any
specific numbers,

265
00:17:06,000 --> 00:17:12,000
so I'll use generic numbers.
Well, but look,

266
00:17:08,000 --> 00:17:14,000
what do we do if we get a
problem and there are no initial

267
00:17:12,000 --> 00:17:18,000
conditions; does that mean we
can't use the Laplace transform?

268
00:17:15,000 --> 00:17:21,000
No, of course you can use it.
But, you will just have to

269
00:17:18,000 --> 00:17:24,000
assume the initial conditions
are on the numbers.

270
00:17:21,000 --> 00:17:27,000
You'll say it but the initial
conditions be y sub zero

271
00:17:24,000 --> 00:17:30,000
and y zero prime,
or whatever,

272
00:17:26,000 --> 00:17:32,000
a and b, whatever you want to
call it.

273
00:17:30,000 --> 00:17:36,000
And now, the answer,
then, will involve the a and

274
00:17:34,000 --> 00:17:40,000
the b or the y zero and the y
zero prime.

275
00:17:38,000 --> 00:17:44,000
But, you must,
at least, give lip service to

276
00:17:42,000 --> 00:17:48,000
the initial conditions,
whereas before we didn't have

277
00:17:46,000 --> 00:17:52,000
to do that.
Now, depending on your point of

278
00:17:50,000 --> 00:17:56,000
view, that's a grave defect,
or it is, so what?

279
00:17:54,000 --> 00:18:00,000
Let's adopt the so what point
of view.

280
00:17:57,000 --> 00:18:03,000
So, there's our problem.
It's an initial value problem.

281
00:18:03,000 --> 00:18:09,000
How is it solved by the Laplace
transform?

282
00:18:05,000 --> 00:18:11,000
Well, the idea is you take the
Laplace transform of this

283
00:18:09,000 --> 00:18:15,000
differential equation and the
initial conditions.

284
00:18:12,000 --> 00:18:18,000
So, I'm going to explain to you
how to do that.

285
00:18:15,000 --> 00:18:21,000
Not right now,
because we're going to need,

286
00:18:17,000 --> 00:18:23,000
first, the Laplace transform of
a derivative,

287
00:18:20,000 --> 00:18:26,000
the formula for that.
You don't know that yet.

288
00:18:23,000 --> 00:18:29,000
But when you do know it,
you will be able to take the

289
00:18:26,000 --> 00:18:32,000
Laplace transform of the initial
value problem.

290
00:18:29,000 --> 00:18:35,000
So, I'll put the little l here,
and what comes out is,

291
00:18:32,000 --> 00:18:38,000
well, y of t is the
solution to the original

292
00:18:36,000 --> 00:18:42,000
problem.
If y of t is the

293
00:18:39,000 --> 00:18:45,000
function which satisfies that
equation and these initial

294
00:18:43,000 --> 00:18:49,000
conditions, its Laplace
transform, let's call it capital

295
00:18:48,000 --> 00:18:54,000
Y, that's our standard notation,
but it's going to be of a new

296
00:18:52,000 --> 00:18:58,000
variable, s.
So, when I take the Laplace

297
00:18:55,000 --> 00:19:01,000
transform of the differential
equation with the initial

298
00:18:59,000 --> 00:19:05,000
conditions, what comes out is an
algebraic-- the emphasis is on

299
00:19:04,000 --> 00:19:10,000
algebraic: no derivatives,
no transcendental functions,

300
00:19:08,000 --> 00:19:14,000
nothing like that,
an algebraic equation,

301
00:19:11,000 --> 00:19:17,000
m Y of s.

302
00:19:22,000 --> 00:19:28,000
And, now what?
Well, now, in the domain of s,

303
00:19:25,000 --> 00:19:31,000
it's easy to solve this
algebraic equation.

304
00:19:28,000 --> 00:19:34,000
Not all algebraic equations are
easy to solve for the capital Y.

305
00:19:33,000 --> 00:19:39,000
But, the ones you will get will
always be, not because I am

306
00:19:37,000 --> 00:19:43,000
making life easy for you,
but that's the way the Laplace

307
00:19:42,000 --> 00:19:48,000
transform works.
So, you will solve it for Y.

308
00:19:45,000 --> 00:19:51,000
And, the answer will always
come out to be Y equals,

309
00:19:49,000 --> 00:19:55,000
Y of s equals some
rational function,

310
00:19:52,000 --> 00:19:58,000
some quotient of polynomials in
s, a polynomial in s divided by

311
00:19:57,000 --> 00:20:03,000
some other polynomial in s.

312
00:20:09,000 --> 00:20:15,000
And, now what?
Well, this is the Laplace

313
00:20:12,000 --> 00:20:18,000
transform of the answer.
This is the Laplace transform

314
00:20:17,000 --> 00:20:23,000
of the solution we are looking
for.

315
00:20:20,000 --> 00:20:26,000
So, the final step is to go
backwards by taking the inverse

316
00:20:25,000 --> 00:20:31,000
Laplace transform of this guy.
And, what will you get?

317
00:20:30,000 --> 00:20:36,000
Well, you will get y equals the
y of t that we are

318
00:20:35,000 --> 00:20:41,000
looking for.
It's really a wildly improbable

319
00:20:39,000 --> 00:20:45,000
procedure.
In other words,

320
00:20:41,000 --> 00:20:47,000
instead of going from here to
here, you have to imagine

321
00:20:45,000 --> 00:20:51,000
there's a mountain here.
And, the only way to get around

322
00:20:48,000 --> 00:20:54,000
it is to go, first,
here, and then cross the stream

323
00:20:51,000 --> 00:20:57,000
here, and then go back up,
and go back up.

324
00:20:54,000 --> 00:21:00,000
It looks like a senseless
procedure, what do they call it,

325
00:20:58,000 --> 00:21:04,000
going around Robin Hood's barn,
it was called when I was a,

326
00:21:01,000 --> 00:21:07,000
I don't know why it's called
that.

327
00:21:05,000 --> 00:21:11,000
But that's what we used to call
it; not Laplace transform.

328
00:21:09,000 --> 00:21:15,000
That was just a generic thing
when you had to do something

329
00:21:14,000 --> 00:21:20,000
like this.
But, the answer is that it's

330
00:21:18,000 --> 00:21:24,000
hard to go from here to here,
but trivial to go from here to

331
00:21:23,000 --> 00:21:29,000
here.
This solution step is the

332
00:21:25,000 --> 00:21:31,000
easiest step of all.
This is not very hard.

333
00:21:29,000 --> 00:21:35,000
It's easy, in fact.
This is easy and

334
00:21:33,000 --> 00:21:39,000
straightforward.
This is trivial,

335
00:21:36,000 --> 00:21:42,000
essentially,
yeah, trivial.

336
00:21:38,000 --> 00:21:44,000
But, this step is the hard
step.

337
00:21:41,000 --> 00:21:47,000
This is where you have to use
partial fractions,

338
00:21:45,000 --> 00:21:51,000
look up things in the table to
get back there so that most of

339
00:21:50,000 --> 00:21:56,000
the work of the procedure isn't
going from here to here.

340
00:21:55,000 --> 00:22:01,000
Going from here to there is a
breeze.

341
00:22:00,000 --> 00:22:06,000
Okay, now, in order to
implement this,

342
00:22:02,000 --> 00:22:08,000
what is it we have to do?
Well, the basic thing is I have

343
00:22:05,000 --> 00:22:11,000
to explain to you,
you already know at least a

344
00:22:08,000 --> 00:22:14,000
little bit, a reasonable amount
of technique for taking that

345
00:22:12,000 --> 00:22:18,000
step if you went to recitation
yesterday and practiced a little

346
00:22:16,000 --> 00:22:22,000
bit.
This part, I assure you,

347
00:22:17,000 --> 00:22:23,000
is nothing.
So, all I have to do now is

348
00:22:20,000 --> 00:22:26,000
explain to you how to take the
Laplace transform of the

349
00:22:23,000 --> 00:22:29,000
differential equation.
And, that really means,

350
00:22:26,000 --> 00:22:32,000
how do you take the Laplace
transform of a derivative?

351
00:22:31,000 --> 00:22:37,000
So, that's our problem.
What I want to form,

352
00:22:35,000 --> 00:22:41,000
in other words,
is a formula for the Laplace

353
00:22:39,000 --> 00:22:45,000
transform f prime of t.

354
00:22:43,000 --> 00:22:49,000
Now, in terms of what?
Well, since f is an arbitrary

355
00:22:48,000 --> 00:22:54,000
function, the only thing I could
hope for is somehow to express

356
00:22:54,000 --> 00:23:00,000
the Laplace transform of the
derivative in terms of the

357
00:23:00,000 --> 00:23:06,000
Laplace transform of the
original function.

358
00:23:06,000 --> 00:23:12,000
So, that's what I'm aiming for.
Okay, where are we going to

359
00:23:10,000 --> 00:23:16,000
start?
Well, starting is easy because

360
00:23:12,000 --> 00:23:18,000
we know nothing.
If you don't know anything,

361
00:23:15,000 --> 00:23:21,000
then there's no place to start
but the definition.

362
00:23:18,000 --> 00:23:24,000
Since I know nothing whatever
about the function f of t,

363
00:23:22,000 --> 00:23:28,000
and I want to
calculate the Laplace transform,

364
00:23:26,000 --> 00:23:32,000
I'd better start with the
definition.

365
00:23:30,000 --> 00:23:36,000
Whatever this is,
it's the integral from zero to

366
00:23:33,000 --> 00:23:39,000
infinity of e to the minus s t
times f prime of t dt.

367
00:23:39,000 --> 00:23:45,000
Now, what am I looking for?
I'm looking for somehow to

368
00:23:43,000 --> 00:23:49,000
transform this so that what
appears here is not

369
00:23:47,000 --> 00:23:53,000
f prime of t, which I'm
clueless about, but f of t

370
00:23:50,000 --> 00:23:56,000
because if this were f of t,
this expression would be the

371
00:23:54,000 --> 00:24:00,000
Laplace transform of f of t.

372
00:23:56,000 --> 00:24:02,000
And, I'm assuming I know that.
So, the question is how do I

373
00:24:02,000 --> 00:24:08,000
take this and somehow do
something clever to it that

374
00:24:05,000 --> 00:24:11,000
turns this into f of t
instead of f prime of t?

375
00:24:10,000 --> 00:24:16,000
Now, to first the question that

376
00:24:13,000 --> 00:24:19,000
way, I hope I would get 100%
response on what to do.

377
00:24:16,000 --> 00:24:22,000
But, I'll go for 1%.
So, what should I do?

378
00:24:20,000 --> 00:24:26,000
I want to change that, so that
instead of f prime of t,

379
00:24:23,000 --> 00:24:29,000
f of t
appears there instead.

380
00:24:27,000 --> 00:24:33,000
What should I do?
Integrate by parts,

381
00:24:31,000 --> 00:24:37,000
the most fundamental procedure
in advanced analysis.

382
00:24:35,000 --> 00:24:41,000
Everything important and
interesting depends on

383
00:24:39,000 --> 00:24:45,000
integration by parts.
And, when you consider that

384
00:24:43,000 --> 00:24:49,000
integration by parts is nothing
more than just the formula for

385
00:24:48,000 --> 00:24:54,000
the derivative of a product read
backwards, it's amazing.

386
00:24:53,000 --> 00:24:59,000
It never fails to amaze me,
but it's okay.

387
00:24:56,000 --> 00:25:02,000
That's what mathematics are so
great.

388
00:24:59,000 --> 00:25:05,000
Okay, so let's use integration
by parts.

389
00:25:04,000 --> 00:25:10,000
Integration by parts:
okay, so, we have to decide,

390
00:25:06,000 --> 00:25:12,000
of course, there's no doubt
that this is the factor we want

391
00:25:10,000 --> 00:25:16,000
to integrate,
which means we have to be

392
00:25:12,000 --> 00:25:18,000
willing to differentiate this
factor.

393
00:25:14,000 --> 00:25:20,000
But that will be okay because
it looks practically,

394
00:25:17,000 --> 00:25:23,000
like any exponential,
it looks practically the same

395
00:25:20,000 --> 00:25:26,000
after you've differentiated it.
So, let's do the work.

396
00:25:23,000 --> 00:25:29,000
First step is you don't do the
differentiation.

397
00:25:26,000 --> 00:25:32,000
You only do the integration.
So, the first step is e to the

398
00:25:29,000 --> 00:25:35,000
negative s t.
And, the integral of f prime of

399
00:25:34,000 --> 00:25:40,000
t is just f of t.

400
00:25:36,000 --> 00:25:42,000
And, that's to be evaluated
between the limits zero and

401
00:25:40,000 --> 00:25:46,000
infinity.
And then, minus,

402
00:25:42,000 --> 00:25:48,000
again, before you forget it,
put down that minus sign.

403
00:25:45,000 --> 00:25:51,000
The integral between the limits
of what you get by doing both

404
00:25:49,000 --> 00:25:55,000
operations, both the
differentiation and the

405
00:25:52,000 --> 00:25:58,000
integration.
So, the differentiation will be

406
00:25:55,000 --> 00:26:01,000
by using the chain rule.
Remember, I'm differentiating

407
00:25:59,000 --> 00:26:05,000
with respect to t.
The variable is t here,

408
00:26:03,000 --> 00:26:09,000
not s.
s is just a parameter.

409
00:26:06,000 --> 00:26:12,000
It's just a constant,
a variable constant,

410
00:26:09,000 --> 00:26:15,000
if you get my meaning.
That's not an oxymoron.

411
00:26:13,000 --> 00:26:19,000
A variable constant:
a parameter is a variable

412
00:26:16,000 --> 00:26:22,000
constant, variable because you
can manipulate the little slider

413
00:26:21,000 --> 00:26:27,000
and make a change its value,
right?

414
00:26:24,000 --> 00:26:30,000
That's why it's variable.
It's not a variable.

415
00:26:28,000 --> 00:26:34,000
It's variable,
if you get the distinction.

416
00:26:33,000 --> 00:26:39,000
Okay, well, I mean,
it becomes a variable

417
00:26:35,000 --> 00:26:41,000
[LAUGHTER].
But right now,

418
00:26:37,000 --> 00:26:43,000
it's not a variable.
It's just sitting there in the

419
00:26:41,000 --> 00:26:47,000
integral.
All right, so,

420
00:26:43,000 --> 00:26:49,000
minus s, e to the negative s t,
f of t dt.

421
00:26:47,000 --> 00:26:53,000
Now, this part's easy.

422
00:26:49,000 --> 00:26:55,000
The interesting thing is this
expression.

423
00:26:52,000 --> 00:26:58,000
So, and the most interesting
thing is I have to evaluate it

424
00:26:56,000 --> 00:27:02,000
at infinity.
Now, of course,

425
00:26:58,000 --> 00:27:04,000
that means take the limit as
you go towards,

426
00:27:01,000 --> 00:27:07,000
as you let t goes to infinity.
Now, so what I'm interested in

427
00:27:07,000 --> 00:27:13,000
knowing is what's the limit of
that expression?

428
00:27:10,000 --> 00:27:16,000
I'll write it as f of t divided
by e to the s t.

429
00:27:13,000 --> 00:27:19,000
Remember, s is a positive

430
00:27:16,000 --> 00:27:22,000
number.
s t goes to infinity,

431
00:27:18,000 --> 00:27:24,000
and I want to know what the
limit of that is.

432
00:27:21,000 --> 00:27:27,000
Well, the limit is what it is.
But really, if that limit isn't

433
00:27:25,000 --> 00:27:31,000
zero, I'm in deep trouble since
the whole process is out of

434
00:27:29,000 --> 00:27:35,000
control.
What will make that limit zero?

435
00:27:33,000 --> 00:27:39,000
Well, that f of t
should not grow faster than e to

436
00:27:37,000 --> 00:27:43,000
the s t if s is a big
enough number.

437
00:27:41,000 --> 00:27:47,000
And now, that's just what will
happen if f of t is of

438
00:27:45,000 --> 00:27:51,000
exponential type.
It's for this step right here

439
00:27:48,000 --> 00:27:54,000
that is the most crucial place
at which we need to know that f

440
00:27:53,000 --> 00:27:59,000
of t is of exponential type.
So, that limit is zero since f

441
00:27:57,000 --> 00:28:03,000
of t is of exponential type,
in other words,

442
00:28:01,000 --> 00:28:07,000
that the value,
the absolute value of f of t,

443
00:28:04,000 --> 00:28:10,000
becomes less than,
let's say, put in the c if you

444
00:28:09,000 --> 00:28:15,000
want, but it's not very
important, c e to the k t

445
00:28:13,000 --> 00:28:19,000
efor all values of t.
And, therefore,

446
00:28:18,000 --> 00:28:24,000
this will go to zero as soon as
s becomes bigger than that k.

447
00:28:22,000 --> 00:28:28,000
In other words,

448
00:28:23,000 --> 00:28:29,000
if f of t isn't
growing any faster than e to the

449
00:28:27,000 --> 00:28:33,000
k t ,
then as soon as s is a number,

450
00:28:30,000 --> 00:28:36,000
that parameter has the value
bigger than k,

451
00:28:33,000 --> 00:28:39,000
this ratio is going to go to
zero because the denominator

452
00:28:37,000 --> 00:28:43,000
will always be bigger than the
numerator, and getting bigger

453
00:28:41,000 --> 00:28:47,000
faster.
So, this goes to zero if s is

454
00:28:45,000 --> 00:28:51,000
bigger than k.
At the upper limit,

455
00:28:48,000 --> 00:28:54,000
therefore, this is zero.
Again, assuming that s is

456
00:28:52,000 --> 00:28:58,000
bigger than that k,
the k of the exponential type,

457
00:28:57,000 --> 00:29:03,000
how about at the lower limit?
We're used to seeing zero

458
00:29:01,000 --> 00:29:07,000
there, but we're not going to
get zero.

459
00:29:04,000 --> 00:29:10,000
If I plug in t equals zero,
this factor becomes one.

460
00:29:09,000 --> 00:29:15,000
And, what happens to that one?
f of zero.

461
00:29:13,000 --> 00:29:19,000
You mean, I'm going to have to
know what f of zero is before I

462
00:29:18,000 --> 00:29:24,000
can take the Laplace transform
of this derivative?

463
00:29:22,000 --> 00:29:28,000
The answer is yes,
and that's why you have to have

464
00:29:26,000 --> 00:29:32,000
an initial value problem.
You have to know in advance

465
00:29:30,000 --> 00:29:36,000
what the value of the function
that you are looking for is at

466
00:29:34,000 --> 00:29:40,000
zero because it enters into the
formula.

467
00:29:37,000 --> 00:29:43,000
I didn't make up these rules;
I'm just following them.

468
00:29:40,000 --> 00:29:46,000
So, what's the rest?
The two negatives cancel,

469
00:29:43,000 --> 00:29:49,000
and you get plus s.
It's just a parameter,

470
00:29:47,000 --> 00:29:53,000
so I can pull it out of the
integral.

471
00:29:49,000 --> 00:29:55,000
I'm integrating with respect to
t, and what's left is,

472
00:29:53,000 --> 00:29:59,000
well, what is left?
If I take out minus s,

473
00:29:56,000 --> 00:30:02,000
combine it there,
I get what's left is just the

474
00:29:59,000 --> 00:30:05,000
Laplace transform of the
function I started with.

475
00:30:04,000 --> 00:30:10,000
So, it's F of s.
And, that's the magic formula

476
00:30:10,000 --> 00:30:16,000
for the Laplace transform of the
derivative.

477
00:30:15,000 --> 00:30:21,000
So, it's worth putting up on
our little list.

478
00:30:20,000 --> 00:30:26,000
So, f prime of t,
assuming it's of exponential

479
00:30:26,000 --> 00:30:32,000
type, has as its Laplace
transform, well,

480
00:30:31,000 --> 00:30:37,000
what is it?
Let's put down the major part

481
00:30:35,000 --> 00:30:41,000
of it is s times whatever the
Laplace transform of the

482
00:30:39,000 --> 00:30:45,000
original function,
F of t,

483
00:30:41,000 --> 00:30:47,000
was.
So, I take the original Laplace

484
00:30:44,000 --> 00:30:50,000
transform.
When I multiply it by s,

485
00:30:46,000 --> 00:30:52,000
that corresponds to taking the
derivative.

486
00:30:49,000 --> 00:30:55,000
But there's also that little
extra piece.

487
00:30:51,000 --> 00:30:57,000
I must know the value of the
starting value of the function.

488
00:30:55,000 --> 00:31:01,000
This is the formula you'll used
to take a Laplace transform of

489
00:31:00,000 --> 00:31:06,000
the differential equation.
Now, but you see I'm not done

490
00:31:05,000 --> 00:31:11,000
yet because that will take care
of the term a y prime.

491
00:31:09,000 --> 00:31:15,000
But, I don't know what the
Laplace transform of the second

492
00:31:13,000 --> 00:31:19,000
derivative is.
Okay, so, we need a formula for

493
00:31:16,000 --> 00:31:22,000
the Laplace transform of a
second derivative as well as the

494
00:31:20,000 --> 00:31:26,000
first.
Now, the hack method is to say,

495
00:31:23,000 --> 00:31:29,000
secondary, all right.
I've got to do this.

496
00:31:25,000 --> 00:31:31,000
I'll second derivative here,
second derivative here,

497
00:31:29,000 --> 00:31:35,000
what do I do with that?
Ah-ha, I integrate by parts

498
00:31:33,000 --> 00:31:39,000
twice.
Yes, you can do that.

499
00:31:34,000 --> 00:31:40,000
But that's a hack method.
And, of course,

500
00:31:40,000 --> 00:31:46,000
I know you're too smart to do
that.

501
00:31:45,000 --> 00:31:51,000
What you would do instead is--
How are we going to fill that

502
00:31:53,000 --> 00:31:59,000
in?
Well, a second derivative is

503
00:31:58,000 --> 00:32:04,000
also a first derivative.
A second derivative is the

504
00:32:07,000 --> 00:32:13,000
first derivative of the first
derivative.

505
00:32:14,000 --> 00:32:20,000
Okay, now, we'll just call this
glop, something.

506
00:32:22,000 --> 00:32:28,000
So, it's glop prime.
What is the formula for the

507
00:32:31,000 --> 00:32:37,000
Laplace transform of glop prime?
It is, well,

508
00:32:39,000 --> 00:32:45,000
I have my formula.
It is the glop prime.

509
00:32:43,000 --> 00:32:49,000
The formula for it is s times
the Laplace transform of glop,

510
00:32:50,000 --> 00:32:56,000
okay, glop.
Well, glop is f prime of t.

511
00:32:55,000 --> 00:33:01,000
I'm not done yet,

512
00:32:58,000 --> 00:33:04,000
minus glop evaluated at zero.
What's glop evaluated at zero?

513
00:33:05,000 --> 00:33:11,000
Well, f prime of zero.

514
00:33:10,000 --> 00:33:16,000
Now, I don't want the formula
in that form,

515
00:33:13,000 --> 00:33:19,000
but I have to have it in that
form because I know what the

516
00:33:17,000 --> 00:33:23,000
Laplace transform of f prime of
t is.

517
00:33:20,000 --> 00:33:26,000
I just calculated that.
So, this is equal to s times

518
00:33:24,000 --> 00:33:30,000
the Laplace transform of f prime
of t, which is s times F of s,

519
00:33:28,000 --> 00:33:34,000
capital F of s,
minus f of zero.

520
00:33:31,000 --> 00:33:37,000
All that bracket stuff

521
00:33:34,000 --> 00:33:40,000
corresponds to this guy.
And, don't forget the stuff

522
00:33:38,000 --> 00:33:44,000
that's tagging along,
minus f prime of zero.

523
00:33:42,000 --> 00:33:48,000
And now, put that all together.

524
00:33:45,000 --> 00:33:51,000
What is it going to be?
Well, there's the principal

525
00:33:49,000 --> 00:33:55,000
term which consists of s squared
multiplied by F of s.

526
00:33:54,000 --> 00:34:00,000
That's the main part of it.

527
00:33:56,000 --> 00:34:02,000
And, the rest is the sort of
fellow travelers.

528
00:34:00,000 --> 00:34:06,000
So, we have minus s times f of
zero,

529
00:34:04,000 --> 00:34:10,000
little term tagging along.
This is a constant times s.

530
00:34:10,000 --> 00:34:16,000
And then, we've got another
one, still another constant.

531
00:34:14,000 --> 00:34:20,000
So, what we have is to
calculate the Laplace transform

532
00:34:18,000 --> 00:34:24,000
of the second derivative,
I need to know both f of zero

533
00:34:22,000 --> 00:34:28,000
and f prime of zero,
exactly the initial

534
00:34:27,000 --> 00:34:33,000
conditions that the problem was
given for the initial value

535
00:34:31,000 --> 00:34:37,000
problem.
But, notice,

536
00:34:33,000 --> 00:34:39,000
there's a principal part of it.
That's the s squared F of s.

537
00:34:37,000 --> 00:34:43,000
That's the guts of it,

538
00:34:39,000 --> 00:34:45,000
so to speak.
The rest of it,

539
00:34:41,000 --> 00:34:47,000
you know, you might hope that
these two numbers are zero.

540
00:34:44,000 --> 00:34:50,000
It could happen,
and often it is made to happen

541
00:34:47,000 --> 00:34:53,000
and problems to simplify them.
And I case, you don't have to

542
00:34:51,000 --> 00:34:57,000
worry; they're not there.
But, if they are there,

543
00:34:54,000 --> 00:35:00,000
you must put them in or you get
the wrong answer.

544
00:34:56,000 --> 00:35:02,000
So, that's the list of
formulas.

545
00:35:00,000 --> 00:35:06,000
So, those formulas on the top
board and these two extra ones,

546
00:35:06,000 --> 00:35:12,000
those are the things you will
be working with on Friday.

547
00:35:12,000 --> 00:35:18,000
But I stress,
the Laplace transform won't be

548
00:35:17,000 --> 00:35:23,000
a big part of the exam.
The exam, of course,

549
00:35:22,000 --> 00:35:28,000
doesn't exist,
let's say a maximum of 20%,

550
00:35:27,000 --> 00:35:33,000
maybe 15.
I don't know,

551
00:35:29,000 --> 00:35:35,000
give or take a few points.
Yeah, what's a point or two?

552
00:35:37,000 --> 00:35:43,000
Okay, let's solve,
yeah, we have time.

553
00:35:41,000 --> 00:35:47,000
We have time to solve a
problem.

554
00:35:44,000 --> 00:35:50,000
Let's solve a problem.
See, I can't touch that.

555
00:35:49,000 --> 00:35:55,000
It's untouchable.
Okay, this, we've got to keep.

556
00:36:14,000 --> 00:36:20,000
Problem? Okay.

557
00:36:39,000 --> 00:36:45,000
Okay, now you know how to solve
this problem by operators.

558
00:36:43,000 --> 00:36:49,000
Let me just briefly remind you
of the basic steps.

559
00:36:47,000 --> 00:36:53,000
You have to do two separate
tasks.

560
00:36:50,000 --> 00:36:56,000
You have to first solve the
homogeneous equation,

561
00:36:54,000 --> 00:37:00,000
putting a zero there.
That's the first thing you

562
00:36:58,000 --> 00:37:04,000
learned to do.
That's easy.

563
00:37:00,000 --> 00:37:06,000
You could almost do that in
your head now.

564
00:37:05,000 --> 00:37:11,000
You get the characteristic
polynomial, get its roots,

565
00:37:08,000 --> 00:37:14,000
get the two functions,
e to the t and e to the

566
00:37:11,000 --> 00:37:17,000
negative t,
which are the solutions.

567
00:37:14,000 --> 00:37:20,000
You make up c1 times one,
and c2 times the other.

568
00:37:18,000 --> 00:37:24,000
That's the complementary
function that solves the

569
00:37:21,000 --> 00:37:27,000
homogeneous problem.
And then you have to find a

570
00:37:24,000 --> 00:37:30,000
particular solution.
Can you see what would happen

571
00:37:27,000 --> 00:37:33,000
if you try to find the
particular solution?

572
00:37:30,000 --> 00:37:36,000
The number here is negative
one, right?

573
00:37:34,000 --> 00:37:40,000
Negative one is a root of the
characteristic polynomial,

574
00:37:37,000 --> 00:37:43,000
so you've got to use that extra
formula.

575
00:37:40,000 --> 00:37:46,000
It's okay.
That's why I gave it to you.

576
00:37:42,000 --> 00:37:48,000
You've used the exponential
input theorem with the extra

577
00:37:46,000 --> 00:37:52,000
formula.
Then, you will get the

578
00:37:48,000 --> 00:37:54,000
particular solution.
And now, you have to make the

579
00:37:51,000 --> 00:37:57,000
general solution.
The particular solution plus

580
00:37:54,000 --> 00:38:00,000
the complementary function,
and now you are ready to put in

581
00:37:58,000 --> 00:38:04,000
the initial conditions.
At the very end,

582
00:38:02,000 --> 00:38:08,000
when you've got the whole
general solution,

583
00:38:05,000 --> 00:38:11,000
now you put in,
not before, you put in the

584
00:38:07,000 --> 00:38:13,000
initial conditions.
You have to calculate the

585
00:38:11,000 --> 00:38:17,000
derivative of that thing and
substitute this.

586
00:38:14,000 --> 00:38:20,000
You take it as it stands to
substitute this.

587
00:38:17,000 --> 00:38:23,000
You get a pair of simultaneous
equations for c1 and c2.

588
00:38:21,000 --> 00:38:27,000
You solve them:
answer.

589
00:38:22,000 --> 00:38:28,000
It's a rather elaborate
procedure, which has at least

590
00:38:26,000 --> 00:38:32,000
three or four separate steps,
all of which,

591
00:38:29,000 --> 00:38:35,000
of course, must be done
correctly.

592
00:38:33,000 --> 00:38:39,000
Now, the Laplace transform,
instead, feeds the entire

593
00:38:37,000 --> 00:38:43,000
problem into the Laplace
transform machine.

594
00:38:40,000 --> 00:38:46,000
You follow that little blue
pattern, and you come out with

595
00:38:44,000 --> 00:38:50,000
the answer.
So, let's do the Laplace

596
00:38:47,000 --> 00:38:53,000
transform way.
Okay, so, the first step is to

597
00:38:51,000 --> 00:38:57,000
say, if here's my unknown
function, y of t,

598
00:38:55,000 --> 00:39:01,000
it obeys this law,
and here are its starting

599
00:38:58,000 --> 00:39:04,000
values, a bit of its derivative.
What I'm going to take is the

600
00:39:03,000 --> 00:39:09,000
Laplace transform of this
equation.

601
00:39:06,000 --> 00:39:12,000
In other words,
I'll take the Laplace transform

602
00:39:09,000 --> 00:39:15,000
of this side,
and this side also.

603
00:39:11,000 --> 00:39:17,000
And, they must be equal because
if they were equal to start

604
00:39:15,000 --> 00:39:21,000
with, the Laplace transforms
also have to be equal.

605
00:39:18,000 --> 00:39:24,000
Okay, so let's take the Laplace
transform of this equation.

606
00:39:22,000 --> 00:39:28,000
Okay, first ID the Laplace
transform of the second

607
00:39:25,000 --> 00:39:31,000
derivative.
Okay, that's going to be,

608
00:39:27,000 --> 00:39:33,000
don't forget the principal
terms.

609
00:39:31,000 --> 00:39:37,000
There is some people who get so
hypnotized by this.

610
00:39:34,000 --> 00:39:40,000
I just know I'm going to forget
this.

611
00:39:36,000 --> 00:39:42,000
So, they read it.
Then they forget this.

612
00:39:38,000 --> 00:39:44,000
But that's everything.
That's the important part.

613
00:39:41,000 --> 00:39:47,000
Okay, so it's s times,
I'm calling the Laplace

614
00:39:44,000 --> 00:39:50,000
transform not capital F but
capital Y because my original

615
00:39:48,000 --> 00:39:54,000
function is called little y.
So, it's s squared Y.

616
00:39:51,000 --> 00:39:57,000
It's Y of s,
but I'm not going to put that,

617
00:39:54,000 --> 00:40:00,000
the of s in because it just
makes the thing look more

618
00:39:58,000 --> 00:40:04,000
complicated.
And now, before you forget,

619
00:40:02,000 --> 00:40:08,000
you have to put in the rest.
So, minus s times the value at

620
00:40:06,000 --> 00:40:12,000
zero, which is one,
minus the value of the

621
00:40:10,000 --> 00:40:16,000
derivative.
But, that's zero.

622
00:40:12,000 --> 00:40:18,000
So, this is not too hard a
problem.

623
00:40:15,000 --> 00:40:21,000
So, minus s minus zero,
so I don't have to put that

624
00:40:19,000 --> 00:40:25,000
in.
So, all this is the Laplace

625
00:40:22,000 --> 00:40:28,000
transform of y double prime.

626
00:40:25,000 --> 00:40:31,000
And now, minus the Laplace
transform of y,

627
00:40:29,000 --> 00:40:35,000
well, that's just capital Y.
What's that equal to?

628
00:40:35,000 --> 00:40:41,000
The Laplace transform of the
right-hand side.

629
00:40:40,000 --> 00:40:46,000
Okay, look up the formula.
It is e to the negative t,

630
00:40:45,000 --> 00:40:51,000
a is minus one,
so, it's one over s minus minus

631
00:40:51,000 --> 00:40:57,000
one;
so, it is s plus one.

632
00:40:57,000 --> 00:41:03,000
This is that.
Okay, the next thing we have to

633
00:41:02,000 --> 00:41:08,000
do is solve for Y.
That doesn't look too hard.

634
00:41:05,000 --> 00:41:11,000
Solve it for y.
Okay, the best thing to do is

635
00:41:08,000 --> 00:41:14,000
put s squared,
group all the Y terms together

636
00:41:11,000 --> 00:41:17,000
unless you're really quite a
good calculator.

637
00:41:14,000 --> 00:41:20,000
Maybe make one extra line out
of it.

638
00:41:17,000 --> 00:41:23,000
Yeah, definitely do this.
And then, the extra garbage I

639
00:41:21,000 --> 00:41:27,000
refer to as the garbage,
this stuff, and this stuff,

640
00:41:25,000 --> 00:41:31,000
the stuff, the linear
polynomials which are tagging

641
00:41:28,000 --> 00:41:34,000
along move to the right-hand
side because they don't involve

642
00:41:32,000 --> 00:41:38,000
capital Y.
So, this we will move to the

643
00:41:37,000 --> 00:41:43,000
other side.
And so, that's equal to (one

644
00:41:40,000 --> 00:41:46,000
over (s plus one)) plus s.

645
00:41:44,000 --> 00:41:50,000
Now, you have a basic choice.
About half the time,

646
00:41:48,000 --> 00:41:54,000
it's a good idea to combine
these terms.

647
00:41:51,000 --> 00:41:57,000
The other half of the time,
it's not a good idea to combine

648
00:41:56,000 --> 00:42:02,000
those terms.
So, how do we know whether to

649
00:41:59,000 --> 00:42:05,000
do it or not to do it?
Experience, which you will get

650
00:42:03,000 --> 00:42:09,000
by solving many,
many problems.

651
00:42:06,000 --> 00:42:12,000
Okay, I'm going to combine them
because I think it's a good

652
00:42:11,000 --> 00:42:17,000
thing to do here.
So, what is that?

653
00:42:15,000 --> 00:42:21,000
That's s squared plus s plus
one.

654
00:42:20,000 --> 00:42:26,000
So, it's s squared plus s plus
one divided by s plus one,

655
00:42:25,000 --> 00:42:31,000
okay?
I'm still not done because now

656
00:42:28,000 --> 00:42:34,000
we have to know,
what's Y?

657
00:42:31,000 --> 00:42:37,000
All right, now we have to
think.

658
00:42:35,000 --> 00:42:41,000
What we're going to do is get Y
in this form.

659
00:42:38,000 --> 00:42:44,000
But, I want it in the form in
which it's most suited for using

660
00:42:42,000 --> 00:42:48,000
partial fractions.
In other words,

661
00:42:44,000 --> 00:42:50,000
I want the denominator as
factored as I possibly can be.

662
00:42:48,000 --> 00:42:54,000
Okay, well, the numerator is
going to be just what it was.

663
00:42:52,000 --> 00:42:58,000
How should I write the
denominator?

664
00:42:55,000 --> 00:43:01,000
Well, the denominator is going
to have the factor s plus one

665
00:42:59,000 --> 00:43:05,000
in it from here.
But after I divide through,

666
00:43:04,000 --> 00:43:10,000
the other factor will be s
squared minus one,

667
00:43:09,000 --> 00:43:15,000
right?
But, s squared minus one is s

668
00:43:12,000 --> 00:43:18,000
minus one times s plus one.

669
00:43:17,000 --> 00:43:23,000
So, I have to divide this by s
squared minus one.

670
00:43:23,000 --> 00:43:29,000
Factored, it's this.
So, the end result is there are

671
00:43:27,000 --> 00:43:33,000
two of these and one of the
other.

672
00:43:32,000 --> 00:43:38,000
And now, it's ready to be used.
It's better to be a partial

673
00:43:36,000 --> 00:43:42,000
fraction.
So, the final step is to use a

674
00:43:40,000 --> 00:43:46,000
partial fraction's decomposition
on this so that you can

675
00:43:44,000 --> 00:43:50,000
calculate its inverse Laplace
transform.

676
00:43:48,000 --> 00:43:54,000
So, let's do that.
Okay, (s squared plus s plus

677
00:43:51,000 --> 00:43:57,000
one) divided by that thing,
(s plus one) squared times (s

678
00:43:56,000 --> 00:44:02,000
minus one) equals s plus

679
00:44:01,000 --> 00:44:07,000
one squared plus s
plus one plus s minus one.

680
00:44:07,000 --> 00:44:13,000
In the top will be constants,

681
00:44:12,000 --> 00:44:18,000
just constants.
Let's do it this way first,

682
00:44:15,000 --> 00:44:21,000
and I'll say at the very end,
something else.

683
00:44:18,000 --> 00:44:24,000
Maybe now.
Many of you are upset.

684
00:44:21,000 --> 00:44:27,000
Some of you are upset.
I know this for a fact because

685
00:44:25,000 --> 00:44:31,000
in high school,
or wherever you learned to do

686
00:44:28,000 --> 00:44:34,000
this before, there weren't two
terms here.

687
00:44:33,000 --> 00:44:39,000
There was just one term,
s plus one squared.

688
00:44:36,000 --> 00:44:42,000
If you do it that way,

689
00:44:39,000 --> 00:44:45,000
then it's all right.
Then, it's all right,

690
00:44:42,000 --> 00:44:48,000
but I don't recommend it.
In that case,

691
00:44:45,000 --> 00:44:51,000
the numerators will not be
constants.

692
00:44:48,000 --> 00:44:54,000
But, if you just have that,
then because this is a

693
00:44:52,000 --> 00:44:58,000
quadratic polynomial all by
itself.

694
00:44:54,000 --> 00:45:00,000
You've got to have a linear
polynomial, a s plus b

695
00:44:59,000 --> 00:45:05,000
in the top, see?

696
00:45:02,000 --> 00:45:08,000
So, you must have a s plus b
here,

697
00:45:04,000 --> 00:45:10,000
as I'm sure you remember if
that's the way you learned to do

698
00:45:08,000 --> 00:45:14,000
it. But, to do cover-up,

699
00:45:09,000 --> 00:45:15,000
the best way as much as
possible to separate out the

700
00:45:12,000 --> 00:45:18,000
terms.
If this were a cubic term,

701
00:45:14,000 --> 00:45:20,000
God forbid, s plus one cubed,

702
00:45:16,000 --> 00:45:22,000
then you'd have to have s plus
one cubed,

703
00:45:20,000 --> 00:45:26,000
s plus one squared.

704
00:45:23,000 --> 00:45:29,000
Okay, I won't give you anything
bigger than quadratic.

705
00:45:26,000 --> 00:45:32,000
[LAUGHTER]
You can trust me.

706
00:45:29,000 --> 00:45:35,000
Okay, now, what can we find by
the cover up method?

707
00:45:33,000 --> 00:45:39,000
Well, surely this.
Cover up the s minus one,

708
00:45:37,000 --> 00:45:43,000
put s equals one,
and I get three divided by two

709
00:45:42,000 --> 00:45:48,000
squared, four.
So, this is three quarters.

710
00:45:45,000 --> 00:45:51,000
Now, in this,

711
00:45:48,000 --> 00:45:54,000
you can always find the highest
power by cover-up because,

712
00:45:52,000 --> 00:45:58,000
cover it up,
put s equals negative one,

713
00:45:56,000 --> 00:46:02,000
and you get one minus one
plus one.

714
00:46:02,000 --> 00:46:08,000
So, one up there,
negative one here makes

715
00:46:04,000 --> 00:46:10,000
negative two here.
So, one over negative two.

716
00:46:07,000 --> 00:46:13,000
So, it's minus one half.

717
00:46:09,000 --> 00:46:15,000
Now, this you cannot determine

718
00:46:12,000 --> 00:46:18,000
by cover-up because you'd want
to cover-up just one of these s

719
00:46:16,000 --> 00:46:22,000
plus ones.
But then you can't put s equals

720
00:46:19,000 --> 00:46:25,000
negative one because
you get infinity.

721
00:46:22,000 --> 00:46:28,000
You get zero there,
makes infinity.

722
00:46:24,000 --> 00:46:30,000
So, this must be determined
some other way,

723
00:46:27,000 --> 00:46:33,000
either by undetermined
coefficients,

724
00:46:29,000 --> 00:46:35,000
or if you've just got one
thing, for heaven's sake,

725
00:46:32,000 --> 00:46:38,000
just make a substitution.
See, this is supposed to be

726
00:46:37,000 --> 00:46:43,000
true.
This is an algebraic identity,

727
00:46:40,000 --> 00:46:46,000
true for all values of the
variable, and therefore,

728
00:46:43,000 --> 00:46:49,000
it ought to be true when s
equals zero,

729
00:46:47,000 --> 00:46:53,000
for instance.
Why zero?

730
00:46:48,000 --> 00:46:54,000
Well, because I haven't used it
yet.

731
00:46:51,000 --> 00:46:57,000
I used negative one and
positive one,

732
00:46:53,000 --> 00:46:59,000
but I didn't use zero.
Okay, let's use zero.

733
00:46:56,000 --> 00:47:02,000
Put s equals zero.
What do we get?

734
00:47:00,000 --> 00:47:06,000
Well, on the left-hand side,
I get one divided by one

735
00:47:03,000 --> 00:47:09,000
squared, negative.
So, I get minus one on the left

736
00:47:06,000 --> 00:47:12,000
hand side equals,
what do I get on the right?

737
00:47:09,000 --> 00:47:15,000
Put s equals zero,
you get negative one half.

738
00:47:12,000 --> 00:47:18,000
Well, this is the number I'm

739
00:47:15,000 --> 00:47:21,000
trying to find.
So, let's write that simply as

740
00:47:18,000 --> 00:47:24,000
plus c, putting s equals zero.
s equals zero here gives me

741
00:47:21,000 --> 00:47:27,000
negative three quarters.

742
00:47:23,000 --> 00:47:29,000
Okay, what's c?
This is minus a half,

743
00:47:26,000 --> 00:47:32,000
minus three quarters,
is minus five quarters.

744
00:47:29,000 --> 00:47:35,000
Put it on the other side,

745
00:47:32,000 --> 00:47:38,000
minus one plus five quarters is
plus one quarter.

746
00:47:35,000 --> 00:47:41,000
So, c equals one quarter.

747
00:47:39,000 --> 00:47:45,000
And now, we are ready to do the

748
00:47:42,000 --> 00:47:48,000
final step.
Take the inverse Laplace

749
00:47:44,000 --> 00:47:50,000
transform.
You see what I said when I said

750
00:47:47,000 --> 00:47:53,000
that all the work is in this
last step?

751
00:47:49,000 --> 00:47:55,000
Just look how much of the work,
how much of the board is

752
00:47:53,000 --> 00:47:59,000
devoted to the first two steps,
and how much is going to be

753
00:47:57,000 --> 00:48:03,000
devoted to the last step?
Okay, so we get e to the

754
00:48:01,000 --> 00:48:07,000
inverse Laplace transform.
Well, the first term is the

755
00:48:05,000 --> 00:48:11,000
hardest.
Let's let that go for the

756
00:48:08,000 --> 00:48:14,000
moment.
So, I leave a space for it,

757
00:48:10,000 --> 00:48:16,000
and then we will have one
quarter.

758
00:48:13,000 --> 00:48:19,000
Well, one over s plus one is,

759
00:48:16,000 --> 00:48:22,000
that's just the exponential
formula.

760
00:48:19,000 --> 00:48:25,000
One over s plus one would be e
to the negative t,e to the minus

761
00:48:24,000 --> 00:48:30,000
one times t.
So, it's one quarter e to the

762
00:48:28,000 --> 00:48:34,000
minus one times t.

763
00:48:32,000 --> 00:48:38,000
And, how about the next thing
would be three quarters times,

764
00:48:36,000 --> 00:48:42,000
well, here it's negative one,
so that's e to the plus t.

765
00:48:41,000 --> 00:48:47,000
Notice how those signs work.

766
00:48:44,000 --> 00:48:50,000
And, that just leaves us the
Laplace transform of this thing.

767
00:48:49,000 --> 00:48:55,000
Now, you look at it and you
say, this Laplace transform

768
00:48:54,000 --> 00:49:00,000
happened in two steps.
I took something and I got,

769
00:48:58,000 --> 00:49:04,000
essentially,
one over s squared.

770
00:49:03,000 --> 00:49:09,000
And then, I changed s to s plus
one.

771
00:49:08,000 --> 00:49:14,000
All right, what gives one over
s squared?

772
00:49:13,000 --> 00:49:19,000
The Laplace transform of what
is one over s squared?

773
00:49:18,000 --> 00:49:24,000
t, you say to yourself,
one over s to some power is

774
00:49:23,000 --> 00:49:29,000
essentially some power of t.
And then, you look at the

775
00:49:28,000 --> 00:49:34,000
formula.
Notice at the top is one

776
00:49:31,000 --> 00:49:37,000
factorial, which is one,
of course.

777
00:49:34,000 --> 00:49:40,000
Okay, now, then how do I
convert this to one over s plus

778
00:49:39,000 --> 00:49:45,000
one squared?
That's the exponential shift

779
00:49:44,000 --> 00:49:50,000
formula.
If you know what the Laplace

780
00:49:47,000 --> 00:49:53,000
transform, so the first formula
in the middle of the board on

781
00:49:51,000 --> 00:49:57,000
the top, there,
if you know what,

782
00:49:54,000 --> 00:50:00,000
change s to s plus one,
corresponds to

783
00:49:58,000 --> 00:50:04,000
multiplying by e to the t.

784
00:50:03,000 --> 00:50:09,000
So, it is t times e to the
negative t.

785
00:50:06,000 --> 00:50:12,000
Sorry, that corresponds to
this.

786
00:50:08,000 --> 00:50:14,000
So, this is the exponential
shift formula.

787
00:50:11,000 --> 00:50:17,000
If t goes to one over s squared,
then t e to the

788
00:50:15,000 --> 00:50:21,000
minus t goes to one
over s plus one squared.

789
00:50:20,000 --> 00:50:26,000
Okay, but there's a constant

790
00:50:22,000 --> 00:50:28,000
out front.
So, it's minus one half t e to

791
00:50:25,000 --> 00:50:31,000
the negative t.

792
00:50:27,000 --> 00:50:33,000
Now, tell me,
what parts of this solution,

793
00:50:30,000 --> 00:50:36,000
oh boy, we're over time.
But, notice,

794
00:50:32,000 --> 00:50:38,000
this is what would have been
the particular solution,

795
00:50:36,000 --> 00:50:42,000
(y)p before,
and this is the stuff that

796
00:50:39,000 --> 00:50:45,000
occurs in the complementary
function, but already the

797
00:50:42,000 --> 00:50:48,000
appropriate constants have been
supplied for the coefficients.

798
00:50:49,000 --> 00:50:55,000
You don't have to calculate
them separately.

799
00:50:52,000 --> 00:50:58,000
They were built into the
method.

800
00:50:54,000 --> 00:51:00,000
Okay, good luck on Friday,
and see you there.