1
00:00:18 --> 00:00:24
Okay, those are the formulas.
You will get all of those on
2
00:00:23 --> 00:00:29
the test, plus a couple more
that I will give you today.
3
00:00:29 --> 00:00:35
Those will be the basic
formulas of the Laplace
4
00:00:33 --> 00:00:39
transform.
If I think you need anything
5
00:00:37 --> 00:00:43
else, I'll give you other stuff,
too.
6
00:00:42 --> 00:00:48
So, I'm going to leave those on
the board all period.
7
00:00:46 --> 00:00:52
The basic test for today is to
see how Laplace transforms are
8
00:00:51 --> 00:00:57
used to solve linear
differential equations with
9
00:00:54 --> 00:01:00
constant coefficients.
Now, to do that,
10
00:00:57 --> 00:01:03
we're going to have to take the
Laplace transform of a
11
00:01:02 --> 00:01:08
derivative.
And, in order to make sense of
12
00:01:06 --> 00:01:12
that procedure,
we're going to have to ask,
13
00:01:09 --> 00:01:15
I apologize in advance,
but a slightly theoretical
14
00:01:12 --> 00:01:18
question, namely,
we have to have some guarantee
15
00:01:15 --> 00:01:21
in advance that the Laplace
transform is going to exist.
16
00:01:19 --> 00:01:25
Now, how could the Laplace
transform fail to exist?
17
00:01:23 --> 00:01:29
Can't I always calculate this?
And the answer is,
18
00:01:26 --> 00:01:32
no, you can't always calculate
it because this is an improper
19
00:01:30 --> 00:01:36
integral.
I'm integrating all the way up
20
00:01:33 --> 00:01:39
to infinity, and you know that
improper integrals don't always
21
00:01:38 --> 00:01:44
converge.
You know, if the integrand for
22
00:01:42 --> 00:01:48
example just didn't have the
exponential factor there,
23
00:01:46 --> 00:01:52
were simply t dt,
that it might look like it made
24
00:01:50 --> 00:01:56
sense, but that integral doesn't
converge.
25
00:01:53 --> 00:01:59
And, anyway,
it has no value.
26
00:01:56 --> 00:02:02
So, I need conditions in
advance, which guarantee that
27
00:02:00 --> 00:02:06
the Laplace transforms will
exist.
28
00:02:04 --> 00:02:10
Only under those circumstances
will the formulas make any
29
00:02:07 --> 00:02:13
sense.
Now, there is a standard
30
00:02:09 --> 00:02:15
condition that's in your book.
But, I didn't get a chance to
31
00:02:13 --> 00:02:19
talk about it last time.
So, I thought I'd better spent
32
00:02:16 --> 00:02:22
the first few minutes today
talking about the condition
33
00:02:20 --> 00:02:26
because it's what we're going to
need in order to be able to
34
00:02:23 --> 00:02:29
solve differential equations.
The condition that makes the
35
00:02:27 --> 00:02:33
Laplace transform definitely
exist for a function is that f
36
00:02:31 --> 00:02:37
of t shouldn't grow too
rapidly.
37
00:02:35 --> 00:02:41
It can grow rapidly.
It can grow because the e to
38
00:02:38 --> 00:02:44
the minus s t is
pulling it down,
39
00:02:41 --> 00:02:47
trying hard to pull it down to
zero to make the integral
40
00:02:45 --> 00:02:51
converge.
All we have to do is to
41
00:02:47 --> 00:02:53
guarantee that it doesn't grow
so rapidly that the e to the
42
00:02:52 --> 00:02:58
minus s t is powerless to pull
it down.
43
00:02:54 --> 00:03:00
Now, the condition is it's
what's called a growth
44
00:02:58 --> 00:03:04
condition.
These are very important in
45
00:03:02 --> 00:03:08
applications,
and unfortunately,
46
00:03:04 --> 00:03:10
it's always taught in 18.01,
but it's not always taught in
47
00:03:09 --> 00:03:15
high school calculus.
And, it's a question of how
48
00:03:13 --> 00:03:19
fast the function is allowed to
grow.
49
00:03:16 --> 00:03:22
And, the condition is
universally said this way,
50
00:03:20 --> 00:03:26
should be of exponential type.
So, what I'm defining is the
51
00:03:24 --> 00:03:30
phrase "exponential type." I'll
put it in quotation marks for
52
00:03:29 --> 00:03:35
that reason.
What does this mean?
53
00:03:33 --> 00:03:39
It's a condition,
a growth condition on a
54
00:03:36 --> 00:03:42
function, says how fast it can
get big.
55
00:03:40 --> 00:03:46
It says that f of t in
size, since f of t might get
56
00:03:44 --> 00:03:50
negatively very large,
and that would hurt,
57
00:03:48 --> 00:03:54
make the integral hard to
converge, not likely to
58
00:03:52 --> 00:03:58
converge, use the absolute
value.
59
00:03:54 --> 00:04:00
In other words,
I don't care if f of t is going
60
00:03:58 --> 00:04:04
up or going down very low.
Whichever way it goes,
61
00:04:02 --> 00:04:08
its size should not be bigger
than a rapidly growing
62
00:04:06 --> 00:04:12
exponential.
And, here's a rapidly growing
63
00:04:10 --> 00:04:16
exponential.
c is some positive constant,
64
00:04:14 --> 00:04:20
for some positive constant c
and some positive constant k.
65
00:04:18 --> 00:04:24
And, this should be true for
all values of t.
66
00:04:20 --> 00:04:26
All t greater than or equal to
zero.
67
00:04:22 --> 00:04:28
I don't have to worry about
negative values of t because the
68
00:04:26 --> 00:04:32
integral doesn't care about
them.
69
00:04:28 --> 00:04:34
I'm only doing the integration
as t runs from zero to infinity.
70
00:04:33 --> 00:04:39
In other words,
f of t could have been
71
00:04:38 --> 00:04:44
an extremely wild function,
sewn a lot of oats or whatever
72
00:04:44 --> 00:04:50
functions do for negative values
of t, and we don't care.
73
00:04:51 --> 00:04:57
It's only what's happening from
now from time zero onto
74
00:04:57 --> 00:05:03
infinity.
As long as it behaves now,
75
00:05:01 --> 00:05:07
from now on,
it's okay.
76
00:05:05 --> 00:05:11
All right, so,
the way it should behave is by
77
00:05:07 --> 00:05:13
being an exponential type.
Now, to try to give you some
78
00:05:11 --> 00:05:17
feeling for what this means,
these functions,
79
00:05:14 --> 00:05:20
for example, if k is 100,
80
00:05:16 --> 00:05:22
do you have any idea what the
plot of e to the 100t
81
00:05:20 --> 00:05:26
looks like?
It goes straight up.
82
00:05:23 --> 00:05:29
On every computer you try to
plot it on, e to the 100t
83
00:05:26 --> 00:05:32
goes like that
unless, of course,
84
00:05:29 --> 00:05:35
you make the scale t equals
zero to, over here,
85
00:05:32 --> 00:05:38
is one millionth.
Well, even that won't do.
86
00:05:37 --> 00:05:43
Okay, so these functions really
can grow quite rapidly.
87
00:05:41 --> 00:05:47
Let's take an example and see
what's of exponential type,
88
00:05:46 --> 00:05:52
and then perhaps even more
interestingly,
89
00:05:50 --> 00:05:56
what isn't.
The function sine t,
90
00:05:53 --> 00:05:59
is that of exponential type?
Well, sure.
91
00:05:57 --> 00:06:03
Its absolute value is always
less than or equal to one.
92
00:06:01 --> 00:06:07
So, it's also this paradigm.
If I take c equal to one,
93
00:06:06 --> 00:06:12
and what should I take k to be?
Zero.
94
00:06:11 --> 00:06:17
Take k to be zero,
c equals one,
95
00:06:13 --> 00:06:19
and in fact sine t
plays that condition.
96
00:06:18 --> 00:06:24
Here's one that's more
interesting, t to the n.
97
00:06:23 --> 00:06:29
Think of t to the 100th power.
98
00:06:26 --> 00:06:32
Is that smaller than some
exponential with maybe a
99
00:06:31 --> 00:06:37
constant out front?
Well, t to the 100th power
100
00:06:35 --> 00:06:41
goes straight up,
also.
101
00:06:37 --> 00:06:43
Well, we feel that if we make
the exponential big enough,
102
00:06:41 --> 00:06:47
maybe it will win out.
In fact, you don't have to make
103
00:06:45 --> 00:06:51
the exponential big.
k equals one is good
104
00:06:48 --> 00:06:54
enough.
In other words,
105
00:06:49 --> 00:06:55
I don't have to put absolute
value signs around the t to the
106
00:06:53 --> 00:06:59
n because I'm only
thinking about t as being a
107
00:06:57 --> 00:07:03
positive number,
anyway.
108
00:07:00 --> 00:07:06
I say that that's less than or
equal to some constant M,
109
00:07:04 --> 00:07:10
positive constant M times e to
the t will be good
110
00:07:10 --> 00:07:16
enough for some M and all t.
Now, why is that?
111
00:07:14 --> 00:07:20
Why is that?
The way to think of that,
112
00:07:17 --> 00:07:23
so, what this proves is that,
therefore, t to the n
113
00:07:22 --> 00:07:28
is of exponential type,
which we could have guessed
114
00:07:27 --> 00:07:33
because after all we were able
to calculate its Laplace
115
00:07:31 --> 00:07:37
transform.
Now, just because you can
116
00:07:36 --> 00:07:42
calculate the Laplace transform
doesn't mean it's of exponential
117
00:07:41 --> 00:07:47
type, but in practical matters,
it almost always does.
118
00:07:46 --> 00:07:52
So, t to the n is of
exponential type.
119
00:07:50 --> 00:07:56
How do you prove that?
Well, the weighted secret is to
120
00:07:55 --> 00:08:01
look at t to the n divided by e
to the t.
121
00:07:59 --> 00:08:05
In other words,
look at the quotient.
122
00:08:04 --> 00:08:10
What I'd like to argue is that
this is bounded by some number,
123
00:08:11 --> 00:08:17
capital M.
That's the question I'm asking.
124
00:08:15 --> 00:08:21
Now, why is this so?
Well, I think I can convince
125
00:08:21 --> 00:08:27
you of it without having to work
very hard.
126
00:08:26 --> 00:08:32
What does the graph of this
function look like?
127
00:08:33 --> 00:08:39
It starts here,
so I'm graphing this function,
128
00:08:37 --> 00:08:43
this ratio.
When t is equal to zero,
129
00:08:41 --> 00:08:47
its value is zero,
right, because of the
130
00:08:45 --> 00:08:51
numerator.
What happens as t goes to
131
00:08:48 --> 00:08:54
infinity?
What happens to this?
132
00:08:51 --> 00:08:57
What does it approach?
Zero.
133
00:08:54 --> 00:09:00
And, why?
By L'Hop.
134
00:08:56 --> 00:09:02
By L'Hopital's rule.
Just keep differentiating,
135
00:09:01 --> 00:09:07
reapply the rule over and over,
keep differentiating it n
136
00:09:06 --> 00:09:12
times, and finally you'll have
won the numerator down to t to
137
00:09:12 --> 00:09:18
the zero,
which isn't doing anything
138
00:09:17 --> 00:09:23
much.
And, the denominator,
139
00:09:21 --> 00:09:27
no matter how many times you
differentiate it,
140
00:09:25 --> 00:09:31
it's still t,
to the t all the time.
141
00:09:27 --> 00:09:33
So, by using Lopital's rule n
times, you change the top to one
142
00:09:32 --> 00:09:38
or n factorial,
actually; the bottom stays e to
143
00:09:36 --> 00:09:42
the t,
and the ratio clearly
144
00:09:39 --> 00:09:45
approaches zero,
and therefore,
145
00:09:41 --> 00:09:47
it approached zero to start
with.
146
00:09:45 --> 00:09:51
So, I don't know what this
function's doing in between.
147
00:09:48 --> 00:09:54
It's a positive function.
It's continuous because the top
148
00:09:51 --> 00:09:57
and bottom are continuous,
and the bottom is never zero.
149
00:09:54 --> 00:10:00
So, it's a continuous function
which starts out at zero and is
150
00:09:58 --> 00:10:04
positive, and as t goes to
infinity, it gets closer and
151
00:10:01 --> 00:10:07
closer to the t-axis,
again.
152
00:10:03 --> 00:10:09
Well, what does t to the n
do?
153
00:10:06 --> 00:10:12
It might wave around.
It doesn't actually.
154
00:10:10 --> 00:10:16
But, the point is,
because it's continuous,
155
00:10:15 --> 00:10:21
starts at zero,
ends at zero,
156
00:10:19 --> 00:10:25
it's bounded.
It has a maximum somewhere.
157
00:10:23 --> 00:10:29
And, that maximum is M.
So, it has a maximum.
158
00:10:30 --> 00:10:36
All you have to know is where
it starts, and where it ends up,
159
00:10:34 --> 00:10:40
and the fact that it's
continuous.
160
00:10:37 --> 00:10:43
That guarantees that it has a
maximum.
161
00:10:39 --> 00:10:45
So, it is less than some
maximum, and that shows that
162
00:10:43 --> 00:10:49
it's of exponential type.
Now, of course,
163
00:10:46 --> 00:10:52
before you get the idea that
everything's of exponential
164
00:10:50 --> 00:10:56
type, let's see what isn't.
I'll give you two functions
165
00:10:54 --> 00:11:00
that are not of exponential
type, for different reasons.
166
00:11:00 --> 00:11:06
One over t is not of
exponential type.
167
00:11:03 --> 00:11:09
Well, of course,
it's not defined that t equals
168
00:11:06 --> 00:11:12
zero. But, you know,
169
00:11:08 --> 00:11:14
it's okay for an integral not
to be defined at one point
170
00:11:12 --> 00:11:18
because you're measuring an
area, and when you measure an
171
00:11:16 --> 00:11:22
area, what happened to one point
doesn't really matter much.
172
00:11:20 --> 00:11:26
That's not the thing.
What's wrong with one over t is
173
00:11:24 --> 00:11:30
that the integral doesn't
converge at zero times one over
174
00:11:28 --> 00:11:34
t dt.
That integral,
175
00:11:32 --> 00:11:38
when t is near zero,
this is approximately equal to
176
00:11:36 --> 00:11:42
one, right?
If t is zero,
177
00:11:38 --> 00:11:44
this is one.
So, it's like the function,
178
00:11:41 --> 00:11:47
integral from zero to infinity
of one over t,
179
00:11:46 --> 00:11:52
near zero it's close to,
180
00:11:50 --> 00:11:56
it's like the integral from
zero to someplace of no
181
00:11:54 --> 00:12:00
importance, dt over t.
182
00:11:57 --> 00:12:03
But, this does not converge.
This is like log t,
183
00:12:01 --> 00:12:07
and log zero is minus infinity.
So, it doesn't converge.
184
00:12:08 --> 00:12:14
So, one over t is not
of exponential type.
185
00:12:11 --> 00:12:17
So, what's the Laplace
transform of one over t?
186
00:12:14 --> 00:12:20
It doesn't have a Laplace
187
00:12:17 --> 00:12:23
transform.
Well, what if I put t equals
188
00:12:20 --> 00:12:26
negative n?
What about t to the minus one?
189
00:12:23 --> 00:12:29
Well, that only works for
190
00:12:26 --> 00:12:32
positive integers,
not negative integers.
191
00:12:30 --> 00:12:36
Okay, so it's not of
exponential type.
192
00:12:33 --> 00:12:39
However, that's because it
never really gets started
193
00:12:37 --> 00:12:43
properly.
It's more fun to look at a
194
00:12:40 --> 00:12:46
function which is not of
exponential type because it
195
00:12:45 --> 00:12:51
grows too fast.
Now, what's a function that
196
00:12:49 --> 00:12:55
grows faster that it grows so
rapidly that you can't find any
197
00:12:54 --> 00:13:00
function e to the k t
which bounds it?
198
00:12:58 --> 00:13:04
A function which grows too
rapidly, a simple one is e to
199
00:13:03 --> 00:13:09
the t squared,
grows too rapidly to be of
200
00:13:07 --> 00:13:13
exponential type.
And, the argument is simple.
201
00:13:13 --> 00:13:19
No matter what you propose,
it's always,
202
00:13:17 --> 00:13:23
for the K, no matter how big a
number, use Avogadro's number,
203
00:13:22 --> 00:13:28
use anything you want.
Ultimately, this is going to be
204
00:13:27 --> 00:13:33
bigger than k t no matter how
big k is, no matter how big k
205
00:13:32 --> 00:13:38
is.
When is this going to happen?
206
00:13:36 --> 00:13:42
This will happen if t squared
is bigger than k t.
207
00:13:40 --> 00:13:46
In other words,
as soon as t becomes bigger
208
00:13:43 --> 00:13:49
than k, you might have to wait
quite a while for that to
209
00:13:47 --> 00:13:53
happen, but, as soon as t gets
bigger than 10 to the 10 to the
210
00:13:52 --> 00:13:58
23,
this e to the t squared will be
211
00:13:55 --> 00:14:01
bigger than e to the 10 to the
10 to the 23 times t.
212
00:14:00 --> 00:14:06
So, e to the t squared,
213
00:14:04 --> 00:14:10
it's a simple
function, a simple elementary
214
00:14:07 --> 00:14:13
function.
It grows so rapidly it doesn't
215
00:14:10 --> 00:14:16
have a Laplace transform.
Okay, so how are we going to
216
00:14:13 --> 00:14:19
solve differential equations if
e to the t squared?
217
00:14:16 --> 00:14:22
I won't give you any.
And, the reason I won't give
218
00:14:19 --> 00:14:25
you any: because I never saw one
occur in real life.
219
00:14:22 --> 00:14:28
Nature, like sines,
cosines, exponentials,
220
00:14:25 --> 00:14:31
are fine, I've never seen a
physical, you know,
221
00:14:28 --> 00:14:34
this is just my ignorance.
But, I've never seen a physical
222
00:14:31 --> 00:14:37
problem that involved a function
growing as rapidly as e to the t
223
00:14:35 --> 00:14:41
squared.
That may be just my ignorance.
224
00:14:40 --> 00:14:46
But, I do know the Laplace
transform won't be used to solve
225
00:14:44 --> 00:14:50
differential equations involving
such a function.
226
00:14:48 --> 00:14:54
How about e to the minus t
squared?
227
00:14:52 --> 00:14:58
That's different.
It looks almost the same,
228
00:14:55 --> 00:15:01
but e to the minus t squared
does this.
229
00:14:58 --> 00:15:04
It's very well-behaved.
That's the curve,
230
00:15:02 --> 00:15:08
of course, that you're all
afraid of.
231
00:15:04 --> 00:15:10
Don't panic.
Okay.
232
00:15:07 --> 00:15:13
So, I'd like to explain to you
now how differential equations,
233
00:15:13 --> 00:15:19
maybe I should save-- I'll tell
you what.
234
00:15:17 --> 00:15:23
We need more formulas.
So, I'll put them,
235
00:15:20 --> 00:15:26
why don't I save this board,
and instead,
236
00:15:24 --> 00:15:30
I'll describe to you the basic
way Laplace transforms are used
237
00:15:30 --> 00:15:36
to solve differential equations,
what are they called,
238
00:15:35 --> 00:15:41
a paradigm.
I'll show you the paradigm,
239
00:15:39 --> 00:15:45
and then we'll fill in the
holes so you have some overall
240
00:15:43 --> 00:15:49
view of how the procedure goes,
and then you'll understand
241
00:15:46 --> 00:15:52
where the various pieces fit
into it.
242
00:15:48 --> 00:15:54
I think you'll understand it
better that way.
243
00:15:51 --> 00:15:57
So, what do we do?
Start with the differential
244
00:15:54 --> 00:16:00
equation.
But, right away,
245
00:15:55 --> 00:16:01
there's a fundamental
difference between what the
246
00:15:58 --> 00:16:04
Laplace transform does,
and what we've been doing up
247
00:16:01 --> 00:16:07
until now, namely,
what you have to start with is
248
00:16:04 --> 00:16:10
not merely the differential
equation.
249
00:16:08 --> 00:16:14
Let's say we have linear with
constant coefficients.
250
00:16:12 --> 00:16:18
It's almost never used to solve
any other type of problem.
251
00:16:16 --> 00:16:22
And, let's take second order so
I don't have to do,
252
00:16:20 --> 00:16:26
because that's the kind we've
been working with all term.
253
00:16:25 --> 00:16:31
But, it's allowed to be
inhomogeneous,
254
00:16:28 --> 00:16:34
so, f of t.
Let's call the something else,
255
00:16:32 --> 00:16:38
another letter,
h of t.
256
00:16:36 --> 00:16:42
I'll want f of t for
the function I'm taking the
257
00:16:39 --> 00:16:45
Laplace transform of.
All right, now,
258
00:16:42 --> 00:16:48
the difference is that up to
now, you know techniques for
259
00:16:45 --> 00:16:51
solving this just as it stands.
The Laplace transform does not
260
00:16:50 --> 00:16:56
know how to solve this just
doesn't stands.
261
00:16:52 --> 00:16:58
The Laplace transform must have
an initial value problem.
262
00:16:56 --> 00:17:02
In other words,
you must supply from the
263
00:16:59 --> 00:17:05
beginning the initial conditions
that the y is to satisfy.
264
00:17:04 --> 00:17:10
Now, I don't want to say any
specific numbers,
265
00:17:06 --> 00:17:12
so I'll use generic numbers.
Well, but look,
266
00:17:08 --> 00:17:14
what do we do if we get a
problem and there are no initial
267
00:17:12 --> 00:17:18
conditions; does that mean we
can't use the Laplace transform?
268
00:17:15 --> 00:17:21
No, of course you can use it.
But, you will just have to
269
00:17:18 --> 00:17:24
assume the initial conditions
are on the numbers.
270
00:17:21 --> 00:17:27
You'll say it but the initial
conditions be y sub zero
271
00:17:24 --> 00:17:30
and y zero prime,
or whatever,
272
00:17:26 --> 00:17:32
a and b, whatever you want to
call it.
273
00:17:30 --> 00:17:36
And now, the answer,
then, will involve the a and
274
00:17:34 --> 00:17:40
the b or the y zero and the y
zero prime.
275
00:17:38 --> 00:17:44
But, you must,
at least, give lip service to
276
00:17:42 --> 00:17:48
the initial conditions,
whereas before we didn't have
277
00:17:46 --> 00:17:52
to do that.
Now, depending on your point of
278
00:17:50 --> 00:17:56
view, that's a grave defect,
or it is, so what?
279
00:17:54 --> 00:18:00
Let's adopt the so what point
of view.
280
00:17:57 --> 00:18:03
So, there's our problem.
It's an initial value problem.
281
00:18:03 --> 00:18:09
How is it solved by the Laplace
transform?
282
00:18:05 --> 00:18:11
Well, the idea is you take the
Laplace transform of this
283
00:18:09 --> 00:18:15
differential equation and the
initial conditions.
284
00:18:12 --> 00:18:18
So, I'm going to explain to you
how to do that.
285
00:18:15 --> 00:18:21
Not right now,
because we're going to need,
286
00:18:17 --> 00:18:23
first, the Laplace transform of
a derivative,
287
00:18:20 --> 00:18:26
the formula for that.
You don't know that yet.
288
00:18:23 --> 00:18:29
But when you do know it,
you will be able to take the
289
00:18:26 --> 00:18:32
Laplace transform of the initial
value problem.
290
00:18:29 --> 00:18:35
So, I'll put the little l here,
and what comes out is,
291
00:18:32 --> 00:18:38
well, y of t is the
solution to the original
292
00:18:36 --> 00:18:42
problem.
If y of t is the
293
00:18:39 --> 00:18:45
function which satisfies that
equation and these initial
294
00:18:43 --> 00:18:49
conditions, its Laplace
transform, let's call it capital
295
00:18:48 --> 00:18:54
Y, that's our standard notation,
but it's going to be of a new
296
00:18:52 --> 00:18:58
variable, s.
So, when I take the Laplace
297
00:18:55 --> 00:19:01
transform of the differential
equation with the initial
298
00:18:59 --> 00:19:05
conditions, what comes out is an
algebraic-- the emphasis is on
299
00:19:04 --> 00:19:10
algebraic: no derivatives,
no transcendental functions,
300
00:19:08 --> 00:19:14
nothing like that,
an algebraic equation,
301
00:19:11 --> 00:19:17
m Y of s.
302
00:19:14 --> 00:19:20
303
00:19:22 --> 00:19:28
And, now what?
Well, now, in the domain of s,
304
00:19:25 --> 00:19:31
it's easy to solve this
algebraic equation.
305
00:19:28 --> 00:19:34
Not all algebraic equations are
easy to solve for the capital Y.
306
00:19:33 --> 00:19:39
But, the ones you will get will
always be, not because I am
307
00:19:37 --> 00:19:43
making life easy for you,
but that's the way the Laplace
308
00:19:42 --> 00:19:48
transform works.
So, you will solve it for Y.
309
00:19:45 --> 00:19:51
And, the answer will always
come out to be Y equals,
310
00:19:49 --> 00:19:55
Y of s equals some
rational function,
311
00:19:52 --> 00:19:58
some quotient of polynomials in
s, a polynomial in s divided by
312
00:19:57 --> 00:20:03
some other polynomial in s.
313
00:20:01 --> 00:20:07
314
00:20:09 --> 00:20:15
And, now what?
Well, this is the Laplace
315
00:20:12 --> 00:20:18
transform of the answer.
This is the Laplace transform
316
00:20:17 --> 00:20:23
of the solution we are looking
for.
317
00:20:20 --> 00:20:26
So, the final step is to go
backwards by taking the inverse
318
00:20:25 --> 00:20:31
Laplace transform of this guy.
And, what will you get?
319
00:20:30 --> 00:20:36
Well, you will get y equals the
y of t that we are
320
00:20:35 --> 00:20:41
looking for.
It's really a wildly improbable
321
00:20:39 --> 00:20:45
procedure.
In other words,
322
00:20:41 --> 00:20:47
instead of going from here to
here, you have to imagine
323
00:20:45 --> 00:20:51
there's a mountain here.
And, the only way to get around
324
00:20:48 --> 00:20:54
it is to go, first,
here, and then cross the stream
325
00:20:51 --> 00:20:57
here, and then go back up,
and go back up.
326
00:20:54 --> 00:21:00
It looks like a senseless
procedure, what do they call it,
327
00:20:58 --> 00:21:04
going around Robin Hood's barn,
it was called when I was a,
328
00:21:01 --> 00:21:07
I don't know why it's called
that.
329
00:21:05 --> 00:21:11
But that's what we used to call
it; not Laplace transform.
330
00:21:09 --> 00:21:15
That was just a generic thing
when you had to do something
331
00:21:14 --> 00:21:20
like this.
But, the answer is that it's
332
00:21:18 --> 00:21:24
hard to go from here to here,
but trivial to go from here to
333
00:21:23 --> 00:21:29
here.
This solution step is the
334
00:21:25 --> 00:21:31
easiest step of all.
This is not very hard.
335
00:21:29 --> 00:21:35
It's easy, in fact.
This is easy and
336
00:21:33 --> 00:21:39
straightforward.
This is trivial,
337
00:21:36 --> 00:21:42
essentially,
yeah, trivial.
338
00:21:38 --> 00:21:44
But, this step is the hard
step.
339
00:21:41 --> 00:21:47
This is where you have to use
partial fractions,
340
00:21:45 --> 00:21:51
look up things in the table to
get back there so that most of
341
00:21:50 --> 00:21:56
the work of the procedure isn't
going from here to here.
342
00:21:55 --> 00:22:01
Going from here to there is a
breeze.
343
00:22:00 --> 00:22:06
Okay, now, in order to
implement this,
344
00:22:02 --> 00:22:08
what is it we have to do?
Well, the basic thing is I have
345
00:22:05 --> 00:22:11
to explain to you,
you already know at least a
346
00:22:08 --> 00:22:14
little bit, a reasonable amount
of technique for taking that
347
00:22:12 --> 00:22:18
step if you went to recitation
yesterday and practiced a little
348
00:22:16 --> 00:22:22
bit.
This part, I assure you,
349
00:22:17 --> 00:22:23
is nothing.
So, all I have to do now is
350
00:22:20 --> 00:22:26
explain to you how to take the
Laplace transform of the
351
00:22:23 --> 00:22:29
differential equation.
And, that really means,
352
00:22:26 --> 00:22:32
how do you take the Laplace
transform of a derivative?
353
00:22:31 --> 00:22:37
So, that's our problem.
What I want to form,
354
00:22:35 --> 00:22:41
in other words,
is a formula for the Laplace
355
00:22:39 --> 00:22:45
transform f prime of t.
356
00:22:43 --> 00:22:49
Now, in terms of what?
Well, since f is an arbitrary
357
00:22:48 --> 00:22:54
function, the only thing I could
hope for is somehow to express
358
00:22:54 --> 00:23:00
the Laplace transform of the
derivative in terms of the
359
00:23:00 --> 00:23:06
Laplace transform of the
original function.
360
00:23:06 --> 00:23:12
So, that's what I'm aiming for.
Okay, where are we going to
361
00:23:10 --> 00:23:16
start?
Well, starting is easy because
362
00:23:12 --> 00:23:18
we know nothing.
If you don't know anything,
363
00:23:15 --> 00:23:21
then there's no place to start
but the definition.
364
00:23:18 --> 00:23:24
Since I know nothing whatever
about the function f of t,
365
00:23:22 --> 00:23:28
and I want to
calculate the Laplace transform,
366
00:23:26 --> 00:23:32
I'd better start with the
definition.
367
00:23:30 --> 00:23:36
Whatever this is,
it's the integral from zero to
368
00:23:33 --> 00:23:39
infinity of e to the minus s t
times f prime of t dt.
369
00:23:37 --> 00:23:43
370
00:23:39 --> 00:23:45
Now, what am I looking for?
I'm looking for somehow to
371
00:23:43 --> 00:23:49
transform this so that what
appears here is not
372
00:23:47 --> 00:23:53
f prime of t, which I'm
clueless about, but f of t
373
00:23:50 --> 00:23:56
because if this were f of t,
this expression would be the
374
00:23:54 --> 00:24:00
Laplace transform of f of t.
375
00:23:56 --> 00:24:02
And, I'm assuming I know that.
So, the question is how do I
376
00:24:02 --> 00:24:08
take this and somehow do
something clever to it that
377
00:24:05 --> 00:24:11
turns this into f of t
instead of f prime of t?
378
00:24:10 --> 00:24:16
Now, to first the question that
379
00:24:13 --> 00:24:19
way, I hope I would get 100%
response on what to do.
380
00:24:16 --> 00:24:22
But, I'll go for 1%.
So, what should I do?
381
00:24:20 --> 00:24:26
I want to change that, so that
instead of f prime of t,
382
00:24:23 --> 00:24:29
f of t
appears there instead.
383
00:24:27 --> 00:24:33
What should I do?
Integrate by parts,
384
00:24:31 --> 00:24:37
the most fundamental procedure
in advanced analysis.
385
00:24:35 --> 00:24:41
Everything important and
interesting depends on
386
00:24:39 --> 00:24:45
integration by parts.
And, when you consider that
387
00:24:43 --> 00:24:49
integration by parts is nothing
more than just the formula for
388
00:24:48 --> 00:24:54
the derivative of a product read
backwards, it's amazing.
389
00:24:53 --> 00:24:59
It never fails to amaze me,
but it's okay.
390
00:24:56 --> 00:25:02
That's what mathematics are so
great.
391
00:24:59 --> 00:25:05
Okay, so let's use integration
by parts.
392
00:25:04 --> 00:25:10
Integration by parts:
okay, so, we have to decide,
393
00:25:06 --> 00:25:12
of course, there's no doubt
that this is the factor we want
394
00:25:10 --> 00:25:16
to integrate,
which means we have to be
395
00:25:12 --> 00:25:18
willing to differentiate this
factor.
396
00:25:14 --> 00:25:20
But that will be okay because
it looks practically,
397
00:25:17 --> 00:25:23
like any exponential,
it looks practically the same
398
00:25:20 --> 00:25:26
after you've differentiated it.
So, let's do the work.
399
00:25:23 --> 00:25:29
First step is you don't do the
differentiation.
400
00:25:26 --> 00:25:32
You only do the integration.
So, the first step is e to the
401
00:25:29 --> 00:25:35
negative s t.
And, the integral of f prime of
402
00:25:34 --> 00:25:40
t is just f of t.
403
00:25:36 --> 00:25:42
And, that's to be evaluated
between the limits zero and
404
00:25:40 --> 00:25:46
infinity.
And then, minus,
405
00:25:42 --> 00:25:48
again, before you forget it,
put down that minus sign.
406
00:25:45 --> 00:25:51
The integral between the limits
of what you get by doing both
407
00:25:49 --> 00:25:55
operations, both the
differentiation and the
408
00:25:52 --> 00:25:58
integration.
So, the differentiation will be
409
00:25:55 --> 00:26:01
by using the chain rule.
Remember, I'm differentiating
410
00:25:59 --> 00:26:05
with respect to t.
The variable is t here,
411
00:26:03 --> 00:26:09
not s.
s is just a parameter.
412
00:26:06 --> 00:26:12
It's just a constant,
a variable constant,
413
00:26:09 --> 00:26:15
if you get my meaning.
That's not an oxymoron.
414
00:26:13 --> 00:26:19
A variable constant:
a parameter is a variable
415
00:26:16 --> 00:26:22
constant, variable because you
can manipulate the little slider
416
00:26:21 --> 00:26:27
and make a change its value,
right?
417
00:26:24 --> 00:26:30
That's why it's variable.
It's not a variable.
418
00:26:28 --> 00:26:34
It's variable,
if you get the distinction.
419
00:26:33 --> 00:26:39
Okay, well, I mean,
it becomes a variable
420
00:26:35 --> 00:26:41
[LAUGHTER].
But right now,
421
00:26:37 --> 00:26:43
it's not a variable.
It's just sitting there in the
422
00:26:41 --> 00:26:47
integral.
All right, so,
423
00:26:43 --> 00:26:49
minus s, e to the negative s t,
f of t dt.
424
00:26:47 --> 00:26:53
Now, this part's easy.
425
00:26:49 --> 00:26:55
The interesting thing is this
expression.
426
00:26:52 --> 00:26:58
So, and the most interesting
thing is I have to evaluate it
427
00:26:56 --> 00:27:02
at infinity.
Now, of course,
428
00:26:58 --> 00:27:04
that means take the limit as
you go towards,
429
00:27:01 --> 00:27:07
as you let t goes to infinity.
Now, so what I'm interested in
430
00:27:07 --> 00:27:13
knowing is what's the limit of
that expression?
431
00:27:10 --> 00:27:16
I'll write it as f of t divided
by e to the s t.
432
00:27:13 --> 00:27:19
Remember, s is a positive
433
00:27:16 --> 00:27:22
number.
s t goes to infinity,
434
00:27:18 --> 00:27:24
and I want to know what the
limit of that is.
435
00:27:21 --> 00:27:27
Well, the limit is what it is.
But really, if that limit isn't
436
00:27:25 --> 00:27:31
zero, I'm in deep trouble since
the whole process is out of
437
00:27:29 --> 00:27:35
control.
What will make that limit zero?
438
00:27:33 --> 00:27:39
Well, that f of t
should not grow faster than e to
439
00:27:37 --> 00:27:43
the s t if s is a big
enough number.
440
00:27:41 --> 00:27:47
And now, that's just what will
happen if f of t is of
441
00:27:45 --> 00:27:51
exponential type.
It's for this step right here
442
00:27:48 --> 00:27:54
that is the most crucial place
at which we need to know that f
443
00:27:53 --> 00:27:59
of t is of exponential type.
So, that limit is zero since f
444
00:27:57 --> 00:28:03
of t is of exponential type,
in other words,
445
00:28:01 --> 00:28:07
that the value,
the absolute value of f of t,
446
00:28:04 --> 00:28:10
becomes less than,
let's say, put in the c if you
447
00:28:09 --> 00:28:15
want, but it's not very
important, c e to the k t
448
00:28:13 --> 00:28:19
efor all values of t.
And, therefore,
449
00:28:18 --> 00:28:24
this will go to zero as soon as
s becomes bigger than that k.
450
00:28:22 --> 00:28:28
In other words,
451
00:28:23 --> 00:28:29
if f of t isn't
growing any faster than e to the
452
00:28:27 --> 00:28:33
k t ,
then as soon as s is a number,
453
00:28:30 --> 00:28:36
that parameter has the value
bigger than k,
454
00:28:33 --> 00:28:39
this ratio is going to go to
zero because the denominator
455
00:28:37 --> 00:28:43
will always be bigger than the
numerator, and getting bigger
456
00:28:41 --> 00:28:47
faster.
So, this goes to zero if s is
457
00:28:45 --> 00:28:51
bigger than k.
At the upper limit,
458
00:28:48 --> 00:28:54
therefore, this is zero.
Again, assuming that s is
459
00:28:52 --> 00:28:58
bigger than that k,
the k of the exponential type,
460
00:28:57 --> 00:29:03
how about at the lower limit?
We're used to seeing zero
461
00:29:01 --> 00:29:07
there, but we're not going to
get zero.
462
00:29:04 --> 00:29:10
If I plug in t equals zero,
this factor becomes one.
463
00:29:09 --> 00:29:15
And, what happens to that one?
f of zero.
464
00:29:13 --> 00:29:19
You mean, I'm going to have to
know what f of zero is before I
465
00:29:18 --> 00:29:24
can take the Laplace transform
of this derivative?
466
00:29:22 --> 00:29:28
The answer is yes,
and that's why you have to have
467
00:29:26 --> 00:29:32
an initial value problem.
You have to know in advance
468
00:29:30 --> 00:29:36
what the value of the function
that you are looking for is at
469
00:29:34 --> 00:29:40
zero because it enters into the
formula.
470
00:29:37 --> 00:29:43
I didn't make up these rules;
I'm just following them.
471
00:29:40 --> 00:29:46
So, what's the rest?
The two negatives cancel,
472
00:29:43 --> 00:29:49
and you get plus s.
It's just a parameter,
473
00:29:47 --> 00:29:53
so I can pull it out of the
integral.
474
00:29:49 --> 00:29:55
I'm integrating with respect to
t, and what's left is,
475
00:29:53 --> 00:29:59
well, what is left?
If I take out minus s,
476
00:29:56 --> 00:30:02
combine it there,
I get what's left is just the
477
00:29:59 --> 00:30:05
Laplace transform of the
function I started with.
478
00:30:04 --> 00:30:10
So, it's F of s.
And, that's the magic formula
479
00:30:10 --> 00:30:16
for the Laplace transform of the
derivative.
480
00:30:15 --> 00:30:21
So, it's worth putting up on
our little list.
481
00:30:20 --> 00:30:26
So, f prime of t,
assuming it's of exponential
482
00:30:26 --> 00:30:32
type, has as its Laplace
transform, well,
483
00:30:31 --> 00:30:37
what is it?
Let's put down the major part
484
00:30:35 --> 00:30:41
of it is s times whatever the
Laplace transform of the
485
00:30:39 --> 00:30:45
original function,
F of t,
486
00:30:41 --> 00:30:47
was.
So, I take the original Laplace
487
00:30:44 --> 00:30:50
transform.
When I multiply it by s,
488
00:30:46 --> 00:30:52
that corresponds to taking the
derivative.
489
00:30:49 --> 00:30:55
But there's also that little
extra piece.
490
00:30:51 --> 00:30:57
I must know the value of the
starting value of the function.
491
00:30:55 --> 00:31:01
This is the formula you'll used
to take a Laplace transform of
492
00:31:00 --> 00:31:06
the differential equation.
Now, but you see I'm not done
493
00:31:05 --> 00:31:11
yet because that will take care
of the term a y prime.
494
00:31:09 --> 00:31:15
But, I don't know what the
Laplace transform of the second
495
00:31:13 --> 00:31:19
derivative is.
Okay, so, we need a formula for
496
00:31:16 --> 00:31:22
the Laplace transform of a
second derivative as well as the
497
00:31:20 --> 00:31:26
first.
Now, the hack method is to say,
498
00:31:23 --> 00:31:29
secondary, all right.
I've got to do this.
499
00:31:25 --> 00:31:31
I'll second derivative here,
second derivative here,
500
00:31:29 --> 00:31:35
what do I do with that?
Ah-ha, I integrate by parts
501
00:31:33 --> 00:31:39
twice.
Yes, you can do that.
502
00:31:34 --> 00:31:40
But that's a hack method.
And, of course,
503
00:31:40 --> 00:31:46
I know you're too smart to do
that.
504
00:31:45 --> 00:31:51
What you would do instead is--
How are we going to fill that
505
00:31:53 --> 00:31:59
in?
Well, a second derivative is
506
00:31:58 --> 00:32:04
also a first derivative.
A second derivative is the
507
00:32:07 --> 00:32:13
first derivative of the first
derivative.
508
00:32:14 --> 00:32:20
Okay, now, we'll just call this
glop, something.
509
00:32:22 --> 00:32:28
So, it's glop prime.
What is the formula for the
510
00:32:31 --> 00:32:37
Laplace transform of glop prime?
It is, well,
511
00:32:39 --> 00:32:45
I have my formula.
It is the glop prime.
512
00:32:43 --> 00:32:49
The formula for it is s times
the Laplace transform of glop,
513
00:32:50 --> 00:32:56
okay, glop.
Well, glop is f prime of t.
514
00:32:55 --> 00:33:01
I'm not done yet,
515
00:32:58 --> 00:33:04
minus glop evaluated at zero.
What's glop evaluated at zero?
516
00:33:05 --> 00:33:11
Well, f prime of zero.
517
00:33:10 --> 00:33:16
Now, I don't want the formula
in that form,
518
00:33:13 --> 00:33:19
but I have to have it in that
form because I know what the
519
00:33:17 --> 00:33:23
Laplace transform of f prime of
t is.
520
00:33:20 --> 00:33:26
I just calculated that.
So, this is equal to s times
521
00:33:24 --> 00:33:30
the Laplace transform of f prime
of t, which is s times F of s,
522
00:33:28 --> 00:33:34
capital F of s,
minus f of zero.
523
00:33:31 --> 00:33:37
All that bracket stuff
524
00:33:34 --> 00:33:40
corresponds to this guy.
And, don't forget the stuff
525
00:33:38 --> 00:33:44
that's tagging along,
minus f prime of zero.
526
00:33:42 --> 00:33:48
And now, put that all together.
527
00:33:45 --> 00:33:51
What is it going to be?
Well, there's the principal
528
00:33:49 --> 00:33:55
term which consists of s squared
multiplied by F of s.
529
00:33:54 --> 00:34:00
That's the main part of it.
530
00:33:56 --> 00:34:02
And, the rest is the sort of
fellow travelers.
531
00:34:00 --> 00:34:06
So, we have minus s times f of
zero,
532
00:34:04 --> 00:34:10
little term tagging along.
This is a constant times s.
533
00:34:10 --> 00:34:16
And then, we've got another
one, still another constant.
534
00:34:14 --> 00:34:20
So, what we have is to
calculate the Laplace transform
535
00:34:18 --> 00:34:24
of the second derivative,
I need to know both f of zero
536
00:34:22 --> 00:34:28
and f prime of zero,
exactly the initial
537
00:34:27 --> 00:34:33
conditions that the problem was
given for the initial value
538
00:34:31 --> 00:34:37
problem.
But, notice,
539
00:34:33 --> 00:34:39
there's a principal part of it.
That's the s squared F of s.
540
00:34:37 --> 00:34:43
That's the guts of it,
541
00:34:39 --> 00:34:45
so to speak.
The rest of it,
542
00:34:41 --> 00:34:47
you know, you might hope that
these two numbers are zero.
543
00:34:44 --> 00:34:50
It could happen,
and often it is made to happen
544
00:34:47 --> 00:34:53
and problems to simplify them.
And I case, you don't have to
545
00:34:51 --> 00:34:57
worry; they're not there.
But, if they are there,
546
00:34:54 --> 00:35:00
you must put them in or you get
the wrong answer.
547
00:34:56 --> 00:35:02
So, that's the list of
formulas.
548
00:35:00 --> 00:35:06
So, those formulas on the top
board and these two extra ones,
549
00:35:06 --> 00:35:12
those are the things you will
be working with on Friday.
550
00:35:12 --> 00:35:18
But I stress,
the Laplace transform won't be
551
00:35:17 --> 00:35:23
a big part of the exam.
The exam, of course,
552
00:35:22 --> 00:35:28
doesn't exist,
let's say a maximum of 20%,
553
00:35:27 --> 00:35:33
maybe 15.
I don't know,
554
00:35:29 --> 00:35:35
give or take a few points.
Yeah, what's a point or two?
555
00:35:37 --> 00:35:43
Okay, let's solve,
yeah, we have time.
556
00:35:41 --> 00:35:47
We have time to solve a
problem.
557
00:35:44 --> 00:35:50
Let's solve a problem.
See, I can't touch that.
558
00:35:49 --> 00:35:55
It's untouchable.
Okay, this, we've got to keep.
559
00:35:56 --> 00:36:02
560
00:36:14 --> 00:36:20
Problem? Okay.
561
00:36:16 --> 00:36:22
562
00:36:39 --> 00:36:45
Okay, now you know how to solve
this problem by operators.
563
00:36:43 --> 00:36:49
Let me just briefly remind you
of the basic steps.
564
00:36:47 --> 00:36:53
You have to do two separate
tasks.
565
00:36:50 --> 00:36:56
You have to first solve the
homogeneous equation,
566
00:36:54 --> 00:37:00
putting a zero there.
That's the first thing you
567
00:36:58 --> 00:37:04
learned to do.
That's easy.
568
00:37:00 --> 00:37:06
You could almost do that in
your head now.
569
00:37:05 --> 00:37:11
You get the characteristic
polynomial, get its roots,
570
00:37:08 --> 00:37:14
get the two functions,
e to the t and e to the
571
00:37:11 --> 00:37:17
negative t,
which are the solutions.
572
00:37:14 --> 00:37:20
You make up c1 times one,
and c2 times the other.
573
00:37:18 --> 00:37:24
That's the complementary
function that solves the
574
00:37:21 --> 00:37:27
homogeneous problem.
And then you have to find a
575
00:37:24 --> 00:37:30
particular solution.
Can you see what would happen
576
00:37:27 --> 00:37:33
if you try to find the
particular solution?
577
00:37:30 --> 00:37:36
The number here is negative
one, right?
578
00:37:34 --> 00:37:40
Negative one is a root of the
characteristic polynomial,
579
00:37:37 --> 00:37:43
so you've got to use that extra
formula.
580
00:37:40 --> 00:37:46
It's okay.
That's why I gave it to you.
581
00:37:42 --> 00:37:48
You've used the exponential
input theorem with the extra
582
00:37:46 --> 00:37:52
formula.
Then, you will get the
583
00:37:48 --> 00:37:54
particular solution.
And now, you have to make the
584
00:37:51 --> 00:37:57
general solution.
The particular solution plus
585
00:37:54 --> 00:38:00
the complementary function,
and now you are ready to put in
586
00:37:58 --> 00:38:04
the initial conditions.
At the very end,
587
00:38:02 --> 00:38:08
when you've got the whole
general solution,
588
00:38:05 --> 00:38:11
now you put in,
not before, you put in the
589
00:38:07 --> 00:38:13
initial conditions.
You have to calculate the
590
00:38:11 --> 00:38:17
derivative of that thing and
substitute this.
591
00:38:14 --> 00:38:20
You take it as it stands to
substitute this.
592
00:38:17 --> 00:38:23
You get a pair of simultaneous
equations for c1 and c2.
593
00:38:21 --> 00:38:27
You solve them:
answer.
594
00:38:22 --> 00:38:28
It's a rather elaborate
procedure, which has at least
595
00:38:26 --> 00:38:32
three or four separate steps,
all of which,
596
00:38:29 --> 00:38:35
of course, must be done
correctly.
597
00:38:33 --> 00:38:39
Now, the Laplace transform,
instead, feeds the entire
598
00:38:37 --> 00:38:43
problem into the Laplace
transform machine.
599
00:38:40 --> 00:38:46
You follow that little blue
pattern, and you come out with
600
00:38:44 --> 00:38:50
the answer.
So, let's do the Laplace
601
00:38:47 --> 00:38:53
transform way.
Okay, so, the first step is to
602
00:38:51 --> 00:38:57
say, if here's my unknown
function, y of t,
603
00:38:55 --> 00:39:01
it obeys this law,
and here are its starting
604
00:38:58 --> 00:39:04
values, a bit of its derivative.
What I'm going to take is the
605
00:39:03 --> 00:39:09
Laplace transform of this
equation.
606
00:39:06 --> 00:39:12
In other words,
I'll take the Laplace transform
607
00:39:09 --> 00:39:15
of this side,
and this side also.
608
00:39:11 --> 00:39:17
And, they must be equal because
if they were equal to start
609
00:39:15 --> 00:39:21
with, the Laplace transforms
also have to be equal.
610
00:39:18 --> 00:39:24
Okay, so let's take the Laplace
transform of this equation.
611
00:39:22 --> 00:39:28
Okay, first ID the Laplace
transform of the second
612
00:39:25 --> 00:39:31
derivative.
Okay, that's going to be,
613
00:39:27 --> 00:39:33
don't forget the principal
terms.
614
00:39:31 --> 00:39:37
There is some people who get so
hypnotized by this.
615
00:39:34 --> 00:39:40
I just know I'm going to forget
this.
616
00:39:36 --> 00:39:42
So, they read it.
Then they forget this.
617
00:39:38 --> 00:39:44
But that's everything.
That's the important part.
618
00:39:41 --> 00:39:47
Okay, so it's s times,
I'm calling the Laplace
619
00:39:44 --> 00:39:50
transform not capital F but
capital Y because my original
620
00:39:48 --> 00:39:54
function is called little y.
So, it's s squared Y.
621
00:39:51 --> 00:39:57
It's Y of s,
but I'm not going to put that,
622
00:39:54 --> 00:40:00
the of s in because it just
makes the thing look more
623
00:39:58 --> 00:40:04
complicated.
And now, before you forget,
624
00:40:02 --> 00:40:08
you have to put in the rest.
So, minus s times the value at
625
00:40:06 --> 00:40:12
zero, which is one,
minus the value of the
626
00:40:10 --> 00:40:16
derivative.
But, that's zero.
627
00:40:12 --> 00:40:18
So, this is not too hard a
problem.
628
00:40:15 --> 00:40:21
So, minus s minus zero,
so I don't have to put that
629
00:40:19 --> 00:40:25
in.
So, all this is the Laplace
630
00:40:22 --> 00:40:28
transform of y double prime.
631
00:40:25 --> 00:40:31
And now, minus the Laplace
transform of y,
632
00:40:29 --> 00:40:35
well, that's just capital Y.
What's that equal to?
633
00:40:35 --> 00:40:41
The Laplace transform of the
right-hand side.
634
00:40:40 --> 00:40:46
Okay, look up the formula.
It is e to the negative t,
635
00:40:45 --> 00:40:51
a is minus one,
so, it's one over s minus minus
636
00:40:51 --> 00:40:57
one;
so, it is s plus one.
637
00:40:57 --> 00:41:03
This is that.
Okay, the next thing we have to
638
00:41:02 --> 00:41:08
do is solve for Y.
That doesn't look too hard.
639
00:41:05 --> 00:41:11
Solve it for y.
Okay, the best thing to do is
640
00:41:08 --> 00:41:14
put s squared,
group all the Y terms together
641
00:41:11 --> 00:41:17
unless you're really quite a
good calculator.
642
00:41:14 --> 00:41:20
Maybe make one extra line out
of it.
643
00:41:17 --> 00:41:23
Yeah, definitely do this.
And then, the extra garbage I
644
00:41:21 --> 00:41:27
refer to as the garbage,
this stuff, and this stuff,
645
00:41:25 --> 00:41:31
the stuff, the linear
polynomials which are tagging
646
00:41:28 --> 00:41:34
along move to the right-hand
side because they don't involve
647
00:41:32 --> 00:41:38
capital Y.
So, this we will move to the
648
00:41:37 --> 00:41:43
other side.
And so, that's equal to (one
649
00:41:40 --> 00:41:46
over (s plus one)) plus s.
650
00:41:44 --> 00:41:50
Now, you have a basic choice.
About half the time,
651
00:41:48 --> 00:41:54
it's a good idea to combine
these terms.
652
00:41:51 --> 00:41:57
The other half of the time,
it's not a good idea to combine
653
00:41:56 --> 00:42:02
those terms.
So, how do we know whether to
654
00:41:59 --> 00:42:05
do it or not to do it?
Experience, which you will get
655
00:42:03 --> 00:42:09
by solving many,
many problems.
656
00:42:06 --> 00:42:12
Okay, I'm going to combine them
because I think it's a good
657
00:42:11 --> 00:42:17
thing to do here.
So, what is that?
658
00:42:15 --> 00:42:21
That's s squared plus s plus
one.
659
00:42:20 --> 00:42:26
So, it's s squared plus s plus
one divided by s plus one,
660
00:42:25 --> 00:42:31
okay?
I'm still not done because now
661
00:42:28 --> 00:42:34
we have to know,
what's Y?
662
00:42:31 --> 00:42:37
All right, now we have to
think.
663
00:42:35 --> 00:42:41
What we're going to do is get Y
in this form.
664
00:42:38 --> 00:42:44
But, I want it in the form in
which it's most suited for using
665
00:42:42 --> 00:42:48
partial fractions.
In other words,
666
00:42:44 --> 00:42:50
I want the denominator as
factored as I possibly can be.
667
00:42:48 --> 00:42:54
Okay, well, the numerator is
going to be just what it was.
668
00:42:52 --> 00:42:58
How should I write the
denominator?
669
00:42:55 --> 00:43:01
Well, the denominator is going
to have the factor s plus one
670
00:42:59 --> 00:43:05
in it from here.
But after I divide through,
671
00:43:04 --> 00:43:10
the other factor will be s
squared minus one,
672
00:43:09 --> 00:43:15
right?
But, s squared minus one is s
673
00:43:12 --> 00:43:18
minus one times s plus one.
674
00:43:17 --> 00:43:23
So, I have to divide this by s
squared minus one.
675
00:43:23 --> 00:43:29
Factored, it's this.
So, the end result is there are
676
00:43:27 --> 00:43:33
two of these and one of the
other.
677
00:43:32 --> 00:43:38
And now, it's ready to be used.
It's better to be a partial
678
00:43:36 --> 00:43:42
fraction.
So, the final step is to use a
679
00:43:40 --> 00:43:46
partial fraction's decomposition
on this so that you can
680
00:43:44 --> 00:43:50
calculate its inverse Laplace
transform.
681
00:43:48 --> 00:43:54
So, let's do that.
Okay, (s squared plus s plus
682
00:43:51 --> 00:43:57
one) divided by that thing,
(s plus one) squared times (s
683
00:43:56 --> 00:44:02
minus one) equals s plus
684
00:44:01 --> 00:44:07
one squared plus s
plus one plus s minus one.
685
00:44:07 --> 00:44:13
In the top will be constants,
686
00:44:12 --> 00:44:18
just constants.
Let's do it this way first,
687
00:44:15 --> 00:44:21
and I'll say at the very end,
something else.
688
00:44:18 --> 00:44:24
Maybe now.
Many of you are upset.
689
00:44:21 --> 00:44:27
Some of you are upset.
I know this for a fact because
690
00:44:25 --> 00:44:31
in high school,
or wherever you learned to do
691
00:44:28 --> 00:44:34
this before, there weren't two
terms here.
692
00:44:33 --> 00:44:39
There was just one term,
s plus one squared.
693
00:44:36 --> 00:44:42
If you do it that way,
694
00:44:39 --> 00:44:45
then it's all right.
Then, it's all right,
695
00:44:42 --> 00:44:48
but I don't recommend it.
In that case,
696
00:44:45 --> 00:44:51
the numerators will not be
constants.
697
00:44:48 --> 00:44:54
But, if you just have that,
then because this is a
698
00:44:52 --> 00:44:58
quadratic polynomial all by
itself.
699
00:44:54 --> 00:45:00
You've got to have a linear
polynomial, a s plus b
700
00:44:59 --> 00:45:05
in the top, see?
701
00:45:02 --> 00:45:08
So, you must have a s plus b
here,
702
00:45:04 --> 00:45:10
as I'm sure you remember if
that's the way you learned to do
703
00:45:08 --> 00:45:14
it. But, to do cover-up,
704
00:45:09 --> 00:45:15
the best way as much as
possible to separate out the
705
00:45:12 --> 00:45:18
terms.
If this were a cubic term,
706
00:45:14 --> 00:45:20
God forbid, s plus one cubed,
707
00:45:16 --> 00:45:22
then you'd have to have s plus
one cubed,
708
00:45:20 --> 00:45:26
s plus one squared.
709
00:45:23 --> 00:45:29
Okay, I won't give you anything
bigger than quadratic.
710
00:45:26 --> 00:45:32
[LAUGHTER]
You can trust me.
711
00:45:29 --> 00:45:35
Okay, now, what can we find by
the cover up method?
712
00:45:33 --> 00:45:39
Well, surely this.
Cover up the s minus one,
713
00:45:37 --> 00:45:43
put s equals one,
and I get three divided by two
714
00:45:42 --> 00:45:48
squared, four.
So, this is three quarters.
715
00:45:45 --> 00:45:51
Now, in this,
716
00:45:48 --> 00:45:54
you can always find the highest
power by cover-up because,
717
00:45:52 --> 00:45:58
cover it up,
put s equals negative one,
718
00:45:56 --> 00:46:02
and you get one minus one
plus one.
719
00:46:02 --> 00:46:08
So, one up there,
negative one here makes
720
00:46:04 --> 00:46:10
negative two here.
So, one over negative two.
721
00:46:07 --> 00:46:13
So, it's minus one half.
722
00:46:09 --> 00:46:15
Now, this you cannot determine
723
00:46:12 --> 00:46:18
by cover-up because you'd want
to cover-up just one of these s
724
00:46:16 --> 00:46:22
plus ones.
But then you can't put s equals
725
00:46:19 --> 00:46:25
negative one because
you get infinity.
726
00:46:22 --> 00:46:28
You get zero there,
makes infinity.
727
00:46:24 --> 00:46:30
So, this must be determined
some other way,
728
00:46:27 --> 00:46:33
either by undetermined
coefficients,
729
00:46:29 --> 00:46:35
or if you've just got one
thing, for heaven's sake,
730
00:46:32 --> 00:46:38
just make a substitution.
See, this is supposed to be
731
00:46:37 --> 00:46:43
true.
This is an algebraic identity,
732
00:46:40 --> 00:46:46
true for all values of the
variable, and therefore,
733
00:46:43 --> 00:46:49
it ought to be true when s
equals zero,
734
00:46:47 --> 00:46:53
for instance.
Why zero?
735
00:46:48 --> 00:46:54
Well, because I haven't used it
yet.
736
00:46:51 --> 00:46:57
I used negative one and
positive one,
737
00:46:53 --> 00:46:59
but I didn't use zero.
Okay, let's use zero.
738
00:46:56 --> 00:47:02
Put s equals zero.
What do we get?
739
00:47:00 --> 00:47:06
Well, on the left-hand side,
I get one divided by one
740
00:47:03 --> 00:47:09
squared, negative.
So, I get minus one on the left
741
00:47:06 --> 00:47:12
hand side equals,
what do I get on the right?
742
00:47:09 --> 00:47:15
Put s equals zero,
you get negative one half.
743
00:47:12 --> 00:47:18
Well, this is the number I'm
744
00:47:15 --> 00:47:21
trying to find.
So, let's write that simply as
745
00:47:18 --> 00:47:24
plus c, putting s equals zero.
s equals zero here gives me
746
00:47:21 --> 00:47:27
negative three quarters.
747
00:47:23 --> 00:47:29
Okay, what's c?
This is minus a half,
748
00:47:26 --> 00:47:32
minus three quarters,
is minus five quarters.
749
00:47:29 --> 00:47:35
Put it on the other side,
750
00:47:32 --> 00:47:38
minus one plus five quarters is
plus one quarter.
751
00:47:35 --> 00:47:41
So, c equals one quarter.
752
00:47:39 --> 00:47:45
And now, we are ready to do the
753
00:47:42 --> 00:47:48
final step.
Take the inverse Laplace
754
00:47:44 --> 00:47:50
transform.
You see what I said when I said
755
00:47:47 --> 00:47:53
that all the work is in this
last step?
756
00:47:49 --> 00:47:55
Just look how much of the work,
how much of the board is
757
00:47:53 --> 00:47:59
devoted to the first two steps,
and how much is going to be
758
00:47:57 --> 00:48:03
devoted to the last step?
Okay, so we get e to the
759
00:48:01 --> 00:48:07
inverse Laplace transform.
Well, the first term is the
760
00:48:05 --> 00:48:11
hardest.
Let's let that go for the
761
00:48:08 --> 00:48:14
moment.
So, I leave a space for it,
762
00:48:10 --> 00:48:16
and then we will have one
quarter.
763
00:48:13 --> 00:48:19
Well, one over s plus one is,
764
00:48:16 --> 00:48:22
that's just the exponential
formula.
765
00:48:19 --> 00:48:25
One over s plus one would be e
to the negative t,e to the minus
766
00:48:24 --> 00:48:30
one times t.
So, it's one quarter e to the
767
00:48:28 --> 00:48:34
minus one times t.
768
00:48:32 --> 00:48:38
And, how about the next thing
would be three quarters times,
769
00:48:36 --> 00:48:42
well, here it's negative one,
so that's e to the plus t.
770
00:48:41 --> 00:48:47
Notice how those signs work.
771
00:48:44 --> 00:48:50
And, that just leaves us the
Laplace transform of this thing.
772
00:48:49 --> 00:48:55
Now, you look at it and you
say, this Laplace transform
773
00:48:54 --> 00:49:00
happened in two steps.
I took something and I got,
774
00:48:58 --> 00:49:04
essentially,
one over s squared.
775
00:49:03 --> 00:49:09
And then, I changed s to s plus
one.
776
00:49:08 --> 00:49:14
All right, what gives one over
s squared?
777
00:49:13 --> 00:49:19
The Laplace transform of what
is one over s squared?
778
00:49:18 --> 00:49:24
t, you say to yourself,
one over s to some power is
779
00:49:23 --> 00:49:29
essentially some power of t.
And then, you look at the
780
00:49:28 --> 00:49:34
formula.
Notice at the top is one
781
00:49:31 --> 00:49:37
factorial, which is one,
of course.
782
00:49:34 --> 00:49:40
Okay, now, then how do I
convert this to one over s plus
783
00:49:39 --> 00:49:45
one squared?
That's the exponential shift
784
00:49:44 --> 00:49:50
formula.
If you know what the Laplace
785
00:49:47 --> 00:49:53
transform, so the first formula
in the middle of the board on
786
00:49:51 --> 00:49:57
the top, there,
if you know what,
787
00:49:54 --> 00:50:00
change s to s plus one,
corresponds to
788
00:49:58 --> 00:50:04
multiplying by e to the t.
789
00:50:03 --> 00:50:09
So, it is t times e to the
negative t.
790
00:50:06 --> 00:50:12
Sorry, that corresponds to
this.
791
00:50:08 --> 00:50:14
So, this is the exponential
shift formula.
792
00:50:11 --> 00:50:17
If t goes to one over s squared,
then t e to the
793
00:50:15 --> 00:50:21
minus t goes to one
over s plus one squared.
794
00:50:20 --> 00:50:26
Okay, but there's a constant
795
00:50:22 --> 00:50:28
out front.
So, it's minus one half t e to
796
00:50:25 --> 00:50:31
the negative t.
797
00:50:27 --> 00:50:33
Now, tell me,
what parts of this solution,
798
00:50:30 --> 00:50:36
oh boy, we're over time.
But, notice,
799
00:50:32 --> 00:50:38
this is what would have been
the particular solution,
800
00:50:36 --> 00:50:42
(y)p before,
and this is the stuff that
801
00:50:39 --> 00:50:45
occurs in the complementary
function, but already the
802
00:50:42 --> 00:50:48
appropriate constants have been
supplied for the coefficients.
803
00:50:49 --> 00:50:55
You don't have to calculate
them separately.
804
00:50:52 --> 00:50:58
They were built into the
method.
805
00:50:54 --> 00:51:00
Okay, good luck on Friday,
and see you there.