Laplace/step function differential equation

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.150
Let's apply everything we've
learned to an actual

00:00:03.150 --> 00:00:04.390
differential equation.

00:00:04.390 --> 00:00:07.850
Instead of just taking Laplace
transforms and taking their

00:00:07.850 --> 00:00:09.940
inverse, let's actually
solve a problem.

00:00:09.940 --> 00:00:13.880
So let's say that I have the
second derivative of my

00:00:13.880 --> 00:00:22.550
function y plus 4 times my
function y is equal to sine of

00:00:22.550 --> 00:00:36.370
t minus the unit step function
0 up until 2 pi of t times

00:00:36.370 --> 00:00:42.890
sine of t minus 2 pi.

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Let's solve this differential
equation, an

00:00:45.120 --> 00:00:45.870
interpretation of it.

00:00:45.870 --> 00:00:47.950
And I actually do a whole
playlist on interpretations of

00:00:47.950 --> 00:00:50.440
differential equations and how
you model it, but you know,

00:00:50.440 --> 00:00:52.490
you can kind of view this
is a forcing function.

00:00:52.490 --> 00:00:55.830
That it's a weird forcing
function of this being applied

00:00:55.830 --> 00:00:58.280
to some weight with, you
know, this is the

00:00:58.280 --> 00:00:59.470
acceleration term, right?

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The second derivative with
respect to time is the

00:01:01.490 --> 00:01:02.290
acceleration.

00:01:02.290 --> 00:01:09.540
So the mass would be 1 whatever
units, and then as a

00:01:09.540 --> 00:01:11.340
function of its position, this
is probably some type of

00:01:11.340 --> 00:01:12.150
spring constant.

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Anyway, I won't go there.

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I don't want to waste your time
with the interpretation

00:01:15.650 --> 00:01:17.860
of it, but let's solve it.

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We can do more about
interpretations later.

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So we're going take the Laplace
transform of both

00:01:21.600 --> 00:01:23.340
sides of this equation.

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So what's the Laplace
transform of

00:01:25.550 --> 00:01:26.610
the left-hand side?

00:01:26.610 --> 00:01:30.680
So the Laplace transform of the
second derivative of y is

00:01:30.680 --> 00:01:34.530
just s squared, so now I'm
taking the Laplace transform

00:01:34.530 --> 00:01:35.880
of just that.

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The Laplace transform of s
squared times the Laplace

00:01:38.570 --> 00:01:45.110
transform of y minus-- lower the
degree there once-- minus

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s times y of 0 minus
y prime of 0.

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So clearly, I must have to
give you some initial

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conditions in order to
do this properly.

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And then plus 4 times the
Laplace transform of y is

00:02:05.490 --> 00:02:10.979
equal to-- what's the Laplace
transform of sine of t?

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That should be second
nature by now.

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It's just 1 over s
squared plus 1.

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And then we have minus
the Laplace

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transform of this thing.

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And I'll do a little side note
here to figure out the Laplace

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transform of this thing
right here.

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And we know, I showed it to you
a couple of videos ago, we

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showed that the Laplace
transform-- actually I could

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just write it out here.

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This is going to be the same
thing as the Laplace transform

00:02:45.290 --> 00:02:49.510
of sine of t, but we're going to
have to multiply it by e to

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the minus-- if you remember that
last formula-- e to the

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minus cs, where c is 2 pi.

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Actually, let me write
that down.

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I decided to write it down,
then I decided, oh, no, I

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don't want to do this.

00:03:02.080 --> 00:03:03.110
But let me write that.

00:03:03.110 --> 00:03:06.940
So the Laplace transform of the
unit step function that

00:03:06.940 --> 00:03:12.950
goes up to c times some function
shifted by c is equal

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to e to the minus cs times the
Laplace transform of just the

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original function times the
Laplace transform of f of t.

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So if we're taking the Laplace
transform of this

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thing, our c is 2 pi.

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Our f of t is just
sine of t, right?

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So then this is just going to
be equal to-- if we just do

00:03:41.240 --> 00:03:45.630
this piece right here-- it's
going to be equal to e to the

00:03:45.630 --> 00:03:53.480
minus cs-- our c is 2 pi-- e to
the minus 2 pi s times the

00:03:53.480 --> 00:03:57.270
Laplace transform of f of t. f
of t is just sine of t before

00:03:57.270 --> 00:03:58.120
we shifted.

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This is f of t minus 2 pi.

00:04:00.340 --> 00:04:03.090
So f of t is just going
to be sine of t.

00:04:03.090 --> 00:04:07.340
So it's going to be times
1 over s squared plus 1.

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This is the Laplace transform
of sine of t.

00:04:09.860 --> 00:04:14.000
So let's go back to where
we had left off.

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So we've taken the Laplace
transform of both sides of

00:04:16.490 --> 00:04:17.490
this equation.

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And clearly, I have some initial
conditions here, so

00:04:20.870 --> 00:04:23.420
the problem must have given me
some and I just forgot to

00:04:23.420 --> 00:04:25.290
write them down.

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So let's see, the initial
conditions I'm given, and they

00:04:28.190 --> 00:04:31.150
are written kind of in the
margin here, they tell us--

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I'll do it in orange, they tell
us that y of 0 is equal

00:04:37.340 --> 00:04:44.310
to 0, and y prime of
0 is equal to 0.

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That makes the math easy.

00:04:46.090 --> 00:04:49.110
That's 0 and that's 0.

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So let's see if I can simplify
my equation.

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So the left-hand side,
let's factor

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out the Laplace transform.

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So let's factor out this
term and that term.

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So we get the Laplace transform
of y times this plus

00:05:06.760 --> 00:05:13.740
this times s squared plus 4 is
equal to the right-hand side.

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And what's the right-hand
side?

00:05:18.270 --> 00:05:19.570
We could simplify this.

00:05:19.570 --> 00:05:20.650
Well, I'll just write it out.

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I don't want to do too
many steps at once.

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It's 1 over s squared plus 1
and then plus-- or minus

00:05:30.180 --> 00:05:32.420
actually, this is a minus--
minus the Laplace transfer of

00:05:32.420 --> 00:05:39.010
this thing, which was e to
the minus 2 pi s over s

00:05:39.010 --> 00:05:41.720
squared plus 1.

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So if we divide both sides of
this equation by the s squared

00:05:48.190 --> 00:05:55.650
plus 4, then we get the Laplace
transform of y is

00:05:55.650 --> 00:06:00.620
equal to-- and actually, I
can just merge these two.

00:06:00.620 --> 00:06:01.810
They're the same denominator.

00:06:01.810 --> 00:06:04.700
So before I even divide by
s squared plus 4, that

00:06:04.700 --> 00:06:07.090
right-hand side will
look like this.

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It will look like with a
denominator of s squared plus

00:06:11.410 --> 00:06:19.280
1 and you have a numerator of 1
minus e to the minus 2 pi s.

00:06:19.280 --> 00:06:21.150
And, of course, we're dividing
both sides of this equation by

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s squared plus 4, so we're going
to have to stick that s

00:06:23.260 --> 00:06:27.150
squared plus 4 over here.

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Now, we're at the hard part.

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In order to figure out why, we
have to take the inverse

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Laplace transform
of this thing.

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So how do we take the
inverse Laplace

00:06:35.620 --> 00:06:36.720
transform of this thing?

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That's where the hard part is
always, you know, it makes

00:06:39.400 --> 00:06:41.060
solving the differential
equation's easy if you know

00:06:41.060 --> 00:06:45.213
the Laplace transforms. So it
looks like we're going to have

00:06:45.213 --> 00:06:48.040
to do some partial fraction
expansion.

00:06:48.040 --> 00:06:49.290
So let's see if we
can do that.

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So we can rewrite this
equation right here.

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Actually, let's write it as
this, because this'll kind of

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simplify our work.

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Let's factor this
whole thing out.

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So we're going to write it as 1
minus e to the minus 2 pi s,

00:07:08.370 --> 00:07:13.340
all of that times-- I'll do it
in orange-- all of that times

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1 over s squared plus 1 times
s squared plus 4.

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Now, we need to do some partial
fraction expansion to

00:07:24.410 --> 00:07:27.430
simplify this thing
right here.

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We're going to do this
on the side.

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Maybe I should do this over
on the right here.

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This thing-- let me rewrite it--
1 over s squared plus 1

00:07:38.300 --> 00:07:44.990
times s squared plus 4 should be
able to be rewritten as two

00:07:44.990 --> 00:07:50.100
separate fractions, s squared
plus 1 and s squared plus 4,

00:07:50.100 --> 00:07:52.680
with the numerators.

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This one would be As plus B.

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It's going to have to have
degree 1, because

00:07:57.770 --> 00:07:58.600
this is degree 2.

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Here And then we'd
have Cs plus D.

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And so when you add these two
things up, you get As plus B

00:08:11.360 --> 00:08:23.000
times s squared plus 4 plus Cs
plus D times s squared plus 1,

00:08:23.000 --> 00:08:27.870
all of that over the
common denominator.

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We've seen this story before.

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We just have to do some
algebra here.

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As you can tell, these
differential equations

00:08:35.520 --> 00:08:37.190
problems, they require
a lot of stamina.

00:08:37.190 --> 00:08:40.830
You kind of just have to say I
will keep moving forward and

00:08:40.830 --> 00:08:43.700
do the algebra that I need to do
in order to get the answer.

00:08:43.700 --> 00:08:45.950
And you kind of have to get
excited about that notion that

00:08:45.950 --> 00:08:48.570
you have all this
algebra to do.

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So let's figure it out.

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So this top can be simplified
to As to the third plus Bs

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squared plus 4As plus 4B.

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And then this one, you end up
with Cs to the third plus Ds

00:09:19.330 --> 00:09:27.850
squared plus Cs plus D.

00:09:27.850 --> 00:09:32.350
So when you add of these up
together, you get-- and this

00:09:32.350 --> 00:09:35.410
is all the algebra that we have
to do, for better, for

00:09:35.410 --> 00:09:43.510
worse-- A plus C over s to the
third plus B plus D times s

00:09:43.510 --> 00:09:49.920
squared plus 4A plus C times s--
let's scroll over a little

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bit-- plus 4B plus D.

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And now we just have to say,
OK, all of this is equal to

00:09:59.730 --> 00:10:00.710
this thing up here.

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This is the numerator.

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We just simplified
the numerator.

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This is the numerator.

00:10:05.050 --> 00:10:06.840
That's the numerator
right there.

00:10:06.840 --> 00:10:11.740
And all of this is going to be
over your original s squared

00:10:11.740 --> 00:10:15.280
plus 1 times your s
squared plus 4.

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And we established that this
thing should be-- let me just

00:10:18.150 --> 00:10:23.610
write this-- that 1 over s
squared plus 1 times s squared

00:10:23.610 --> 00:10:26.240
plus 4 should equal
this thing.

00:10:26.240 --> 00:10:29.010
And then you just pattern match
on the coefficients.

00:10:29.010 --> 00:10:33.420
This is all just intense partial
fraction expansion.

00:10:33.420 --> 00:10:36.190
And you say, look, A plus C is
the coefficient of the s cubed

00:10:36.190 --> 00:10:38.610
terms. I don't see any
s cubed terms here.

00:10:38.610 --> 00:10:43.180
So A plus C must
be equal to 0.

00:10:43.180 --> 00:10:46.070
And then you see, OK, B plus D
is the coefficient of the s

00:10:46.070 --> 00:10:48.010
squared terms. I don't see any
s squared terms there.

00:10:48.010 --> 00:10:53.670
So B plus D must
be equal to 0.

00:10:53.670 --> 00:10:56.220
4A plus C, the coefficient of
the s terms. I don't see any s

00:10:56.220 --> 00:10:57.070
terms over here.

00:10:57.070 --> 00:11:01.080
So 4A plus C must
be equal to 0.

00:11:01.080 --> 00:11:02.490
And then we're almost done.

00:11:02.490 --> 00:11:05.490
4B plus D must be the constant
terms. There is a constant

00:11:05.490 --> 00:11:06.530
term there.

00:11:06.530 --> 00:11:10.465
So 4B plus D is equal to 1.

00:11:10.465 --> 00:11:12.490
So let's see if we can
do anything here.

00:11:12.490 --> 00:11:17.680
If we subtract this from that,
we get minus 3A is equal to 0,

00:11:17.680 --> 00:11:19.160
or A is equal to 0.

00:11:19.160 --> 00:11:22.840
If A is equal to 0, then
C is equals to 0.

00:11:22.840 --> 00:11:24.720
And let's see what
we can get here.

00:11:24.720 --> 00:11:31.010
If we subtract this from
that, we get minus 3B.

00:11:31.010 --> 00:11:32.020
The D's cancel out.

00:11:32.020 --> 00:11:36.290
It's equal to minus 1,
or B is equal to 1/3.

00:11:36.290 --> 00:11:40.470
And then, of course, we have D
is equal to minus B, if you

00:11:40.470 --> 00:11:44.550
subtract B from both sides.
so D is equal to 1/3.

00:11:44.550 --> 00:11:46.960
So all of that work, and
we actually have a

00:11:46.960 --> 00:11:48.400
pretty simple result.

00:11:52.190 --> 00:12:00.660
Our equation, this thing here,
can be rewritten as-- the A

00:12:00.660 --> 00:12:01.630
disappeared.

00:12:01.630 --> 00:12:06.130
It's 1/3 over s squared
plus 1.

00:12:06.130 --> 00:12:10.490
B was the coefficient on the--
let me make it very clear.

00:12:10.490 --> 00:12:14.230
B was the coefficient on the--
or it was a term on top of the

00:12:14.230 --> 00:12:17.400
s squared plus 1, so that's
why I'm using B there.

00:12:17.400 --> 00:12:22.730
And then D is minus B,
so D is minus 1.

00:12:22.730 --> 00:12:23.510
So let me make sure
I have that.

00:12:23.510 --> 00:12:29.060
B is 1/3 minus-- let me make
sure I get that right.

00:12:29.060 --> 00:12:30.260
D is 1/3.

00:12:30.260 --> 00:12:36.830
So, sorry, B as in boy is
1/3, so D is minus 1/3.

00:12:36.830 --> 00:12:40.930
So B, there's a term on top
of the s squared plus 1.

00:12:40.930 --> 00:12:45.560
And then you have minus D over
the minus 1/3 over s

00:12:45.560 --> 00:12:50.510
squared plus 4.

00:12:50.510 --> 00:12:53.560
This takes a lot of stamina
to record this video.

00:12:53.560 --> 00:12:56.370
I hope you appreciate it.

00:12:56.370 --> 00:12:59.090
OK, so let me rewrite
everything, just so we can get

00:12:59.090 --> 00:13:01.450
back to the problem because
when you take the partial

00:13:01.450 --> 00:13:05.740
fraction detour, you forget--
not even to speak of the

00:13:05.740 --> 00:13:07.620
problem, you forget
what day it is.

00:13:07.620 --> 00:13:12.810
Let's see, so you get the
Laplace transform of y is

00:13:12.810 --> 00:13:20.060
equal to 1 minus e to the minus
2 pi s times what that

00:13:20.060 --> 00:13:23.673
mess that we just solved
for, times-- and I'll

00:13:23.673 --> 00:13:24.360
write it like this.

00:13:24.360 --> 00:13:34.440
1/3 times 1 over s squared
plus 1 minus 1/3 times--

00:13:34.440 --> 00:13:35.470
actually, let me write
it this way.

00:13:35.470 --> 00:13:38.190
Because I have this s squared
plus 4, so I really want to

00:13:38.190 --> 00:13:39.880
have a 2 there.

00:13:39.880 --> 00:13:41.680
So I want to have a 2 in the
numerator, so you want to have

00:13:41.680 --> 00:13:44.825
a 2 over s squared plus 4.

00:13:44.825 --> 00:13:48.690
So if I put a 2 in the numerator
there, I have to

00:13:48.690 --> 00:13:51.440
divide this by 2 as well.

00:13:51.440 --> 00:13:55.140
So let me change this to a 6.

00:13:55.140 --> 00:13:58.690
Minus 1/6 times 2
is minus 1/3.

00:13:58.690 --> 00:14:00.800
So I did that just so I get
this in the form of the

00:14:00.800 --> 00:14:03.860
Laplace transform
of sine of t.

00:14:03.860 --> 00:14:06.920
Now, let's see if there's
anything that

00:14:06.920 --> 00:14:09.090
I can do from here.

00:14:09.090 --> 00:14:11.640
This is an epic problem.

00:14:11.640 --> 00:14:13.750
I'll be amazed if I don't
make a careless

00:14:13.750 --> 00:14:16.320
mistake while I do this.

00:14:16.320 --> 00:14:19.170
So we can rewrite everything.

00:14:19.170 --> 00:14:21.280
Let's see if we can
simplify this.

00:14:21.280 --> 00:14:23.880
And by simplifying it, I'm just
going to make it longer.

00:14:23.880 --> 00:14:28.400
We can write the Laplace
transform of y is equal to--

00:14:28.400 --> 00:14:30.640
I'm just going to multiply the
1 out, and then I'm going to

00:14:30.640 --> 00:14:32.660
multiply the e to the
minus 2 pi s out.

00:14:32.660 --> 00:14:39.540
So if you multiply the 1 out,
you get 1/3 times 1 over s

00:14:39.540 --> 00:14:43.840
squared plus 1-- I'm just
multiplying the 1 out-- minus

00:14:43.840 --> 00:14:50.060
1/6-- these are all the 1's
times the 1-- times 2 over s

00:14:50.060 --> 00:14:52.020
squared plus 4.

00:14:52.020 --> 00:14:55.660
And then I'm going to multiply
the minus e.

00:14:55.660 --> 00:14:58.670
Let me just switch colors,
do the minus e.

00:14:58.670 --> 00:15:06.820
So then you get minus e to the
minus 2 pi s over 3 times 1

00:15:06.820 --> 00:15:09.430
over s squared plus 1.

00:15:09.430 --> 00:15:11.860
And then the minus and the minus
cancel out, so you get

00:15:11.860 --> 00:15:20.680
plus e to the minus 2 pi
s over 6 times 2 over s

00:15:20.680 --> 00:15:22.680
squared plus 4.

00:15:22.680 --> 00:15:25.630
Now, taking the inverse Laplace
transform of these

00:15:25.630 --> 00:15:26.800
things are pretty
straightforward.

00:15:26.800 --> 00:15:27.580
So let's do that.

00:15:27.580 --> 00:15:31.000
Let's take the inverse Laplace
transform of the whole thing.

00:15:31.000 --> 00:15:36.280
And we get y is equal to the
inverse Laplace transform of

00:15:36.280 --> 00:15:44.390
this guy right here, is just 1/3
sine of t-- I don't have

00:15:44.390 --> 00:15:48.330
to write a parentheses there--
sine of t, and then this is

00:15:48.330 --> 00:15:56.590
minus 1/6 times-- this
is the Laplace

00:15:56.590 --> 00:15:57.840
transform of sine of 2t.

00:16:00.820 --> 00:16:02.730
That's that term right there.

00:16:02.730 --> 00:16:05.410
Now, these are almost the same,
but we have this little

00:16:05.410 --> 00:16:06.700
pesky character over here.

00:16:06.700 --> 00:16:10.250
We have this e to the
minus 2 pi s.

00:16:10.250 --> 00:16:12.486
And there, we just have to
remind ourselves-- I'll write

00:16:12.486 --> 00:16:13.550
it here in the bottom.

00:16:13.550 --> 00:16:15.520
We just have to remind ourselves
that the Laplace

00:16:15.520 --> 00:16:21.380
transform of the unit step
function-- I'll put the pi

00:16:21.380 --> 00:16:28.470
there, just 2 pi times f of t
minus 2 pi-- I should put as

00:16:28.470 --> 00:16:36.260
the step function of t-- is
equal to e to the minus 2 pi s

00:16:36.260 --> 00:16:39.865
times the Laplace transform of
just-- or let me just write it

00:16:39.865 --> 00:16:45.860
this way-- times the Laplace
transform of f of t.

00:16:45.860 --> 00:16:50.040
So if we view f of t as just
sine of t or sine of 2t, then

00:16:50.040 --> 00:16:52.290
we can kind of backwards
pattern match.

00:16:52.290 --> 00:16:54.790
And we'll have to shift
it and multiply it by

00:16:54.790 --> 00:16:58.500
the unit step function.

00:16:58.500 --> 00:17:00.050
So I want to make that clear.

00:17:00.050 --> 00:17:03.690
If you didn't have this guy
here, the inverse Laplace

00:17:03.690 --> 00:17:06.010
transform of this guy would be
the same thing as this guy.

00:17:06.010 --> 00:17:07.110
It'd just be sine of t.

00:17:07.110 --> 00:17:09.210
The inverse Laplace transform
of this guy

00:17:09.210 --> 00:17:10.579
would be sine of 2t.

00:17:10.579 --> 00:17:13.450
But we have this pesky
character here, which

00:17:13.450 --> 00:17:16.079
essentially, instead of having
the inverse Laplace transform

00:17:16.079 --> 00:17:19.190
just being our f of t, it's
going to be our f of t shifted

00:17:19.190 --> 00:17:23.510
by 2 pi times the unit
step function, where

00:17:23.510 --> 00:17:26.050
it steps up at 2pi.

00:17:26.050 --> 00:17:34.340
So this is going to be minus
1/3 times the unit step

00:17:34.340 --> 00:17:41.010
function, where c is 2 pi of t
times-- instead of sine of t--

00:17:41.010 --> 00:17:46.150
sine of t minus 2pi.

00:17:46.150 --> 00:17:49.170
And then we're almost done.

00:17:49.170 --> 00:17:51.660
I'll do it in magenta
to celebrate it.

00:17:51.660 --> 00:17:58.030
Plus this very last term, which
is 1/6 times the unit

00:17:58.030 --> 00:18:03.550
step function 2 pi of t, the
unit step function that steps

00:18:03.550 --> 00:18:13.780
up at 2 pi times sine of-- and
we have to be careful here.

00:18:13.780 --> 00:18:16.990
Wherever we had a t before,
we're going to replace it with

00:18:16.990 --> 00:18:18.750
a t minus 2 pi.

00:18:18.750 --> 00:18:20.850
So sine of, instead of
2t, is going to be 2

00:18:20.850 --> 00:18:25.270
times t minus 2 pi.

00:18:25.270 --> 00:18:26.540
And there you have it.

00:18:26.540 --> 00:18:31.650
We finally have solved our
very hairy problem.

00:18:31.650 --> 00:18:34.600
We could take some time
if we want to simplify

00:18:34.600 --> 00:18:35.250
this a little bit.

00:18:35.250 --> 00:18:36.780
In fact, we might as well.

00:18:36.780 --> 00:18:39.580
At the risk of making a careless
mistake at the last

00:18:39.580 --> 00:18:42.690
moment, let me see if I can make
any simplifications here.

00:18:48.350 --> 00:18:52.470
Well, we could factor out this
guy right here, but other than

00:18:52.470 --> 00:18:54.090
that, that seems about as
simple as we can get.

00:18:54.090 --> 00:18:58.180
So this is our function of t
that satisfies our otherwise

00:18:58.180 --> 00:19:01.330
simple-looking differential
equation that we had up here.

00:19:01.330 --> 00:19:03.850
This looked fairly
straightforward, but we got

00:19:03.850 --> 00:19:08.340
this big mess to actually
satisfy that equation, given

00:19:08.340 --> 00:19:11.130
those initial conditions
that we had initially.


Inverse Laplace examples

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:03.720
A lot of what we do with Laplace
transforms, taking

00:00:03.720 --> 00:00:05.070
them and taking their
inverse, it's a

00:00:05.070 --> 00:00:06.260
lot of pattern matching.

00:00:06.260 --> 00:00:08.380
And it shouldn't just be a
mechanical thing, and that's

00:00:08.380 --> 00:00:10.130
why I've gone through the
exercise of showing

00:00:10.130 --> 00:00:11.270
you why they work.

00:00:11.270 --> 00:00:13.650
But in order to just kind of
make sure we don't get

00:00:13.650 --> 00:00:16.370
confused, I think it might be
useful to review a little bit

00:00:16.370 --> 00:00:17.890
of everything that we've
learned so far.

00:00:17.890 --> 00:00:21.070
So in the last video, we saw
that the Laplace transform--

00:00:21.070 --> 00:00:22.760
well, let me just
write something.

00:00:22.760 --> 00:00:26.930
The Laplace transform of f of
t, let me just get some

00:00:26.930 --> 00:00:31.210
notation down, and we can write
that as big capital F of

00:00:31.210 --> 00:00:33.210
s, and I've told you
that before.

00:00:33.210 --> 00:00:36.090
And so given that, in the last
video I showed you that if we

00:00:36.090 --> 00:00:39.960
have to deal with the unit step
function, so if I said,

00:00:39.960 --> 00:00:43.810
look, the Laplace transform of
the unit step function, it

00:00:43.810 --> 00:00:49.690
becomes 1 at some value c times
some shifted function f

00:00:49.690 --> 00:00:54.420
of t minus c, in the last video,
we saw that this is

00:00:54.420 --> 00:01:01.530
just equal to e to the minus cs
times the Laplace transform

00:01:01.530 --> 00:01:05.079
of just this function right
there, so times the F of s.

00:01:08.005 --> 00:01:10.780
And it's really important not
to get this confused with

00:01:10.780 --> 00:01:14.530
another Laplace transform
property or rule, or whatever

00:01:14.530 --> 00:01:16.310
you want to call it, that
we figured out.

00:01:16.310 --> 00:01:18.840
I think it was one of the videos
that I made last year,

00:01:18.840 --> 00:01:20.690
but if you're just following
these in order I think it's

00:01:20.690 --> 00:01:22.870
three or four videos ago.

00:01:22.870 --> 00:01:31.000
And that one told us that the
Laplace transform of e to the

00:01:31.000 --> 00:01:40.710
at, times f of t, that this is
equal to-- and I want to make

00:01:40.710 --> 00:01:42.340
this distinction very clear.

00:01:42.340 --> 00:01:45.740
Here we shifted the f of t
and we got just kind of a

00:01:45.740 --> 00:01:47.580
regular F of s.

00:01:47.580 --> 00:01:50.660
In this situation, when we
multiply it times a e to the

00:01:50.660 --> 00:01:57.480
positive at, we end up shifting
the actual transform.

00:01:57.480 --> 00:02:01.480
So this becomes F
of s minus a.

00:02:01.480 --> 00:02:04.720
And these two rules, or
properties, or whatever you

00:02:04.720 --> 00:02:06.690
want to call them, they're
very easy to

00:02:06.690 --> 00:02:07.750
confuse with each other.

00:02:07.750 --> 00:02:09.509
So we're going to do a couple
of examples that we're going

00:02:09.509 --> 00:02:12.040
to have to figure out which
one of these two apply.

00:02:12.040 --> 00:02:14.350
Let's write all the other stuff
that we learned as well.

00:02:14.350 --> 00:02:16.740
The very first thing we learned
was that the Laplace

00:02:16.740 --> 00:02:21.740
transform of 1 was
equal to 1/s.

00:02:21.740 --> 00:02:24.270
We know that that's a pretty
straightforward one, easy to

00:02:24.270 --> 00:02:25.320
prove to yourself.

00:02:25.320 --> 00:02:27.930
And more generally, we learned
that the Laplace transform of

00:02:27.930 --> 00:02:34.330
t to the n, where n is a
positive integer, it equaled n

00:02:34.330 --> 00:02:42.120
factorial over s to
the n plus 1.

00:02:42.120 --> 00:02:44.500
And then we had our
trig functions

00:02:44.500 --> 00:02:46.150
that we've gone over.

00:02:46.150 --> 00:02:49.340
Let me do this in a
different color.

00:02:49.340 --> 00:02:50.130
I'll do it right here.

00:02:50.130 --> 00:02:58.980
The Laplace transform of sine
of at is equal to a over s

00:02:58.980 --> 00:03:01.530
squared, plus a squared.

00:03:01.530 --> 00:03:08.760
And the Laplace transform of the
cosine of at is equal to s

00:03:08.760 --> 00:03:11.660
over s squared plus a squared.

00:03:11.660 --> 00:03:15.330
And you'll be amazed by how far
we can go with just what

00:03:15.330 --> 00:03:15.980
I've written here.

00:03:15.980 --> 00:03:20.020
In future videos, we're going
to broaden our toolkit even

00:03:20.020 --> 00:03:22.610
further, but just these right
here, you can already do a

00:03:22.610 --> 00:03:26.170
whole set of Laplace transforms
and inverse Laplace

00:03:26.170 --> 00:03:28.330
transforms. So let's
try to do a few.

00:03:28.330 --> 00:03:30.435
So let's say I were to give
you the Laplace transform.

00:03:30.435 --> 00:03:32.410
And you know, this is
just the hard part.

00:03:32.410 --> 00:03:34.830
I think you know how to solve
a differential equation, if

00:03:34.830 --> 00:03:36.360
you know how to take the
Laplace transforms

00:03:36.360 --> 00:03:37.240
and go back and forth.

00:03:37.240 --> 00:03:39.860
The hard part is just
recognizing or inverting your

00:03:39.860 --> 00:03:42.640
Laplace transforms. So let's
say we had the Laplace

00:03:42.640 --> 00:03:46.760
transform of some
function F of s.

00:03:46.760 --> 00:03:56.260
Let's say it's 3 factorial over
s minus 2 to the fourth.

00:03:56.260 --> 00:04:00.210
Now, your pattern matching, or
your pattern recognition part

00:04:00.210 --> 00:04:01.810
of your brain, should
immediately say, look, I have

00:04:01.810 --> 00:04:03.880
a Laplace transform of something
that has a factorial

00:04:03.880 --> 00:04:05.860
in it, and it's over
an exponent.

00:04:05.860 --> 00:04:08.770
This must be something
related to this

00:04:08.770 --> 00:04:10.540
thing right here, right?

00:04:10.540 --> 00:04:13.240
If I just had the Laplace
transform-- let me write that

00:04:13.240 --> 00:04:18.670
down-- the Laplace transform
of-- you see a 3 factorial and

00:04:18.670 --> 00:04:22.380
a fourth power, so it looks
like n is equal to 3.

00:04:22.380 --> 00:04:28.200
So if you write the Laplace
transform of t to the 3, this

00:04:28.200 --> 00:04:31.530
rule that we showed right here,
this means that it would

00:04:31.530 --> 00:04:38.550
be equal to 3 factorial
over s to the fourth.

00:04:38.550 --> 00:04:41.430
Now, this thing isn't
exactly this thing.

00:04:41.430 --> 00:04:43.530
They're not quite
the same thing.

00:04:43.530 --> 00:04:46.150
You know, I'm doing this to
instruct you, but I find

00:04:46.150 --> 00:04:48.210
these, when I'm actually doing
them on an exam-- I remember

00:04:48.210 --> 00:04:51.370
when I did them when I first
learned this, I would actually

00:04:51.370 --> 00:04:54.290
go through this step because you
definitely don't want to

00:04:54.290 --> 00:04:56.690
make a careless mistake and you
definitely want to make

00:04:56.690 --> 00:04:58.610
sure you have a good handle
on what you're doing.

00:04:58.610 --> 00:05:00.970
So you're like, OK, it's
something related to this, but

00:05:00.970 --> 00:05:04.780
what's the difference between
this expression right here and

00:05:04.780 --> 00:05:07.020
the expression that we're trying
to take the inverse

00:05:07.020 --> 00:05:09.720
Laplace transform of,
and this one here?

00:05:09.720 --> 00:05:12.310
Well, we've shifted our s.

00:05:12.310 --> 00:05:22.600
If we call this expression right
here F of s, then what's

00:05:22.600 --> 00:05:24.110
this expression?

00:05:24.110 --> 00:05:28.200
This expression right here
is F of s minus 2.

00:05:32.100 --> 00:05:34.410
So what are we dealing
with here?

00:05:34.410 --> 00:05:38.730
So you see here, you have
a shifted F of s.

00:05:38.730 --> 00:05:43.760
So in this case, a would
be equal to 2.

00:05:43.760 --> 00:05:48.180
So this is the Laplace transform
of e to the at times

00:05:48.180 --> 00:05:51.390
our f of t.

00:05:51.390 --> 00:05:52.660
So let me write this down.

00:05:52.660 --> 00:05:57.850
This is the Laplace transform
of e to the-- and what's a?

00:05:57.850 --> 00:06:00.440
a is what we shifted by.

00:06:00.440 --> 00:06:03.170
It's what we shifted by minus
a, so you have a positive a,

00:06:03.170 --> 00:06:08.060
so e to the 2t times the
actual function.

00:06:08.060 --> 00:06:10.415
If this was just an F of s,
what would f of t be?

00:06:10.415 --> 00:06:15.550
Well, we figured out, it's t the
3, t to the third power.

00:06:15.550 --> 00:06:19.260
So the Laplace transform of
this is equal to that.

00:06:19.260 --> 00:06:25.250
Or we could write that the
inverse Laplace transform of 3

00:06:25.250 --> 00:06:31.600
factorial over s minus 2 to the
fourth is equal to e to

00:06:31.600 --> 00:06:35.300
the 2t times t to the third.

00:06:35.300 --> 00:06:39.420
Now, if that seemed confusing
to you, you

00:06:39.420 --> 00:06:40.670
can kind of go forward.

00:06:43.150 --> 00:06:45.730
Let's go the other direction,
and maybe this will make it a

00:06:45.730 --> 00:06:47.250
little bit clearer for you.

00:06:47.250 --> 00:06:48.570
So let's go from
this direction.

00:06:48.570 --> 00:06:51.390
If I have to take the Laplace
transform of this thing, I'd

00:06:51.390 --> 00:06:53.370
say, OK, well, the Laplace
transform of t to

00:06:53.370 --> 00:06:54.620
the third is easy.

00:06:57.340 --> 00:07:00.570
I think the tool isn't working
right there properly.

00:07:00.570 --> 00:07:02.630
Let me scroll up a little bit.

00:07:02.630 --> 00:07:04.540
So I could write
it right here.

00:07:04.540 --> 00:07:08.385
So if I wanted to figure out the
Laplace transform of e to

00:07:08.385 --> 00:07:12.600
the 2t times t to the third,
I'll say, well, you know, this

00:07:12.600 --> 00:07:14.950
e to the 2t, I remember that
it shifts something.

00:07:14.950 --> 00:07:20.050
So if I know that the Laplace
transform of t the third, this

00:07:20.050 --> 00:07:20.810
is an easy one.

00:07:20.810 --> 00:07:25.710
It's equal to 3 factorial
over s to the fourth.

00:07:25.710 --> 00:07:27.160
That's 3 plus 1.

00:07:27.160 --> 00:07:30.780
Then the Laplace transform of e
to the 2t times t the third

00:07:30.780 --> 00:07:32.650
is going to be this shifted.

00:07:32.650 --> 00:07:35.570
This is equal to F of s.

00:07:35.570 --> 00:07:41.600
Then this is going to
be f of s minus 2.

00:07:41.600 --> 00:07:43.550
So what's F of s minus 2?

00:07:43.550 --> 00:07:49.550
It's going to be equal to 3
factorial over s minus 2 to

00:07:49.550 --> 00:07:50.580
the fourth.

00:07:50.580 --> 00:07:52.420
I think you're already getting
an appreciation that the

00:07:52.420 --> 00:07:55.480
hardest thing about these
Laplace transform problems are

00:07:55.480 --> 00:07:58.380
really kind of all of these
shifts and kind of recognizing

00:07:58.380 --> 00:08:01.410
the patterns and recognizing
what's your a, and what's your

00:08:01.410 --> 00:08:04.220
c, and being very careful about
it so you don't make a

00:08:04.220 --> 00:08:05.070
careless mistake.

00:08:05.070 --> 00:08:07.430
And I think doing a lot of
examples probably helps a lot,

00:08:07.430 --> 00:08:11.500
so let's do a couple of more
to kind of make sure things

00:08:11.500 --> 00:08:14.710
really get hammered home
in your brain.

00:08:14.710 --> 00:08:16.840
So let's try this
one right here.

00:08:16.840 --> 00:08:19.640
This looks a little bit
more complicated.

00:08:19.640 --> 00:08:23.030
They give us that the Laplace
transform of some function is

00:08:23.030 --> 00:08:32.070
equal to 2 times s minus 1 times
e to the minus 2s, all

00:08:32.070 --> 00:08:39.940
of that over s squared
minus 2s plus 2.

00:08:39.940 --> 00:08:41.600
Now this looks very daunting.

00:08:41.600 --> 00:08:41.990
How do you do this?

00:08:41.990 --> 00:08:42.680
I have an e here.

00:08:42.680 --> 00:08:44.049
I have something shifted here.

00:08:44.049 --> 00:08:47.260
I have this polynomial in
the denominator here.

00:08:47.260 --> 00:08:48.720
What can I do with this?

00:08:48.720 --> 00:08:51.080
So the first thing, when I look
at these polynomials in

00:08:51.080 --> 00:08:53.620
the denominator, I say can
I factor it somehow?

00:08:53.620 --> 00:08:55.800
Can I factor it fairly simply?

00:08:55.800 --> 00:08:58.830
And actually, in the exams
that you'll find in

00:08:58.830 --> 00:09:00.280
differential equation class,
they'll never give you

00:09:00.280 --> 00:09:03.920
something that's factorable
into these weird numbers.

00:09:03.920 --> 00:09:05.240
It tends to be integers.

00:09:05.240 --> 00:09:07.200
So you see, OK, what
two numbers?

00:09:07.200 --> 00:09:09.060
They have to be positive.

00:09:09.060 --> 00:09:10.990
When you give their product,
you get 2.

00:09:10.990 --> 00:09:13.430
And then when you add them, you
get negative 2, or they

00:09:13.430 --> 00:09:14.750
could both be negative.

00:09:14.750 --> 00:09:17.360
But there's no two easy
numbers, not 1 and 2.

00:09:17.360 --> 00:09:18.800
None of those work.

00:09:18.800 --> 00:09:21.820
So if you can't factor this
outright, the next idea is

00:09:21.820 --> 00:09:24.290
maybe we could complete the
square and maybe this will

00:09:24.290 --> 00:09:26.470
match one of the cosine
or the sine formulas.

00:09:26.470 --> 00:09:29.280
So how can we complete the
square in this denominator?

00:09:29.280 --> 00:09:38.640
Well, this can be rewritten
as s squared minus 2s.

00:09:38.640 --> 00:09:42.010
And I'm going to put
a plus 2 out here.

00:09:42.010 --> 00:09:44.230
And you can watch, I have
a bunch of videos on the

00:09:44.230 --> 00:09:45.650
completing of the square,
if all of this

00:09:45.650 --> 00:09:46.830
looks foreign to you.

00:09:46.830 --> 00:09:49.090
And to complete the square, we
just want to turn this into a

00:09:49.090 --> 00:09:50.130
perfect square.

00:09:50.130 --> 00:09:54.850
So to turn this into a perfect
square-- so something when I

00:09:54.850 --> 00:09:58.720
add it to itself twice becomes
minus 2, and so that when I

00:09:58.720 --> 00:10:03.030
square it, when I add it to
itself twice, it becomes minus

00:10:03.030 --> 00:10:04.300
2, it's minus 1.

00:10:04.300 --> 00:10:07.700
And when I square it,
it'll become plus 1.

00:10:07.700 --> 00:10:10.160
I can't just add plus 1
arbitrarily to some

00:10:10.160 --> 00:10:11.650
expression, I have to
make it neutral.

00:10:11.650 --> 00:10:12.920
So let me subtract 1.

00:10:12.920 --> 00:10:13.730
I haven't changed this.

00:10:13.730 --> 00:10:15.630
I added 1 and I subtracted 1.

00:10:15.630 --> 00:10:17.890
A little bit of a primer on
completing the square.

00:10:17.890 --> 00:10:21.610
But by doing this, I now can
call this expression right

00:10:21.610 --> 00:10:27.780
here, I can now say that this
thing is s minus 1 squared.

00:10:27.780 --> 00:10:32.430
And then this stuff out here,
this out here is 2 minus 1.

00:10:32.430 --> 00:10:34.410
This is just plus 1.

00:10:34.410 --> 00:10:41.100
So I can rewrite my entire
expression now as 2 times s

00:10:41.100 --> 00:10:45.560
minus 1 times e to the minus
2s-- make sure I'm not

00:10:45.560 --> 00:10:51.420
clipping off at the top-- e to
the minus 2s, all of that over

00:10:51.420 --> 00:10:56.620
s minus 1 squared plus 1.

00:10:56.620 --> 00:11:01.880
So a couple of interesting
things seem to

00:11:01.880 --> 00:11:06.020
be happening here.

00:11:06.020 --> 00:11:09.710
Let's just do a couple of test
Laplace transforms. So if a

00:11:09.710 --> 00:11:15.610
Laplace transform of cosine of
t, we know that this is equal

00:11:15.610 --> 00:11:23.050
to s over s squared plus 1,
which this kind of looks like

00:11:23.050 --> 00:11:26.180
if this was an s and this
was an s squared plus 1.

00:11:26.180 --> 00:11:30.960
If this was F of s,
then what is this?

00:11:30.960 --> 00:11:34.130
Well, let's ignore this guy
right here for a little bit.

00:11:34.130 --> 00:11:34.920
So what is it?

00:11:34.920 --> 00:11:36.930
We know, actually, from
the last video.

00:11:36.930 --> 00:11:43.720
We saw, well, what if we took
the Laplace transform of e to

00:11:43.720 --> 00:11:45.540
the-- I'll call it 1t.

00:11:45.540 --> 00:11:49.000
But let's say e to the-- yeah
I'll just write it e to the 1t

00:11:49.000 --> 00:11:52.250
times cosine of t?

00:11:52.250 --> 00:11:55.410
Well, then this will just
shift this Laplace

00:11:55.410 --> 00:11:56.995
transform by 1.

00:11:56.995 --> 00:12:00.100
It will shift it by
1 to the right.

00:12:00.100 --> 00:12:03.270
Wherever you see an s, you
would put an s minus a 1.

00:12:03.270 --> 00:12:08.900
So this will be equal to s
minus 1 over s minus 1

00:12:08.900 --> 00:12:11.200
squared plus 1.

00:12:11.200 --> 00:12:13.270
We're getting close.

00:12:13.270 --> 00:12:18.390
We now figured out this
part right here.

00:12:18.390 --> 00:12:22.440
Now, in the previous video, I
think it was two videos ago,

00:12:22.440 --> 00:12:25.040
or maybe it was the last
video, I forget.

00:12:25.040 --> 00:12:27.000
Memory fails me.

00:12:27.000 --> 00:12:31.060
I showed you that if you have
the Laplace transform of the

00:12:31.060 --> 00:12:38.970
unit step function of t times
some f of t shifted by some

00:12:38.970 --> 00:12:50.160
value of c, then that this is
equal to e to the minus cs

00:12:50.160 --> 00:12:54.260
times F of s.

00:12:54.260 --> 00:12:58.260
OK, And this can get
very confusing.

00:12:58.260 --> 00:13:00.000
This can get very confusing,
so I want to be

00:13:00.000 --> 00:13:01.290
very careful here.

00:13:01.290 --> 00:13:03.190
Let's ignore all of this.

00:13:03.190 --> 00:13:06.660
I called this F of s before, but
now I'm going to backtrack

00:13:06.660 --> 00:13:07.130
a little bit.

00:13:07.130 --> 00:13:09.980
And let's just ignore this,
because I'm going to redefine

00:13:09.980 --> 00:13:10.670
our F of s.

00:13:10.670 --> 00:13:12.670
So let's just ignore
that for a second.

00:13:12.670 --> 00:13:15.940
Let's define our new
f of t to be this.

00:13:15.940 --> 00:13:17.260
Let's say that that is f of t.

00:13:17.260 --> 00:13:23.690
Let's say f of t is equal to
e to the t cosine of t.

00:13:23.690 --> 00:13:27.200
Then if you take the Laplace
transform of that, that means

00:13:27.200 --> 00:13:35.850
that F of s is equal to s
minus 1 over s minus 1

00:13:35.850 --> 00:13:38.050
squared plus 1.

00:13:38.050 --> 00:13:39.040
Nothing fancy there.

00:13:39.040 --> 00:13:41.190
I just defined our f of
t as this, and then

00:13:41.190 --> 00:13:44.400
our F of s is that.

00:13:44.400 --> 00:13:48.390
Now, we have a situation here.

00:13:48.390 --> 00:13:49.880
Let's ignore the 2 here.

00:13:49.880 --> 00:13:52.410
The 2 is just kind of
a scaling factor.

00:13:52.410 --> 00:13:57.180
This expression right here, we
can rewrite as that expression

00:13:57.180 --> 00:13:59.610
is equal to-- this
is our F of s.

00:13:59.610 --> 00:14:05.600
This expression right here is
equal to 2 times our F of s

00:14:05.600 --> 00:14:09.150
times e to the minus 2s.

00:14:09.150 --> 00:14:10.230
Or let me just write it.

00:14:10.230 --> 00:14:13.440
Let me switch the order, just
so we make it look right.

00:14:13.440 --> 00:14:19.140
2 times e to the minus
2s times F of s.

00:14:19.140 --> 00:14:22.720
Well, that looks just like
this if our 2 was

00:14:22.720 --> 00:14:23.970
equal to our c.

00:14:27.000 --> 00:14:28.570
So what does that tell us?

00:14:28.570 --> 00:14:31.590
That tells us that the inverse
Laplace transform, if we take

00:14:31.590 --> 00:14:33.660
the inverse Laplace
transform-- and

00:14:33.660 --> 00:14:37.060
let's ignore the 2.

00:14:37.060 --> 00:14:39.540
Let's do the inverse Laplace
transform of the whole thing.

00:14:39.540 --> 00:14:44.020
The inverse Laplace transform
of this thing is going to be

00:14:44.020 --> 00:14:47.070
equal to-- we can just write
the 2 there as a scaling

00:14:47.070 --> 00:14:52.000
factor, 2 there times
this thing times

00:14:52.000 --> 00:14:55.010
the unit step function.

00:14:55.010 --> 00:14:56.580
What's our c?

00:14:56.580 --> 00:14:57.650
You can just pattern match.

00:14:57.650 --> 00:14:58.330
You have a 2 here.

00:14:58.330 --> 00:15:01.780
You have a c, a minus c,
a minus 2, so c is 2.

00:15:01.780 --> 00:15:06.380
The unit step function is zero
until it gets to 2 times t, or

00:15:06.380 --> 00:15:12.180
of t, so, then it becomes 1
after t is equal to 2, times

00:15:12.180 --> 00:15:19.110
our function shifted by 2.

00:15:19.110 --> 00:15:21.990
So this is our inverse
Laplace transform.

00:15:21.990 --> 00:15:24.950
Now, what was our function?

00:15:24.950 --> 00:15:29.140
Our function was this
thing right here.

00:15:29.140 --> 00:15:32.090
So if our inverse Laplace
transform of that thing that I

00:15:32.090 --> 00:15:38.170
had written is this thing, an
f of t, f of t is equal to e

00:15:38.170 --> 00:15:41.650
to the t cosine of t.

00:15:41.650 --> 00:15:44.370
Then our inverse-- let me
write all of this down.

00:15:44.370 --> 00:15:47.180
Let me write our big result.

00:15:47.180 --> 00:15:51.470
We established that the inverse
Laplace transform of

00:15:51.470 --> 00:15:55.300
that big thing that I had
written before, 2 times s

00:15:55.300 --> 00:16:05.890
minus 1 times e to the minus 2--
sorry, e to the minus 2s

00:16:05.890 --> 00:16:13.820
over s squared minus 2s plus 2
is equal to this thing where f

00:16:13.820 --> 00:16:14.770
of t is this.

00:16:14.770 --> 00:16:20.700
Or we could just rewrite this
as 2 times the unit step

00:16:20.700 --> 00:16:24.130
function starting at 2, where
that's when it becomes

00:16:24.130 --> 00:16:28.630
non-zero of t times f of t minus
2. f of t minus 2 is

00:16:28.630 --> 00:16:32.190
this with t being replaced
by t minus 2.

00:16:32.190 --> 00:16:35.120
I'll do it in another color,
just to ease the monotony.

00:16:35.120 --> 00:16:43.920
So it would be e to the t minus
2 cosine of t minus 2.

00:16:43.920 --> 00:16:45.910
Now, you might be thinking, Sal,
you know, he must have

00:16:45.910 --> 00:16:47.990
taken all these baby steps with
this problem, because

00:16:47.990 --> 00:16:49.790
he's trying to explain
it to me.

00:16:49.790 --> 00:16:53.370
But I'm taking baby steps with
this problem so that I myself

00:16:53.370 --> 00:16:54.320
don't get confused.

00:16:54.320 --> 00:16:56.750
And I think it's essential
that you do

00:16:56.750 --> 00:16:58.110
take these baby steps.

00:16:58.110 --> 00:17:00.320
And let's just think about
what baby steps we took.

00:17:00.320 --> 00:17:01.350
And I really want
to review this.

00:17:01.350 --> 00:17:02.800
This is actually a surprisingly
good problem.

00:17:02.800 --> 00:17:05.930
I didn't realize it when I
first decided to do it.

00:17:05.930 --> 00:17:07.020
We solved this thing.

00:17:07.020 --> 00:17:09.930
We wanted to get this
denominator into some form

00:17:09.930 --> 00:17:14.560
that is vaguely useful to us,
so I completed the square

00:17:14.560 --> 00:17:17.940
there and then we rewrote
our Laplace transform,

00:17:17.940 --> 00:17:19.770
our f of s like this.

00:17:19.770 --> 00:17:21.339
And then we used a little
pattern recognition.

00:17:21.339 --> 00:17:23.869
We said, look, if I take the
Laplace transform of cosine of

00:17:23.869 --> 00:17:27.750
t, I'd get s over s
squared plus 1.

00:17:27.750 --> 00:17:29.500
But this isn't s over
s squared plus 1.

00:17:29.500 --> 00:17:33.180
It's s minus 1 over s minus
1 squared plus 1.

00:17:33.180 --> 00:17:36.450
So we said, oh, well, that means
that we're multiplying

00:17:36.450 --> 00:17:38.300
our original time
domain function.

00:17:38.300 --> 00:17:43.850
We're multiplying our f of
t times e to the 1t.

00:17:43.850 --> 00:17:46.200
And that's what we got there.

00:17:46.200 --> 00:17:49.460
So the Laplace transform of e to
the t cosine of t became s

00:17:49.460 --> 00:17:53.180
minus 1 over s minus
1 squared plus 1.

00:17:53.180 --> 00:17:56.970
And then we had this e to the
minus 2s this entire time.

00:17:56.970 --> 00:18:01.470
And that's where we said, hey,
if we have e to the minus 2s

00:18:01.470 --> 00:18:04.610
in our Laplace transform, when
you take the inverse Laplace

00:18:04.610 --> 00:18:09.760
transform, it must be the step
function times the shifted

00:18:09.760 --> 00:18:11.020
version of that function.

00:18:11.020 --> 00:18:12.200
And that's why I was
very careful.

00:18:12.200 --> 00:18:14.320
And you had this 2 hanging out
the whole time, and I could

00:18:14.320 --> 00:18:15.030
have used that any time.

00:18:15.030 --> 00:18:19.710
But the simple constants
just scale.

00:18:19.710 --> 00:18:21.960
A function is equal to two times
the Laplace transform of

00:18:21.960 --> 00:18:23.410
that function and vice versa.

00:18:23.410 --> 00:18:25.250
So the 2's are very easy to
deal with, so I kind of

00:18:25.250 --> 00:18:27.470
ignored that most of the time.

00:18:27.470 --> 00:18:28.780
But that's why I was
very careful.

00:18:28.780 --> 00:18:34.420
I redefined f of t to be this,
F of s to be this, and said,

00:18:34.420 --> 00:18:39.920
gee, if F of s is this, and if
I'm multiplying it times e to

00:18:39.920 --> 00:18:45.040
the minus 2s, then what I'm
essentially doing, I'm fitting

00:18:45.040 --> 00:18:46.730
this pattern right here.

00:18:46.730 --> 00:18:49.070
And so the answer to my problem
is going to be the

00:18:49.070 --> 00:18:52.450
unit step function-- I just
throw the 2 out there-- the 2

00:18:52.450 --> 00:18:57.960
times the unit step function
times my f of t shifted by c.

00:18:57.960 --> 00:19:00.360
And we established this was
our f of t, so we just

00:19:00.360 --> 00:19:02.500
shifted it by c.

00:19:02.500 --> 00:19:05.570
We shifted it by 2, and we
got our final answer.

00:19:05.570 --> 00:19:09.220
So this is about as hard up to
this point as you'll see an

00:19:09.220 --> 00:19:11.780
inverse Laplace transform
problem.

00:19:11.780 --> 00:19:14.450
So, hopefully, you found that
pretty interesting.


Laplace transform of the unit step function

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:03.540
The whole point in learning
differential equations is that

00:00:03.540 --> 00:00:06.270
eventually we want to model real
physical systems. I know

00:00:06.270 --> 00:00:09.330
everything we've done so far has
really just been a toolkit

00:00:09.330 --> 00:00:12.390
of being able to solve them, but
the whole reason is that

00:00:12.390 --> 00:00:15.410
because differential equations
can describe a lot of systems,

00:00:15.410 --> 00:00:17.320
and then we can actually
model them that way.

00:00:17.320 --> 00:00:20.750
And we know that in the real
world, everything isn't these

00:00:20.750 --> 00:00:23.820
nice continuous functions, so
over the next couple of videos

00:00:23.820 --> 00:00:25.890
we're going to talk about
functions that are a little

00:00:25.890 --> 00:00:28.880
bit more discontinuous than what
you might be used to even

00:00:28.880 --> 00:00:31.460
in kind of a traditional
calculus or traditional

00:00:31.460 --> 00:00:32.880
Precalculus class.

00:00:32.880 --> 00:00:36.910
And the first one is the
unit step function.

00:00:36.910 --> 00:00:39.790
Let's write it as u, and
then I'll put a little

00:00:39.790 --> 00:00:43.290
subscript c here of t.

00:00:43.290 --> 00:00:48.110
And it's defined as when t is--
let me put it this way.

00:00:48.110 --> 00:00:52.150
It's defined as 0 when t is less
than-- whatever subscript

00:00:52.150 --> 00:00:55.220
I put here-- when t
is less than c.

00:00:55.220 --> 00:00:58.380
And it's defined as 1-- that's
why we call it the unit step

00:00:58.380 --> 00:01:02.900
function-- when t is greater
than or equal to c.

00:01:02.900 --> 00:01:05.700
And if I had to graph this, and
you could graph it as well

00:01:05.700 --> 00:01:07.030
but it's not too difficult
to graph.

00:01:07.030 --> 00:01:10.622
Let me draw my x-axis
right here.

00:01:10.622 --> 00:01:13.010
And I'll do a little
thicker line.

00:01:13.010 --> 00:01:15.490
That's my x-axis right there.

00:01:15.490 --> 00:01:20.430
This is my y-axis right there.

00:01:22.990 --> 00:01:25.380
And when we talk about Laplace
transforms, which we'll talk

00:01:25.380 --> 00:01:28.400
about shortly, we only care
about t is greater than 0.

00:01:28.400 --> 00:01:30.090
Because we saw, in our
definition of the Laplace

00:01:30.090 --> 00:01:32.510
transform, we're always taking
the integral from 0 to

00:01:32.510 --> 00:01:34.360
infinity, so we're
only dealing with

00:01:34.360 --> 00:01:36.470
the positive x-axis.

00:01:36.470 --> 00:01:39.160
But anyway, by this definition,
it would be zero

00:01:39.160 --> 00:01:43.970
all the way until you get to
some value c, so you'd be zero

00:01:43.970 --> 00:01:45.400
until you get to c.

00:01:45.400 --> 00:01:49.460
And then at c, you jump, and the
point c is included x is

00:01:49.460 --> 00:01:51.220
equal to c here.

00:01:51.220 --> 00:01:53.070
So it's included, so I'll put
a dot there, because it's

00:01:53.070 --> 00:01:54.300
greater than or equal to c.

00:01:54.300 --> 00:01:57.040
You're at 1, so this
is 1 right here.

00:01:57.040 --> 00:01:59.790
And then you go forward
for all of time.

00:01:59.790 --> 00:02:01.960
And you're like, Sal, you just
said that differential

00:02:01.960 --> 00:02:05.100
equations, we're learning to
model things, why is this type

00:02:05.100 --> 00:02:06.760
of a function useful?

00:02:06.760 --> 00:02:08.580
Well, in the real world,
sometimes you do have

00:02:08.580 --> 00:02:11.100
something that essentially jolts
something, that moves it

00:02:11.100 --> 00:02:12.790
from this position
to that position.

00:02:12.790 --> 00:02:15.450
And obviously, nothing can move
it immediately like this,

00:02:15.450 --> 00:02:18.650
but you might have some
system, it could be an

00:02:18.650 --> 00:02:22.580
electrical system or mechanical
system, where maybe

00:02:22.580 --> 00:02:25.000
the behavior looks something
like this, where maybe it

00:02:25.000 --> 00:02:26.680
moves it like that
or something.

00:02:26.680 --> 00:02:30.090
And this function is a pretty
good analytic approximation

00:02:30.090 --> 00:02:32.490
for some type of real world
behavior like this when

00:02:32.490 --> 00:02:33.920
something just gets moved.

00:02:33.920 --> 00:02:36.010
Whenever we solve these
differential equations

00:02:36.010 --> 00:02:38.540
analytically, we're really
just trying to get a pure

00:02:38.540 --> 00:02:39.730
model of something.

00:02:39.730 --> 00:02:42.610
Eventually, we'll see that it
doesn't perfectly describe

00:02:42.610 --> 00:02:44.730
things, but it helps describe
it enough for us to get a

00:02:44.730 --> 00:02:47.040
sense of what's going
to happen.

00:02:47.040 --> 00:02:49.820
Sometimes it will completely
describe things, but anyway,

00:02:49.820 --> 00:02:53.220
we can ignore that for now, so
let me get rid of these things

00:02:53.220 --> 00:02:54.030
right there.

00:02:54.030 --> 00:02:56.180
So the first question is,
well, you know, what if

00:02:56.180 --> 00:02:58.000
something doesn't jar
just like that?

00:02:58.000 --> 00:03:03.500
What if I want to construct more
fancy unit functions or

00:03:03.500 --> 00:03:05.270
more fancy step functions?

00:03:05.270 --> 00:03:07.470
Let's say I wanted to construct
something that

00:03:07.470 --> 00:03:09.350
looked like this.

00:03:09.350 --> 00:03:13.940
Let me say this is my y-axis.

00:03:13.940 --> 00:03:16.310
This is my x-axis.

00:03:16.310 --> 00:03:18.700
And let's say I wanted to
construct something that is

00:03:18.700 --> 00:03:23.820
at-- and let me do it in
a different color.

00:03:23.820 --> 00:03:29.935
Let's say it's at 2
until I get to pi.

00:03:33.450 --> 00:03:40.410
And then from pi until forever
it just stays at zero.

00:03:40.410 --> 00:03:45.290
So how could I construct this
function right here using my

00:03:45.290 --> 00:03:47.350
unit step function?

00:03:47.350 --> 00:03:51.940
So what if I had written
it as-- so my unit step

00:03:51.940 --> 00:03:57.630
function's at zero initially, so
what if I make it 2 minus a

00:03:57.630 --> 00:04:04.100
unit step function that
starts at pi?

00:04:06.770 --> 00:04:10.710
So if I define my function
here, will this work?

00:04:10.710 --> 00:04:13.250
Well, this unit step function,
when we pass pi, is only going

00:04:13.250 --> 00:04:14.970
to be equal to 1, but
we want this thing

00:04:14.970 --> 00:04:15.900
to be equal to zero.

00:04:15.900 --> 00:04:18.820
So it has to be 2 minus 2, so
I'll have to put at 2 here,

00:04:18.820 --> 00:04:20.010
and this should work.

00:04:20.010 --> 00:04:26.730
When we're at any value below
pi, when t is less than pi

00:04:26.730 --> 00:04:29.970
here, this becomes a zero,
so our function will just

00:04:29.970 --> 00:04:32.870
evaluate to 2, which
is right there.

00:04:32.870 --> 00:04:36.490
But as soon as we hit t is equal
to pi, that pi is the c

00:04:36.490 --> 00:04:40.050
in this example, as soon as
we hit that, the unit step

00:04:40.050 --> 00:04:41.150
function becomes 1.

00:04:41.150 --> 00:04:45.120
We multiply that by 2, and we
have 2 minus 2, and then we

00:04:45.120 --> 00:04:48.650
end up here with zero,

00:04:48.650 --> 00:04:50.810
Now, that might be nice and
everything, but let's say you

00:04:50.810 --> 00:04:54.610
wanted for it to go back up.

00:04:54.610 --> 00:05:03.600
Let's say that instead of it
going like this-- let me kind

00:05:03.600 --> 00:05:07.180
of erase that by overdrawing the
x-axis again-- we want the

00:05:07.180 --> 00:05:09.510
function to jump up again.

00:05:09.510 --> 00:05:10.860
We want it to jump up again.

00:05:10.860 --> 00:05:14.150
And lets say at some value,
let's say it's at 2pi, we want

00:05:14.150 --> 00:05:16.210
the function to jump up again.

00:05:16.210 --> 00:05:18.920
How could we construct this?

00:05:18.920 --> 00:05:20.630
And we could make it jump to
anything, but let's say we

00:05:20.630 --> 00:05:22.410
want it to jump back to 2.

00:05:22.410 --> 00:05:25.290
Well, we could just add another
unit step function

00:05:25.290 --> 00:05:28.810
here, something that would have
been zero all along, all

00:05:28.810 --> 00:05:30.270
the way up until this point.

00:05:30.270 --> 00:05:34.160
But then at 2pi, it jumps,
so in this case,

00:05:34.160 --> 00:05:36.475
our c would be 2pi.

00:05:36.475 --> 00:05:38.670
That's our unit step function,
and we want it to jump to 2.

00:05:38.670 --> 00:05:40.430
This would just jump
to 1 by itself.

00:05:40.430 --> 00:05:42.070
So let's multiply it by 2.

00:05:42.070 --> 00:05:43.480
And now we have this function.

00:05:43.480 --> 00:05:45.780
So you could imagine, you
can make an arbitrarily

00:05:45.780 --> 00:05:49.180
complicated function of things
jumping up and down to

00:05:49.180 --> 00:05:52.260
different levels based on
different essentially linear

00:05:52.260 --> 00:05:54.720
combinations of these
unit step functions.

00:05:54.720 --> 00:05:56.090
Now, what if I wanted
to do something

00:05:56.090 --> 00:05:57.030
a little bit fancier?

00:05:57.030 --> 00:05:59.990
What if I wanted to do something
that-- let's say I

00:05:59.990 --> 00:06:05.420
have some function that
looks like this.

00:06:05.420 --> 00:06:07.756
Let me draw some function.

00:06:07.756 --> 00:06:09.340
I should draw straighter
than that.

00:06:09.340 --> 00:06:11.950
I should have some standards.

00:06:11.950 --> 00:06:16.860
So let's say that just
my regular f of t--

00:06:16.860 --> 00:06:18.740
let me, this is x.

00:06:18.740 --> 00:06:20.030
Actually, why am I doing x?

00:06:20.030 --> 00:06:21.700
This would be the t-axis.

00:06:21.700 --> 00:06:22.940
We're doing the time domain.

00:06:22.940 --> 00:06:24.100
It could have been x.

00:06:24.100 --> 00:06:26.720
And then we'll call
this f of t.

00:06:26.720 --> 00:06:29.590
So let me draw some
arbitrary f of t.

00:06:29.590 --> 00:06:32.160
Let's say my function looks
something crazy like that.

00:06:35.100 --> 00:06:37.980
So this is my f of t.

00:06:37.980 --> 00:06:39.680
What if I'm modeling
a physical system

00:06:39.680 --> 00:06:40.910
that doesn't do this?

00:06:40.910 --> 00:06:45.260
That actually at some point--
well, actually, let's say it

00:06:45.260 --> 00:06:47.330
stays at zero.

00:06:47.330 --> 00:06:51.050
It stays at zero until
some value.

00:06:51.050 --> 00:06:53.405
Let's say it goes to zero
until-- I don't know, I'll

00:06:53.405 --> 00:06:55.190
call that c again.

00:06:55.190 --> 00:06:57.870
And then at c, f of t
kind of starts up.

00:06:57.870 --> 00:07:03.140
So right at c, f of t should
start up, so it just kind of

00:07:03.140 --> 00:07:05.620
goes like this.

00:07:05.620 --> 00:07:09.270
So essentially what we have here
is a combination of zero

00:07:09.270 --> 00:07:13.860
all the way, and then we
have a shifted f of t.

00:07:13.860 --> 00:07:18.600
So at c, we have a shifted f of
t, so it shifts that way.

00:07:18.600 --> 00:07:21.960
So how can we construct this
yellow function, where it's

00:07:21.960 --> 00:07:25.150
essentially a shifted version
of this green function, but

00:07:25.150 --> 00:07:26.960
it's zero below c?

00:07:26.960 --> 00:07:29.020
This green function might
have continued.

00:07:29.020 --> 00:07:31.320
It might have gone something
like this.

00:07:31.320 --> 00:07:34.325
It might have, continued and
done something crazy, but what

00:07:34.325 --> 00:07:38.460
we did is we shifted it from
here to there, and then we

00:07:38.460 --> 00:07:40.440
zeroed out everything
before c.

00:07:40.440 --> 00:07:41.870
So how could we do that?

00:07:41.870 --> 00:07:44.880
Well, just shifting this
function, you've learned in

00:07:44.880 --> 00:07:47.940
your Algebra II or your
precalculus classes, to shift

00:07:47.940 --> 00:07:51.620
a function by c to the right,
you just to replace your t

00:07:51.620 --> 00:07:52.970
with a t minus c.

00:07:52.970 --> 00:07:59.790
So this function right here
is f of t minus c.

00:07:59.790 --> 00:08:02.960
And to make sure I get it right,
what I always do is I

00:08:02.960 --> 00:08:06.270
imagine, OK, what's going to
happen when t is equal to c?

00:08:06.270 --> 00:08:10.400
When t is equal to c, you're
going to have a c minus a c,

00:08:10.400 --> 00:08:13.030
and you're going
to have f of 0.

00:08:13.030 --> 00:08:15.840
So f of 0, it should
be the same.

00:08:15.840 --> 00:08:19.270
So when t is equal to c, this
value, the value of the

00:08:19.270 --> 00:08:22.560
function should be equivalent
to the value of the original

00:08:22.560 --> 00:08:25.850
green function at zero, so it's
equivalent to that value,

00:08:25.850 --> 00:08:27.110
which makes sense.

00:08:27.110 --> 00:08:31.110
If we go up one more above c, so
let's say this is one more

00:08:31.110 --> 00:08:36.650
above c, so we get to this
point, if t is c plus 1, then

00:08:36.650 --> 00:08:40.390
when you put c plus 1 minus c,
you just have f of 1, and f of

00:08:40.390 --> 00:08:42.970
1 is really just this
point right here.

00:08:42.970 --> 00:08:45.650
And so it'll be that f of
1, so it makes sense.

00:08:45.650 --> 00:08:48.390
So as we move one forward here,
we're essentially at the

00:08:48.390 --> 00:08:52.350
same function value as we were
there, so the shift works.

00:08:52.350 --> 00:08:55.320
But I said that we have to
also-- if I just shifted this

00:08:55.320 --> 00:08:58.080
function, you would have all
this other stuff, because you

00:08:58.080 --> 00:08:59.700
would have had all this other
stuff when the function was

00:08:59.700 --> 00:09:02.500
back here still going on.

00:09:02.500 --> 00:09:03.770
The function-- I'll draw it

00:09:03.770 --> 00:09:05.810
lightly-- would still continue.

00:09:05.810 --> 00:09:08.330
But I said I wanted to zero
out this function

00:09:08.330 --> 00:09:10.390
before we reach c.

00:09:10.390 --> 00:09:12.800
So how can I zero out
that function?

00:09:12.800 --> 00:09:14.950
Well, I think it's pretty
obvious to you.

00:09:14.950 --> 00:09:17.750
I started this video talking
about the unit step function.

00:09:17.750 --> 00:09:19.680
So what if I multiply
the unit step

00:09:19.680 --> 00:09:21.280
function times this thing?

00:09:21.280 --> 00:09:22.990
What's going to happen?

00:09:22.990 --> 00:09:26.870
So what if I-- my new function,
I call it the unit

00:09:26.870 --> 00:09:35.130
step function up until c of
t times f of t minus c?

00:09:35.130 --> 00:09:37.210
So what's going to happen?

00:09:37.210 --> 00:09:41.300
Until we get to c, the unit
step function is zero when

00:09:41.300 --> 00:09:42.250
it's less than c.

00:09:42.250 --> 00:09:44.700
So you're going to have zero
times I don't care what this

00:09:44.700 --> 00:09:48.070
is Zero times anything is zero,
so this function is

00:09:48.070 --> 00:09:50.260
going to be zero.

00:09:50.260 --> 00:09:53.900
Once you hit c, the unit step
function becomes 1.

00:09:53.900 --> 00:09:57.710
So once you pass c, this thing
becomes a 1, and you're just

00:09:57.710 --> 00:09:59.320
left with 1 times
your function.

00:09:59.320 --> 00:10:01.520
So then your function can behave
as it would like to

00:10:01.520 --> 00:10:03.450
behave, and you actually
shifted it.

00:10:03.450 --> 00:10:06.410
This t minus c is what actually
shifted this green

00:10:06.410 --> 00:10:08.210
function over to the right.

00:10:08.210 --> 00:10:09.760
And this is actually
going to be a very

00:10:09.760 --> 00:10:14.630
useful constructed function.

00:10:14.630 --> 00:10:17.110
And in a second, wer'e going
to figure out the Laplace

00:10:17.110 --> 00:10:19.820
transform of this, and you're
going to appreciate, I think,

00:10:19.820 --> 00:10:22.220
why this is a useful function
to look at.

00:10:22.220 --> 00:10:26.800
But now you understand at least
what it is and why it

00:10:26.800 --> 00:10:30.000
essentially shifts a function
and zeroes out everything

00:10:30.000 --> 00:10:32.520
before that point.

00:10:32.520 --> 00:10:36.140
Well, I told you that this is
a useful function, so we

00:10:36.140 --> 00:10:39.480
should add its Laplace transform
to our library of

00:10:39.480 --> 00:10:42.720
Laplace transforms.
So let's do that.

00:10:42.720 --> 00:10:49.410
Let's take a the Laplace
transform of this, of the unit

00:10:49.410 --> 00:10:50.670
step function up to c.

00:10:50.670 --> 00:10:52.840
I'm doing it in fairly general
terms. In the next video,

00:10:52.840 --> 00:10:56.200
we'll do a bunch of examples
where we can apply this, but

00:10:56.200 --> 00:10:58.300
we should at least prove to
ourselves what the Laplace

00:10:58.300 --> 00:11:01.000
transform of this thing is.

00:11:01.000 --> 00:11:04.400
Well, the Laplace transform of
anything, or our definition of

00:11:04.400 --> 00:11:09.260
it so far, is the integral from
0 to infinity of e to the

00:11:09.260 --> 00:11:12.630
minus st times our function.

00:11:12.630 --> 00:11:18.240
So our function in this case is
the unit step function, u

00:11:18.240 --> 00:11:24.950
sub c of t times f
of t minus c dt.

00:11:24.950 --> 00:11:26.530
And this seems very general.

00:11:26.530 --> 00:11:29.670
It seems very hard to evaluate
this integral at first, but

00:11:29.670 --> 00:11:32.540
maybe we can make some form of
a substitution to get it into

00:11:32.540 --> 00:11:33.860
a term that we can appreciate.

00:11:33.860 --> 00:11:36.940
So let's make a substitution
here.

00:11:36.940 --> 00:11:40.690
Let me pick a nice variable
to work with.

00:11:40.690 --> 00:11:42.370
I don't know, we're not
using an x anywhere.

00:11:42.370 --> 00:11:43.440
We might as well use an x.

00:11:43.440 --> 00:11:47.980
That's the most fun variable
to work with.

00:11:47.980 --> 00:11:50.325
Sometimes, you'll see in a
lot of math classes, they

00:11:50.325 --> 00:11:53.060
introduce these crazy Latin
alphabets, and that by itself

00:11:53.060 --> 00:11:54.660
makes it hard to understand.

00:11:54.660 --> 00:11:59.510
So I like to stay away from
those crazy Latin alphabets,

00:11:59.510 --> 00:12:01.180
so we'll just use a regular x.

00:12:01.180 --> 00:12:02.120
Let's make a substitute.

00:12:02.120 --> 00:12:08.790
Let's say that x is equal
to t minus c.

00:12:08.790 --> 00:12:11.560
Or you could, if we added t to
both sides, we could say that

00:12:11.560 --> 00:12:14.700
t is equal to x plus c.

00:12:14.700 --> 00:12:16.600
Let's see what happens
to our subsitution.

00:12:16.600 --> 00:12:19.220
And also, if we took the
derivative of both sides of

00:12:19.220 --> 00:12:22.360
this, or I guess the
differential, you would get dx

00:12:22.360 --> 00:12:23.965
is equal to dt.

00:12:23.965 --> 00:12:27.460
Or I mean, if you took dx with
respect to dt, you would get

00:12:27.460 --> 00:12:30.040
that to equal to 1. c
is just a constant.

00:12:30.040 --> 00:12:32.900
Then if you multiply both sides
by dt, you get dx is

00:12:32.900 --> 00:12:35.700
equal to dt, and that's
a nice substitution.

00:12:35.700 --> 00:12:37.610
So what is our integral going
to become with this

00:12:37.610 --> 00:12:39.100
substitution?

00:12:39.100 --> 00:12:42.670
So our integral this
was t equals 0 to

00:12:42.670 --> 00:12:44.520
t is equal to infinity.

00:12:44.520 --> 00:12:51.880
When t is equal to 0, what is
x going to be equal to?

00:12:51.880 --> 00:12:57.080
Well, x is going to be
equal to minus c.

00:12:57.080 --> 00:13:00.170
Actually, before I go there,
let me actually take a step

00:13:00.170 --> 00:13:03.350
back, because we
could progress.

00:13:03.350 --> 00:13:04.720
We could go in this direction.

00:13:04.720 --> 00:13:09.150
But we could actually simplify
it more before we do that.

00:13:09.150 --> 00:13:11.680
Let's go back to out original
integral before we even made

00:13:11.680 --> 00:13:13.010
our substitution.

00:13:13.010 --> 00:13:15.680
If we're taking the integral
from 0 to infinity of this

00:13:15.680 --> 00:13:18.290
thing, we already said what does
this integral look like

00:13:18.290 --> 00:13:20.750
or what does this function
look like?

00:13:20.750 --> 00:13:21.750
It's zero.

00:13:21.750 --> 00:13:26.870
We have this unit step function
sitting right here.

00:13:26.870 --> 00:13:29.650
We have the unit step function
sitting right there.

00:13:29.650 --> 00:13:31.980
So this whole expression
is going to be zero

00:13:31.980 --> 00:13:33.980
until we get to c.

00:13:33.980 --> 00:13:36.220
This whole thing, by definition,
this unit step

00:13:36.220 --> 00:13:38.400
function is zero until
we get to c.

00:13:38.400 --> 00:13:40.400
So this everything's going
to be zeroed out

00:13:40.400 --> 00:13:41.940
until we get to c.

00:13:41.940 --> 00:13:44.890
So we could essentially say,
you know, we don't have to

00:13:44.890 --> 00:13:48.010
take the integral from t equals
0 to t equals infinity.

00:13:48.010 --> 00:13:52.540
We could take the integral--
let me write it here.

00:13:52.540 --> 00:13:54.450
I'll just use that old
integral sign.

00:13:54.450 --> 00:13:59.070
We could just take the integral
from t is equal to c

00:13:59.070 --> 00:14:05.790
to t is equal to infinity of e
to the minus st, the unit step

00:14:05.790 --> 00:14:12.280
function, uc of t times
f of t minus c dt.

00:14:12.280 --> 00:14:16.990
In fact, at this point, this
unit step function, it has no

00:14:16.990 --> 00:14:18.280
use anymore.

00:14:18.280 --> 00:14:23.210
Because before t is equal to c,
it's 0, and now that we're

00:14:23.210 --> 00:14:26.090
only worried about values above
c, it's equal to 1, so

00:14:26.090 --> 00:14:28.440
it equals 1 in this context.

00:14:28.440 --> 00:14:30.240
I want to make that
very clear to you.

00:14:30.240 --> 00:14:31.320
What did I do just here?

00:14:31.320 --> 00:14:35.340
I changed our bottom boundary
from 0 to c.

00:14:35.340 --> 00:14:37.550
And I think you might realize
why I did it when I was

00:14:37.550 --> 00:14:40.270
working with the substitution,
because this will simplify

00:14:40.270 --> 00:14:42.090
things if we do this
ahead of time.

00:14:42.090 --> 00:14:45.550
So if we have this unit step
function, this thing is going

00:14:45.550 --> 00:14:49.960
to zero out this entire integral
before we get to c.

00:14:49.960 --> 00:14:51.950
Remember, this definite integral
is really just the

00:14:51.950 --> 00:14:56.060
area under this curve of this
whole function, of the unit

00:14:56.060 --> 00:14:57.980
step function times
all of this stuff.

00:14:57.980 --> 00:15:00.200
All of this stuff, when we
multiply it, is going to be

00:15:00.200 --> 00:15:03.050
zero until we get
to some value c.

00:15:03.050 --> 00:15:05.620
And then above c, it's going to
be e to the minus st times

00:15:05.620 --> 00:15:06.760
f of t minus c.

00:15:06.760 --> 00:15:08.820
So it's going to start doing
all this crazy stuff.

00:15:08.820 --> 00:15:12.610
So if we want to essentially
find the area under this

00:15:12.610 --> 00:15:16.840
curve, we can ignore all the
stuff that happens before c.

00:15:16.840 --> 00:15:19.520
So instead of going from t
equals 0 to infinity, we can

00:15:19.520 --> 00:15:22.640
go from t is equal to c to
infinity because there was no

00:15:22.640 --> 00:15:25.540
area before t was equal to c.

00:15:25.540 --> 00:15:26.960
So that's all I did here.

00:15:26.960 --> 00:15:28.920
And then the other thing I said
is that the unit step

00:15:28.920 --> 00:15:34.640
function, it's going to be 1
over this entire range of

00:15:34.640 --> 00:15:38.420
potential t-values, so we can
just kind of ignore it.

00:15:38.420 --> 00:15:41.370
It's just going to be 1 this
entire time, so our integral

00:15:41.370 --> 00:15:46.380
simplifies to the definite
integral from t is equal to c

00:15:46.380 --> 00:15:53.440
to t is equal to infinity of e
to the minus st times f of t

00:15:53.440 --> 00:15:56.025
minus is c dt.

00:15:56.025 --> 00:15:58.030
And this will simplify
it a good bit.

00:15:58.030 --> 00:15:59.600
I was going down the other
road when I did the

00:15:59.600 --> 00:16:01.830
substitution first, which would
have worked, but I think

00:16:01.830 --> 00:16:04.750
the argument as to why I could
have changed the boundaries

00:16:04.750 --> 00:16:06.760
would've been a harder
argument to make.

00:16:06.760 --> 00:16:09.900
So now that we had this, let's
go back and make that

00:16:09.900 --> 00:16:13.760
substitution that x is
equal to t minus c.

00:16:13.760 --> 00:16:17.340
So our integral becomes-- I'll
do it in green-- when t is

00:16:17.340 --> 00:16:18.990
equal to c, what is x?

00:16:18.990 --> 00:16:20.950
Then x is 0, right?

00:16:20.950 --> 00:16:22.440
c minus c is 0.

00:16:22.440 --> 00:16:24.680
When t is equal to infinity,
what is x?

00:16:24.680 --> 00:16:26.720
Well x is, you know, infinity
minus any constant is still

00:16:26.720 --> 00:16:29.460
going to be infinity, or if
the limit is t approaches

00:16:29.460 --> 00:16:32.920
infinity, x is still going
to be infinity here.

00:16:32.920 --> 00:16:37.100
And it's the integral of e to
the minus s, but now instead

00:16:37.100 --> 00:16:39.245
of a t, we have the
substitution.

00:16:39.245 --> 00:16:42.530
If we said x is equal to t minus
c, then we can just add

00:16:42.530 --> 00:16:45.660
c to both sides and get t
is equal to x plus c.

00:16:45.660 --> 00:16:53.830
So you get x plus c there, and
then times the function f of t

00:16:53.830 --> 00:16:58.230
minus c, but we said t minus
c is the same thing as x.

00:16:58.230 --> 00:17:00.370
And dt is the same
thing is dx.

00:17:00.370 --> 00:17:05.460
I showed you that right there,
so we can write this as dx.

00:17:05.460 --> 00:17:08.700
Now this is starting to look
a little bit interesting.

00:17:08.700 --> 00:17:10.780
So what is this equal to?

00:17:10.780 --> 00:17:14.730
This is equal to the integral
from 0 to infinity-- let me

00:17:14.730 --> 00:17:23.579
expand this out-- of e to
the minus sx minus sc

00:17:23.579 --> 00:17:28.240
times f of x dx.

00:17:28.240 --> 00:17:30.360
Now, what is the equal to?

00:17:30.360 --> 00:17:34.500
Well, we could factor out an e
to the minus sc and bring it

00:17:34.500 --> 00:17:36.300
outside of the integral, because
this has nothing to do

00:17:36.300 --> 00:17:39.150
with what we're taking the
integral with respect to.

00:17:39.150 --> 00:17:39.970
So let's do that.

00:17:39.970 --> 00:17:44.530
Let me take this guy out, and
maybe just to not confuse you,

00:17:44.530 --> 00:17:45.600
let me rewrite the
whole thing.

00:17:45.600 --> 00:17:46.650
0 to infinity.

00:17:46.650 --> 00:17:50.010
I could rewrite this e term
as e-- actually, let me

00:17:50.010 --> 00:17:51.460
write it this way.

00:17:51.460 --> 00:17:55.930
I'll do what was already in
green as e to the minus sx

00:17:55.930 --> 00:18:00.820
times e to the minus sc.

00:18:00.820 --> 00:18:01.860
Common base.

00:18:01.860 --> 00:18:05.800
So if I were to multiply these
two, I could just add the

00:18:05.800 --> 00:18:10.430
exponents, which you would
get that up there, times

00:18:10.430 --> 00:18:14.580
f of x, d of x.

00:18:14.580 --> 00:18:17.200
This is a constant term with
respect to x, so we can just

00:18:17.200 --> 00:18:18.720
factor it out.

00:18:18.720 --> 00:18:22.350
We can just factor this thing
out right there, so then you

00:18:22.350 --> 00:18:28.290
get e to the minus sc times
the integral from 0 to

00:18:28.290 --> 00:18:37.810
infinity of e to the minus
sx times f of x dx.

00:18:37.810 --> 00:18:40.090
Now, what were we doing
here the whole time?

00:18:40.090 --> 00:18:49.070
We were taking the Laplace
transform of the unit step

00:18:49.070 --> 00:18:51.950
function that goes up to c, and
then it's 0 up to c, and

00:18:51.950 --> 00:18:58.000
it's 1 after that, of t times
some shifted function

00:18:58.000 --> 00:19:01.680
f of t minus c.

00:19:01.680 --> 00:19:05.720
And now we got that as being
equal to this thing, and we

00:19:05.720 --> 00:19:06.710
made a substitution.

00:19:06.710 --> 00:19:08.490
We simplified it a little bit.

00:19:08.490 --> 00:19:16.610
e to the minus sc times the
integral from 0 to infinity of

00:19:16.610 --> 00:19:21.600
e to the minus sx f of x dx.

00:19:21.600 --> 00:19:24.180
Something about the tablet
doesn't work properly right

00:19:24.180 --> 00:19:26.290
around this period.

00:19:26.290 --> 00:19:30.920
But this should look
interesting to you.

00:19:30.920 --> 00:19:32.600
What is this?

00:19:32.600 --> 00:19:35.470
This is the Laplace transform
of f of x.

00:19:35.470 --> 00:19:36.280
Let me write that down.

00:19:36.280 --> 00:19:40.610
What's the Laplace transform
of-- well, I could write it as

00:19:40.610 --> 00:19:42.640
f of t or f of x.

00:19:42.640 --> 00:19:46.910
The Laplace transform of f of
t is equal to the integral

00:19:46.910 --> 00:19:55.680
from 0 to infinity of e to the
minus st times f of t dt.

00:19:55.680 --> 00:19:58.040
This and this are the
exact same thing.

00:19:58.040 --> 00:19:59.520
We're just using a t here.

00:19:59.520 --> 00:20:00.960
We're using an x here.

00:20:00.960 --> 00:20:01.790
No difference.

00:20:01.790 --> 00:20:03.910
They're just letters.

00:20:03.910 --> 00:20:06.490
This is f of t.

00:20:06.490 --> 00:20:08.870
e to the minus st
times f of t dt.

00:20:08.870 --> 00:20:11.270
I could have also rewritten
it as the Laplace

00:20:11.270 --> 00:20:14.970
transform of f of t.

00:20:14.970 --> 00:20:17.790
I could write this as the
integral from 0 to infinity of

00:20:17.790 --> 00:20:24.470
e to the minus sy
times f of y dy.

00:20:24.470 --> 00:20:25.930
I could do it by anything
because this

00:20:25.930 --> 00:20:26.740
is a definite integral.

00:20:26.740 --> 00:20:28.330
The y's are going to disappear,

00:20:28.330 --> 00:20:29.190
and we've seen that.

00:20:29.190 --> 00:20:32.500
All you're left with
is a function of s.

00:20:32.500 --> 00:20:35.910
This ends up being some capital,
well, you know, we

00:20:35.910 --> 00:20:39.140
could write some capital
function of s.

00:20:39.140 --> 00:20:40.060
So this is interesting.

00:20:40.060 --> 00:20:44.680
This is the Laplace transform
of f of t times some scaling

00:20:44.680 --> 00:20:48.060
factor, and that's what
we set out to show.

00:20:48.060 --> 00:20:54.790
So we can now show that the
Laplace transform of the unit

00:20:54.790 --> 00:21:02.700
step function times some
function t minus c is equal to

00:21:02.700 --> 00:21:07.870
this function right here, e to
the minus sc, where this c is

00:21:07.870 --> 00:21:11.650
the same as this c right
here, times the Laplace

00:21:11.650 --> 00:21:13.740
transform of f of t.

00:21:13.740 --> 00:21:17.370
Times the Laplace transform--
I don't know what's going on

00:21:17.370 --> 00:21:21.450
with the tablet right
there-- of f of t.

00:21:21.450 --> 00:21:22.100
Let me write that.

00:21:22.100 --> 00:21:25.450
This is equal to-- because it's
looking funny there-- e

00:21:25.450 --> 00:21:32.540
to the minus sc times the
Laplace transform of f of t.

00:21:32.540 --> 00:21:33.865
So this is our result.

00:21:37.660 --> 00:21:39.740
Now, what does this mean?

00:21:39.740 --> 00:21:42.290
Oh, look it back-filled
it somehow.

00:21:42.290 --> 00:21:43.180
What does this mean?

00:21:43.180 --> 00:21:45.285
What can we do with this?

00:21:45.285 --> 00:21:50.420
Well, let's say we wanted to
figure out the Laplace

00:21:50.420 --> 00:21:53.910
transform of the unit
step function that

00:21:53.910 --> 00:21:56.200
starts up at pi of t.

00:21:56.200 --> 00:21:59.430
And let's say we're taking
something that we know well:

00:21:59.430 --> 00:22:05.510
sine of t minus pi.

00:22:05.510 --> 00:22:08.570
So we shifted it, right?

00:22:08.570 --> 00:22:10.750
This thing is really
malfunctioning at this point

00:22:10.750 --> 00:22:11.450
right here.

00:22:11.450 --> 00:22:14.600
Let me pause it.

00:22:14.600 --> 00:22:15.440
I just paused.

00:22:15.440 --> 00:22:16.890
Sorry if that was a little
disconcerting.

00:22:16.890 --> 00:22:19.420
I just paused the video because
it was having trouble

00:22:19.420 --> 00:22:21.950
recording at some point
on my little board.

00:22:21.950 --> 00:22:25.130
So let me rewrite the result
that we proved just now.

00:22:25.130 --> 00:22:29.430
We showed that the Laplace
transform of the unit step

00:22:29.430 --> 00:22:34.520
function t, and it goes to 1
at some value c times some

00:22:34.520 --> 00:22:38.820
function that's shifted
by c to the right.

00:22:38.820 --> 00:22:43.770
It's equal to e to the minus cs
times the Laplace transform

00:22:43.770 --> 00:22:45.520
of just the unshifted
function.

00:22:45.520 --> 00:22:46.400
That was our result.

00:22:46.400 --> 00:22:48.630
That was the big takeaway
from this video.

00:22:48.630 --> 00:22:51.550
And if this seems like some
Byzantine, hard-to-understand

00:22:51.550 --> 00:22:53.630
result, we can apply it.

00:22:53.630 --> 00:22:55.930
So let's say the Laplace
transform, this is what I was

00:22:55.930 --> 00:23:01.080
doing right before the actual
pen tablet started

00:23:01.080 --> 00:23:01.920
malfunctioning.

00:23:01.920 --> 00:23:04.500
If we want to take the Laplace
transform of the unit step

00:23:04.500 --> 00:23:09.610
function that goes to 1 at pi,
t times the sine function

00:23:09.610 --> 00:23:15.220
shifted by pi to the right, we
know that this is going to be

00:23:15.220 --> 00:23:19.320
equal to e to the minus cs.

00:23:19.320 --> 00:23:24.700
c is pi in this case, so minus
pi s times the Laplace

00:23:24.700 --> 00:23:27.760
transform of the unshifted
function.

00:23:27.760 --> 00:23:29.030
So in this case, it's
the Laplace

00:23:29.030 --> 00:23:32.300
transform of sine of t.

00:23:32.300 --> 00:23:36.460
And we know what the Laplace
transform of sine of t is.

00:23:36.460 --> 00:23:41.250
It's just 1 over s
squared plus 1.

00:23:41.250 --> 00:23:43.770
So the Laplace transform of this
thing here, which before

00:23:43.770 --> 00:23:47.720
this video seemed like something
crazy, we now know

00:23:47.720 --> 00:23:49.090
is this times this.

00:23:49.090 --> 00:23:53.380
So it's e to the minus pi s
times this, or we could just

00:23:53.380 --> 00:23:58.900
write it as e to the minus pi
s over s squared plus 1.

00:23:58.900 --> 00:24:00.990
We'll do a couple more examples
of this in the next

00:24:00.990 --> 00:24:02.900
video, where we go back and
forth between the Laplace

00:24:02.900 --> 00:24:05.950
world and the t and between
the s domain

00:24:05.950 --> 00:24:06.810
and the time domain.

00:24:06.810 --> 00:24:10.730
And I'll show you how this is a
very useful result to take a

00:24:10.730 --> 00:24:13.910
lot of Laplace transforms and
to invert a lot of Laplace

00:24:13.910 --> 00:24:15.160
transforms.


Laplace transform of t^n: L{t^n}

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:05.860
In the last video, I showed the
Laplace transform of t, or

00:00:05.860 --> 00:00:09.410
we could view that as t the
first power, is equal to 1/s

00:00:09.410 --> 00:00:13.410
squared, if we assume that
s is greater than 0.

00:00:13.410 --> 00:00:16.100
In this video, we're going to
see if we can generalize this

00:00:16.100 --> 00:00:20.270
by trying to figure out the
Laplace transform of t to the

00:00:20.270 --> 00:00:24.660
n, where n is any integer power
greater than 0, so n is

00:00:24.660 --> 00:00:27.575
any positive integer
greater than 0.

00:00:27.575 --> 00:00:29.120
So let's try it out.

00:00:29.120 --> 00:00:33.060
So we know from our definition
of the Laplace transform that

00:00:33.060 --> 00:00:38.930
the Laplace transform of t to
the n is equal to the integral

00:00:38.930 --> 00:00:45.080
from 0 to infinity of our
function-- well, let me write

00:00:45.080 --> 00:00:48.270
t to the n-- times, and this is
just the definition of the

00:00:48.270 --> 00:00:54.420
transform, e to the
minus st, dt.

00:00:54.420 --> 00:00:57.280
And similar to when we figured
out this Laplace transform,

00:00:57.280 --> 00:00:59.570
your intuition might be that,
hey, we should use integration

00:00:59.570 --> 00:01:02.240
by parts, and I showed
it in the last video.

00:01:02.240 --> 00:01:04.769
I always forget it, but I just
recorded it, so I do happen to

00:01:04.769 --> 00:01:05.860
remember it.

00:01:05.860 --> 00:01:08.000
So the integration by parts
just tells us that the

00:01:08.000 --> 00:01:15.920
integral of uv prime is equal to
uv minus the integral of--

00:01:15.920 --> 00:01:21.060
I view this as kind of the
swap-- so u prime v.

00:01:21.060 --> 00:01:23.175
So this is just our integration
by parts formula.

00:01:23.175 --> 00:01:26.720
If you ever forget it, you can
derive it in about 30 seconds

00:01:26.720 --> 00:01:27.380
from the product rule.

00:01:27.380 --> 00:01:29.630
And I did it in the last video
because I hadn't used it for

00:01:29.630 --> 00:01:31.330
awhile, so I had
to rederive it.

00:01:31.330 --> 00:01:32.460
So let's apply it here.

00:01:32.460 --> 00:01:34.850
So what do we want to
make our v prime?

00:01:34.850 --> 00:01:37.530
It's always good to use the
exponential function, because

00:01:37.530 --> 00:01:40.140
that's easy to take the
antiderivative of.

00:01:40.140 --> 00:01:44.540
So this is our v prime, in which
case our v is just the

00:01:44.540 --> 00:01:46.130
antiderivative of that.

00:01:46.130 --> 00:01:51.490
So it's e to the minus
st over minus s.

00:01:51.490 --> 00:01:54.550
If we take the derivative of
this, minus s divided by minus

00:01:54.550 --> 00:01:56.660
s cancels out, and you
just get that.

00:01:56.660 --> 00:02:02.080
And then if we make our u-- let
me pick a good color here.

00:02:02.080 --> 00:02:05.940
If we make this equal to our
u, what's our u prime?

00:02:05.940 --> 00:02:12.700
u prime is just going to be n
times t to the n minus 1.

00:02:12.700 --> 00:02:13.280
Fair enough.

00:02:13.280 --> 00:02:16.010
So let's apply the integration
by parts.

00:02:16.010 --> 00:02:19.330
So this is going to
be equal to uv.

00:02:19.330 --> 00:02:25.730
u, I'll use this t to the n, so
u is t to the n, that's our

00:02:25.730 --> 00:02:31.880
u, times v, which is e-- let me
write this down-- so it's

00:02:31.880 --> 00:02:35.075
minus-- there's a minus sign
there, so we put the minus.

00:02:35.075 --> 00:02:36.510
Let me do it in that color.

00:02:39.920 --> 00:02:42.555
I'm just rewriting this.
e to the minus st/s.

00:02:45.930 --> 00:02:47.725
So that's the uv term
right there.

00:02:47.725 --> 00:02:50.180
Let me make that clear.

00:02:50.180 --> 00:02:51.820
And let me pick a
good color here.

00:02:51.820 --> 00:02:57.090
So this term right here is
this term right here.

00:02:57.090 --> 00:03:00.040
And, of course, I'm going to
have to evaluate this from 0

00:03:00.040 --> 00:03:02.615
to infinity, so let me write
that: 0 to infinity.

00:03:02.615 --> 00:03:04.420
I could put a little bracket
there or something, but you

00:03:04.420 --> 00:03:05.990
know we're going to have
to evaluate that.

00:03:05.990 --> 00:03:07.560
And then from that, we're
going to have to

00:03:07.560 --> 00:03:09.460
subtract the integral.

00:03:09.460 --> 00:03:11.410
And let me not forget
our boundaries.

00:03:11.410 --> 00:03:20.520
0 to infinity of u prime is n
times t to the n minus 1--

00:03:20.520 --> 00:03:26.450
that's our u prime-- times v
times minus-- so let me put

00:03:26.450 --> 00:03:34.220
this minus out here-- so minus
e to the minus st/s.

00:03:36.810 --> 00:03:37.470
And then all of that.

00:03:37.470 --> 00:03:42.700
Of course, we have our dt, and
you have a minus minus.

00:03:42.700 --> 00:03:44.540
These things become pluses.

00:03:44.540 --> 00:03:47.430
Let's see if we can simplify
this a little bit.

00:03:47.430 --> 00:03:52.760
So we get our Laplace transform
of t to the n is

00:03:52.760 --> 00:03:57.100
equal to this evaluated at
infinity and evaluated at 0.

00:03:57.100 --> 00:04:00.800
So when you've evaluate-- what's
the limit of this as t

00:04:00.800 --> 00:04:02.430
approaches infinity?

00:04:02.430 --> 00:04:04.900
As t approaches infinity, this
term, you might say, oh, this

00:04:04.900 --> 00:04:05.690
becomes really big.

00:04:05.690 --> 00:04:07.610
And I went over this
in the last video.

00:04:07.610 --> 00:04:09.800
But this term overpowers it,
because you're going to have e

00:04:09.800 --> 00:04:11.980
to the minus infinity, if
we assume that s is

00:04:11.980 --> 00:04:13.470
greater than 0.

00:04:13.470 --> 00:04:15.960
So if s is greater than 0, this
term is going to win out

00:04:15.960 --> 00:04:17.970
and go to 0 much faster
than this term is

00:04:17.970 --> 00:04:18.750
going to go to infinity.

00:04:18.750 --> 00:04:22.770
So when you evaluate it at
infinity, when you evaluate

00:04:22.770 --> 00:04:25.730
this at infinity, you're
going to get 0.

00:04:25.730 --> 00:04:30.020
And then you're going to
subtract this evaluated at 0.

00:04:30.020 --> 00:04:35.600
This evaluated at 0, when it's
evaluated at 0 is just minus 0

00:04:35.600 --> 00:04:40.960
to the n times e to the minus
s times 0/s Well, this

00:04:40.960 --> 00:04:42.980
becomes 0 as well.

00:04:42.980 --> 00:04:46.220
So this whole term evaluated
from 0 to infinity is all 0,

00:04:46.220 --> 00:04:48.720
which is a nice, convenient
thing for us.

00:04:48.720 --> 00:04:53.100
And then we're going to have
this next term right there.

00:04:53.100 --> 00:04:55.470
So let's take out the constant
terms. This n

00:04:55.470 --> 00:04:57.150
and this s are constant.

00:04:57.150 --> 00:04:59.335
They are constant with
respect to t.

00:04:59.335 --> 00:05:06.990
So you have plus n/s times the
integral from 0 to infinity of

00:05:06.990 --> 00:05:16.740
t to the n minus 1 times
e to the minus st, dt.

00:05:16.740 --> 00:05:20.670
Now, this should look reasonably
familiar to you.

00:05:23.170 --> 00:05:26.010
What's the definition of
the Laplace transform?

00:05:26.010 --> 00:05:31.290
The Laplace transform of any
function is equal to the

00:05:31.290 --> 00:05:36.750
integral from 0 to infinity of
that function times e to the

00:05:36.750 --> 00:05:39.030
minus st, dt.

00:05:39.030 --> 00:05:41.390
Well, when we have an e to the
minus st, dt, we're taking the

00:05:41.390 --> 00:05:44.500
integral from 0 to infinity,
so this whole integral is

00:05:44.500 --> 00:05:48.310
equal to the Laplace transform
of this, of t

00:05:48.310 --> 00:05:50.930
to the n minus 1.

00:05:50.930 --> 00:05:54.530
So just that easily, because
this term went to 0, we've

00:05:54.530 --> 00:05:56.130
simplified things.

00:05:56.130 --> 00:06:04.110
We get the Laplace transform
of t to the n is equal to--

00:06:04.110 --> 00:06:08.830
this is all 0-- it's equal to
n/s-- that's right there--

00:06:08.830 --> 00:06:11.640
times this integral right here,
which we just figured

00:06:11.640 --> 00:06:17.000
out was the Laplace transform
of t to the n minus 1.

00:06:17.000 --> 00:06:19.070
Well, this is a nice,
neat simplification.

00:06:19.070 --> 00:06:22.090
We can now figure out the
Laplace transform of a higher

00:06:22.090 --> 00:06:24.645
power in terms of the one power
lower that, but it still

00:06:24.645 --> 00:06:25.975
doesn't give me a generalized
formula.

00:06:25.975 --> 00:06:31.060
So let's see if we can use this
with this information to

00:06:31.060 --> 00:06:32.480
get a generalized formula.

00:06:32.480 --> 00:06:36.790
So the Laplace transform of just
t-- so let me write that

00:06:36.790 --> 00:06:39.030
down; I wrote that at the
beginning of the problem.

00:06:39.030 --> 00:06:42.270
We get the Laplace transform, I
could write this as t to the

00:06:42.270 --> 00:06:47.510
1, which is just t, is equal
to 1/s squared, where s is

00:06:47.510 --> 00:06:52.050
greater than 0.

00:06:52.050 --> 00:06:54.420
Now, what happens if
we take the Laplace

00:06:54.420 --> 00:06:59.480
transform of t squared?

00:06:59.480 --> 00:07:01.160
Well, we can just use this
formula up here.

00:07:01.160 --> 00:07:05.460
The Laplace transform of t
squared is equal to 2/s times

00:07:05.460 --> 00:07:10.760
the Laplace transform of t,
of just t to the 1, right?

00:07:10.760 --> 00:07:11.580
2 minus 1.

00:07:11.580 --> 00:07:16.550
So times the Laplace transform
of t to the 1.

00:07:16.550 --> 00:07:18.520
Well, t, we know what that is.

00:07:18.520 --> 00:07:24.510
This is equal to 2/s times this,
times 1/s squared, which

00:07:24.510 --> 00:07:27.520
is equal to 2/s to the third.

00:07:27.520 --> 00:07:28.310
Interesting.

00:07:28.310 --> 00:07:29.820
Let's see if we can
do another one.

00:07:29.820 --> 00:07:33.950
What is-- I'll do it in the
dark blue-- the Laplace

00:07:33.950 --> 00:07:37.660
transform of t to the third?

00:07:37.660 --> 00:07:39.370
Well, we just use this
formula up here.

00:07:39.370 --> 00:07:40.200
It's n/s.

00:07:40.200 --> 00:07:41.880
In this case, n is 3.

00:07:41.880 --> 00:07:47.630
So it's 3/s times the Laplace
transform of t to the n minus

00:07:47.630 --> 00:07:49.570
1, so t squared.

00:07:52.120 --> 00:07:54.770
We know what the Laplace
transform of this one was.

00:07:54.770 --> 00:07:56.400
This is just this right there.

00:07:56.400 --> 00:08:01.980
So it's equal to 3/s
times this thing.

00:08:01.980 --> 00:08:03.610
And I'm going to actually write
it this way, because I

00:08:03.610 --> 00:08:04.390
think it's interesting.

00:08:04.390 --> 00:08:05.560
So I'll write the numerator.

00:08:05.560 --> 00:08:13.310
Times 2 times 1/s over s
squared, which is-- we could

00:08:13.310 --> 00:08:18.020
write it as 3 factorial
over-- what is this? s

00:08:18.020 --> 00:08:19.270
to the fourth power.

00:08:22.340 --> 00:08:23.340
Let's do another one.

00:08:23.340 --> 00:08:26.940
And I think you already are
getting the idea of

00:08:26.940 --> 00:08:28.470
what's going on.

00:08:28.470 --> 00:08:33.750
The Laplace transform of t to
the fourth power is what?

00:08:33.750 --> 00:08:40.270
It's equal to 4/s times the
Laplace transform of t the

00:08:40.270 --> 00:08:41.710
third power.

00:08:41.710 --> 00:08:43.169
And that's just 4/s
times this.

00:08:43.169 --> 00:08:50.150
So it's 4/s times 3 factorial
over s to the fourth.

00:08:50.150 --> 00:08:54.790
So now 4 times 3 factorial,
that's just 4 factorial over s

00:08:54.790 --> 00:08:56.120
to the fifth.

00:08:56.120 --> 00:08:59.490
And so you can just get
the general principle.

00:08:59.490 --> 00:09:01.100
And we can prove this
by induction.

00:09:01.100 --> 00:09:03.235
It's almost trivial based on
what we've already done.

00:09:03.235 --> 00:09:14.710
That the Laplace transform of
t to the n is equal to n

00:09:14.710 --> 00:09:21.740
factorial over s to
the n plus 1.

00:09:21.740 --> 00:09:25.250
We proved it directly for this
base case right here.

00:09:25.250 --> 00:09:29.290
This is 1 factorial over
s to the 1 plus 1.

00:09:29.290 --> 00:09:33.610
And then if we know it's true
for this, we know it's going

00:09:33.610 --> 00:09:34.940
to be true for the
next increment.

00:09:34.940 --> 00:09:37.850
So induction proof is almost
obvious, but you can even see

00:09:37.850 --> 00:09:38.970
it based on this.

00:09:38.970 --> 00:09:41.330
If you have to figure out the
Laplace transform of t to the

00:09:41.330 --> 00:09:43.500
tenth, you could just keep
doing this over and over

00:09:43.500 --> 00:09:46.000
again, but I think you see the
pattern pretty clearly.

00:09:46.000 --> 00:09:48.940
So anyway, I thought that was
a neat problem in and of

00:09:48.940 --> 00:09:52.730
itself, outside of the fact,
it'll be useful when we figure

00:09:52.730 --> 00:09:55.620
out inverse and Laplace
transforms. But this is a

00:09:55.620 --> 00:09:56.640
pretty neat result.

00:09:56.640 --> 00:10:00.730
The Laplace transform of t to
the n, where n is some integer

00:10:00.730 --> 00:10:04.480
greater than 0 is equal to n
factorial over s to the n plus

00:10:04.480 --> 00:10:06.690
1, where s is also
greater than 0.

00:10:06.690 --> 00:10:09.290
That was an assumption we had to
make early on when we took

00:10:09.290 --> 00:10:12.250
our limits as t approaches
infinity.

00:10:12.250 --> 00:10:15.390
Anyway, hopefully you
found that useful.