WEBVTT
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Okay, those are the formulas.
You will get all of those on
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the test, plus a couple more
that I will give you today.
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Those will be the basic
formulas of the Laplace
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transform.
If I think you need anything
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else, I'll give you other stuff,
too.
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So, I'm going to leave those on
the board all period.
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The basic test for today is to
see how Laplace transforms are
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used to solve linear
differential equations with
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constant coefficients.
Now, to do that,
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we're going to have to take the
Laplace transform of a
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derivative.
And, in order to make sense of
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that procedure,
we're going to have to ask,
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I apologize in advance,
but a slightly theoretical
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question, namely,
we have to have some guarantee
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in advance that the Laplace
transform is going to exist.
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Now, how could the Laplace
transform fail to exist?
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Can't I always calculate this?
And the answer is,
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no, you can't always calculate
it because this is an improper
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integral.
I'm integrating all the way up
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to infinity, and you know that
improper integrals don't always
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converge.
You know, if the integrand for
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example just didn't have the
exponential factor there,
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were simply t dt,
that it might look like it made
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sense, but that integral doesn't
converge.
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And, anyway,
it has no value.
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So, I need conditions in
advance, which guarantee that
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the Laplace transforms will
exist.
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Only under those circumstances
will the formulas make any
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sense.
Now, there is a standard
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condition that's in your book.
But, I didn't get a chance to
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talk about it last time.
So, I thought I'd better spent
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the first few minutes today
talking about the condition
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because it's what we're going to
need in order to be able to
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solve differential equations.
The condition that makes the
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Laplace transform definitely
exist for a function is that f
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of t shouldn't grow too
rapidly.
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It can grow rapidly.
It can grow because the e to
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the minus s t is
pulling it down,
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trying hard to pull it down to
zero to make the integral
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converge.
All we have to do is to
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guarantee that it doesn't grow
so rapidly that the e to the
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minus s t is powerless to pull
it down.
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Now, the condition is it's
what's called a growth
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condition.
These are very important in
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applications,
and unfortunately,
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it's always taught in 18.01,
but it's not always taught in
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high school calculus.
And, it's a question of how
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fast the function is allowed to
grow.
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And, the condition is
universally said this way,
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should be of exponential type.
So, what I'm defining is the
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phrase "exponential type." I'll
put it in quotation marks for
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that reason.
What does this mean?
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It's a condition,
a growth condition on a
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function, says how fast it can
get big.
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It says that f of t in
size, since f of t might get
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negatively very large,
and that would hurt,
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make the integral hard to
converge, not likely to
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converge, use the absolute
value.
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In other words,
I don't care if f of t is going
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up or going down very low.
Whichever way it goes,
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its size should not be bigger
than a rapidly growing
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exponential.
And, here's a rapidly growing
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exponential.
c is some positive constant,
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for some positive constant c
and some positive constant k.
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And, this should be true for
all values of t.
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All t greater than or equal to
zero.
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I don't have to worry about
negative values of t because the
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integral doesn't care about
them.
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I'm only doing the integration
as t runs from zero to infinity.
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In other words,
f of t could have been
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an extremely wild function,
sewn a lot of oats or whatever
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functions do for negative values
of t, and we don't care.
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It's only what's happening from
now from time zero onto
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infinity.
As long as it behaves now,
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from now on,
it's okay.
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All right, so,
the way it should behave is by
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being an exponential type.
Now, to try to give you some
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feeling for what this means,
these functions,
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for example, if k is 100,
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do you have any idea what the
plot of e to the 100t
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looks like?
It goes straight up.
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On every computer you try to
plot it on, e to the 100t
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goes like that
unless, of course,
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you make the scale t equals
zero to, over here,
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is one millionth.
Well, even that won't do.
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Okay, so these functions really
can grow quite rapidly.
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Let's take an example and see
what's of exponential type,
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and then perhaps even more
interestingly,
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what isn't.
The function sine t,
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is that of exponential type?
Well, sure.
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Its absolute value is always
less than or equal to one.
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So, it's also this paradigm.
If I take c equal to one,
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and what should I take k to be?
Zero.
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Take k to be zero,
c equals one,
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and in fact sine t
plays that condition.
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Here's one that's more
interesting, t to the n.
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Think of t to the 100th power.
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Is that smaller than some
exponential with maybe a
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constant out front?
Well, t to the 100th power
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goes straight up,
also.
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Well, we feel that if we make
the exponential big enough,
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maybe it will win out.
In fact, you don't have to make
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the exponential big.
k equals one is good
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enough.
In other words,
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I don't have to put absolute
value signs around the t to the
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n because I'm only
thinking about t as being a
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positive number,
anyway.
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I say that that's less than or
equal to some constant M,
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positive constant M times e to
the t will be good
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enough for some M and all t.
Now, why is that?
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Why is that?
The way to think of that,
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so, what this proves is that,
therefore, t to the n
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is of exponential type,
which we could have guessed
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because after all we were able
to calculate its Laplace
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transform.
Now, just because you can
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calculate the Laplace transform
doesn't mean it's of exponential
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type, but in practical matters,
it almost always does.
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So, t to the n is of
exponential type.
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How do you prove that?
Well, the weighted secret is to
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look at t to the n divided by e
to the t.
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In other words,
look at the quotient.
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What I'd like to argue is that
this is bounded by some number,
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capital M.
That's the question I'm asking.
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Now, why is this so?
Well, I think I can convince
00:08:21.000 --> 00:08:27.000
you of it without having to work
very hard.
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What does the graph of this
function look like?
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It starts here,
so I'm graphing this function,
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this ratio.
When t is equal to zero,
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its value is zero,
right, because of the
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numerator.
What happens as t goes to
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infinity?
What happens to this?
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What does it approach?
Zero.
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And, why?
By L'Hop.
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By L'Hopital's rule.
Just keep differentiating,
00:09:01.000 --> 00:09:07.000
reapply the rule over and over,
keep differentiating it n
00:09:06.000 --> 00:09:12.000
times, and finally you'll have
won the numerator down to t to
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the zero,
which isn't doing anything
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much.
And, the denominator,
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no matter how many times you
differentiate it,
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it's still t,
to the t all the time.
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So, by using Lopital's rule n
times, you change the top to one
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or n factorial,
actually; the bottom stays e to
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the t,
and the ratio clearly
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approaches zero,
and therefore,
00:09:41.000 --> 00:09:47.000
it approached zero to start
with.
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So, I don't know what this
function's doing in between.
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It's a positive function.
It's continuous because the top
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and bottom are continuous,
and the bottom is never zero.
00:09:54.000 --> 00:10:00.000
So, it's a continuous function
which starts out at zero and is
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positive, and as t goes to
infinity, it gets closer and
00:10:01.000 --> 00:10:07.000
closer to the t-axis,
again.
00:10:03.000 --> 00:10:09.000
Well, what does t to the n
do?
00:10:06.000 --> 00:10:12.000
It might wave around.
It doesn't actually.
00:10:10.000 --> 00:10:16.000
But, the point is,
because it's continuous,
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starts at zero,
ends at zero,
00:10:19.000 --> 00:10:25.000
it's bounded.
It has a maximum somewhere.
00:10:23.000 --> 00:10:29.000
And, that maximum is M.
So, it has a maximum.
00:10:30.000 --> 00:10:36.000
All you have to know is where
it starts, and where it ends up,
00:10:34.000 --> 00:10:40.000
and the fact that it's
continuous.
00:10:37.000 --> 00:10:43.000
That guarantees that it has a
maximum.
00:10:39.000 --> 00:10:45.000
So, it is less than some
maximum, and that shows that
00:10:43.000 --> 00:10:49.000
it's of exponential type.
Now, of course,
00:10:46.000 --> 00:10:52.000
before you get the idea that
everything's of exponential
00:10:50.000 --> 00:10:56.000
type, let's see what isn't.
I'll give you two functions
00:10:54.000 --> 00:11:00.000
that are not of exponential
type, for different reasons.
00:11:00.000 --> 00:11:06.000
One over t is not of
exponential type.
00:11:03.000 --> 00:11:09.000
Well, of course,
it's not defined that t equals
00:11:06.000 --> 00:11:12.000
zero. But, you know,
00:11:08.000 --> 00:11:14.000
it's okay for an integral not
to be defined at one point
00:11:12.000 --> 00:11:18.000
because you're measuring an
area, and when you measure an
00:11:16.000 --> 00:11:22.000
area, what happened to one point
doesn't really matter much.
00:11:20.000 --> 00:11:26.000
That's not the thing.
What's wrong with one over t is
00:11:24.000 --> 00:11:30.000
that the integral doesn't
converge at zero times one over
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t dt.
That integral,
00:11:32.000 --> 00:11:38.000
when t is near zero,
this is approximately equal to
00:11:36.000 --> 00:11:42.000
one, right?
If t is zero,
00:11:38.000 --> 00:11:44.000
this is one.
So, it's like the function,
00:11:41.000 --> 00:11:47.000
integral from zero to infinity
of one over t,
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near zero it's close to,
00:11:50.000 --> 00:11:56.000
it's like the integral from
zero to someplace of no
00:11:54.000 --> 00:12:00.000
importance, dt over t.
00:11:57.000 --> 00:12:03.000
But, this does not converge.
This is like log t,
00:12:01.000 --> 00:12:07.000
and log zero is minus infinity.
So, it doesn't converge.
00:12:08.000 --> 00:12:14.000
So, one over t is not
of exponential type.
00:12:11.000 --> 00:12:17.000
So, what's the Laplace
transform of one over t?
00:12:14.000 --> 00:12:20.000
It doesn't have a Laplace
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transform.
Well, what if I put t equals
00:12:20.000 --> 00:12:26.000
negative n?
What about t to the minus one?
00:12:23.000 --> 00:12:29.000
Well, that only works for
00:12:26.000 --> 00:12:32.000
positive integers,
not negative integers.
00:12:30.000 --> 00:12:36.000
Okay, so it's not of
exponential type.
00:12:33.000 --> 00:12:39.000
However, that's because it
never really gets started
00:12:37.000 --> 00:12:43.000
properly.
It's more fun to look at a
00:12:40.000 --> 00:12:46.000
function which is not of
exponential type because it
00:12:45.000 --> 00:12:51.000
grows too fast.
Now, what's a function that
00:12:49.000 --> 00:12:55.000
grows faster that it grows so
rapidly that you can't find any
00:12:54.000 --> 00:13:00.000
function e to the k t
which bounds it?
00:12:58.000 --> 00:13:04.000
A function which grows too
rapidly, a simple one is e to
00:13:03.000 --> 00:13:09.000
the t squared,
grows too rapidly to be of
00:13:07.000 --> 00:13:13.000
exponential type.
And, the argument is simple.
00:13:13.000 --> 00:13:19.000
No matter what you propose,
it's always,
00:13:17.000 --> 00:13:23.000
for the K, no matter how big a
number, use Avogadro's number,
00:13:22.000 --> 00:13:28.000
use anything you want.
Ultimately, this is going to be
00:13:27.000 --> 00:13:33.000
bigger than k t no matter how
big k is, no matter how big k
00:13:32.000 --> 00:13:38.000
is.
When is this going to happen?
00:13:36.000 --> 00:13:42.000
This will happen if t squared
is bigger than k t.
00:13:40.000 --> 00:13:46.000
In other words,
as soon as t becomes bigger
00:13:43.000 --> 00:13:49.000
than k, you might have to wait
quite a while for that to
00:13:47.000 --> 00:13:53.000
happen, but, as soon as t gets
bigger than 10 to the 10 to the
00:13:52.000 --> 00:13:58.000
23,
this e to the t squared will be
00:13:55.000 --> 00:14:01.000
bigger than e to the 10 to the
10 to the 23 times t.
00:14:00.000 --> 00:14:06.000
So, e to the t squared,
00:14:04.000 --> 00:14:10.000
it's a simple
function, a simple elementary
00:14:07.000 --> 00:14:13.000
function.
It grows so rapidly it doesn't
00:14:10.000 --> 00:14:16.000
have a Laplace transform.
Okay, so how are we going to
00:14:13.000 --> 00:14:19.000
solve differential equations if
e to the t squared?
00:14:16.000 --> 00:14:22.000
I won't give you any.
And, the reason I won't give
00:14:19.000 --> 00:14:25.000
you any: because I never saw one
occur in real life.
00:14:22.000 --> 00:14:28.000
Nature, like sines,
cosines, exponentials,
00:14:25.000 --> 00:14:31.000
are fine, I've never seen a
physical, you know,
00:14:28.000 --> 00:14:34.000
this is just my ignorance.
But, I've never seen a physical
00:14:31.000 --> 00:14:37.000
problem that involved a function
growing as rapidly as e to the t
00:14:35.000 --> 00:14:41.000
squared.
That may be just my ignorance.
00:14:40.000 --> 00:14:46.000
But, I do know the Laplace
transform won't be used to solve
00:14:44.000 --> 00:14:50.000
differential equations involving
such a function.
00:14:48.000 --> 00:14:54.000
How about e to the minus t
squared?
00:14:52.000 --> 00:14:58.000
That's different.
It looks almost the same,
00:14:55.000 --> 00:15:01.000
but e to the minus t squared
does this.
00:14:58.000 --> 00:15:04.000
It's very well-behaved.
That's the curve,
00:15:02.000 --> 00:15:08.000
of course, that you're all
afraid of.
00:15:04.000 --> 00:15:10.000
Don't panic.
Okay.
00:15:07.000 --> 00:15:13.000
So, I'd like to explain to you
now how differential equations,
00:15:13.000 --> 00:15:19.000
maybe I should save-- I'll tell
you what.
00:15:17.000 --> 00:15:23.000
We need more formulas.
So, I'll put them,
00:15:20.000 --> 00:15:26.000
why don't I save this board,
and instead,
00:15:24.000 --> 00:15:30.000
I'll describe to you the basic
way Laplace transforms are used
00:15:30.000 --> 00:15:36.000
to solve differential equations,
what are they called,
00:15:35.000 --> 00:15:41.000
a paradigm.
I'll show you the paradigm,
00:15:39.000 --> 00:15:45.000
and then we'll fill in the
holes so you have some overall
00:15:43.000 --> 00:15:49.000
view of how the procedure goes,
and then you'll understand
00:15:46.000 --> 00:15:52.000
where the various pieces fit
into it.
00:15:48.000 --> 00:15:54.000
I think you'll understand it
better that way.
00:15:51.000 --> 00:15:57.000
So, what do we do?
Start with the differential
00:15:54.000 --> 00:16:00.000
equation.
But, right away,
00:15:55.000 --> 00:16:01.000
there's a fundamental
difference between what the
00:15:58.000 --> 00:16:04.000
Laplace transform does,
and what we've been doing up
00:16:01.000 --> 00:16:07.000
until now, namely,
what you have to start with is
00:16:04.000 --> 00:16:10.000
not merely the differential
equation.
00:16:08.000 --> 00:16:14.000
Let's say we have linear with
constant coefficients.
00:16:12.000 --> 00:16:18.000
It's almost never used to solve
any other type of problem.
00:16:16.000 --> 00:16:22.000
And, let's take second order so
I don't have to do,
00:16:20.000 --> 00:16:26.000
because that's the kind we've
been working with all term.
00:16:25.000 --> 00:16:31.000
But, it's allowed to be
inhomogeneous,
00:16:28.000 --> 00:16:34.000
so, f of t.
Let's call the something else,
00:16:32.000 --> 00:16:38.000
another letter,
h of t.
00:16:36.000 --> 00:16:42.000
I'll want f of t for
the function I'm taking the
00:16:39.000 --> 00:16:45.000
Laplace transform of.
All right, now,
00:16:42.000 --> 00:16:48.000
the difference is that up to
now, you know techniques for
00:16:45.000 --> 00:16:51.000
solving this just as it stands.
The Laplace transform does not
00:16:50.000 --> 00:16:56.000
know how to solve this just
doesn't stands.
00:16:52.000 --> 00:16:58.000
The Laplace transform must have
an initial value problem.
00:16:56.000 --> 00:17:02.000
In other words,
you must supply from the
00:16:59.000 --> 00:17:05.000
beginning the initial conditions
that the y is to satisfy.
00:17:04.000 --> 00:17:10.000
Now, I don't want to say any
specific numbers,
00:17:06.000 --> 00:17:12.000
so I'll use generic numbers.
Well, but look,
00:17:08.000 --> 00:17:14.000
what do we do if we get a
problem and there are no initial
00:17:12.000 --> 00:17:18.000
conditions; does that mean we
can't use the Laplace transform?
00:17:15.000 --> 00:17:21.000
No, of course you can use it.
But, you will just have to
00:17:18.000 --> 00:17:24.000
assume the initial conditions
are on the numbers.
00:17:21.000 --> 00:17:27.000
You'll say it but the initial
conditions be y sub zero
00:17:24.000 --> 00:17:30.000
and y zero prime,
or whatever,
00:17:26.000 --> 00:17:32.000
a and b, whatever you want to
call it.
00:17:30.000 --> 00:17:36.000
And now, the answer,
then, will involve the a and
00:17:34.000 --> 00:17:40.000
the b or the y zero and the y
zero prime.
00:17:38.000 --> 00:17:44.000
But, you must,
at least, give lip service to
00:17:42.000 --> 00:17:48.000
the initial conditions,
whereas before we didn't have
00:17:46.000 --> 00:17:52.000
to do that.
Now, depending on your point of
00:17:50.000 --> 00:17:56.000
view, that's a grave defect,
or it is, so what?
00:17:54.000 --> 00:18:00.000
Let's adopt the so what point
of view.
00:17:57.000 --> 00:18:03.000
So, there's our problem.
It's an initial value problem.
00:18:03.000 --> 00:18:09.000
How is it solved by the Laplace
transform?
00:18:05.000 --> 00:18:11.000
Well, the idea is you take the
Laplace transform of this
00:18:09.000 --> 00:18:15.000
differential equation and the
initial conditions.
00:18:12.000 --> 00:18:18.000
So, I'm going to explain to you
how to do that.
00:18:15.000 --> 00:18:21.000
Not right now,
because we're going to need,
00:18:17.000 --> 00:18:23.000
first, the Laplace transform of
a derivative,
00:18:20.000 --> 00:18:26.000
the formula for that.
You don't know that yet.
00:18:23.000 --> 00:18:29.000
But when you do know it,
you will be able to take the
00:18:26.000 --> 00:18:32.000
Laplace transform of the initial
value problem.
00:18:29.000 --> 00:18:35.000
So, I'll put the little l here,
and what comes out is,
00:18:32.000 --> 00:18:38.000
well, y of t is the
solution to the original
00:18:36.000 --> 00:18:42.000
problem.
If y of t is the
00:18:39.000 --> 00:18:45.000
function which satisfies that
equation and these initial
00:18:43.000 --> 00:18:49.000
conditions, its Laplace
transform, let's call it capital
00:18:48.000 --> 00:18:54.000
Y, that's our standard notation,
but it's going to be of a new
00:18:52.000 --> 00:18:58.000
variable, s.
So, when I take the Laplace
00:18:55.000 --> 00:19:01.000
transform of the differential
equation with the initial
00:18:59.000 --> 00:19:05.000
conditions, what comes out is an
algebraic-- the emphasis is on
00:19:04.000 --> 00:19:10.000
algebraic: no derivatives,
no transcendental functions,
00:19:08.000 --> 00:19:14.000
nothing like that,
an algebraic equation,
00:19:11.000 --> 00:19:17.000
m Y of s.
00:19:22.000 --> 00:19:28.000
And, now what?
Well, now, in the domain of s,
00:19:25.000 --> 00:19:31.000
it's easy to solve this
algebraic equation.
00:19:28.000 --> 00:19:34.000
Not all algebraic equations are
easy to solve for the capital Y.
00:19:33.000 --> 00:19:39.000
But, the ones you will get will
always be, not because I am
00:19:37.000 --> 00:19:43.000
making life easy for you,
but that's the way the Laplace
00:19:42.000 --> 00:19:48.000
transform works.
So, you will solve it for Y.
00:19:45.000 --> 00:19:51.000
And, the answer will always
come out to be Y equals,
00:19:49.000 --> 00:19:55.000
Y of s equals some
rational function,
00:19:52.000 --> 00:19:58.000
some quotient of polynomials in
s, a polynomial in s divided by
00:19:57.000 --> 00:20:03.000
some other polynomial in s.
00:20:09.000 --> 00:20:15.000
And, now what?
Well, this is the Laplace
00:20:12.000 --> 00:20:18.000
transform of the answer.
This is the Laplace transform
00:20:17.000 --> 00:20:23.000
of the solution we are looking
for.
00:20:20.000 --> 00:20:26.000
So, the final step is to go
backwards by taking the inverse
00:20:25.000 --> 00:20:31.000
Laplace transform of this guy.
And, what will you get?
00:20:30.000 --> 00:20:36.000
Well, you will get y equals the
y of t that we are
00:20:35.000 --> 00:20:41.000
looking for.
It's really a wildly improbable
00:20:39.000 --> 00:20:45.000
procedure.
In other words,
00:20:41.000 --> 00:20:47.000
instead of going from here to
here, you have to imagine
00:20:45.000 --> 00:20:51.000
there's a mountain here.
And, the only way to get around
00:20:48.000 --> 00:20:54.000
it is to go, first,
here, and then cross the stream
00:20:51.000 --> 00:20:57.000
here, and then go back up,
and go back up.
00:20:54.000 --> 00:21:00.000
It looks like a senseless
procedure, what do they call it,
00:20:58.000 --> 00:21:04.000
going around Robin Hood's barn,
it was called when I was a,
00:21:01.000 --> 00:21:07.000
I don't know why it's called
that.
00:21:05.000 --> 00:21:11.000
But that's what we used to call
it; not Laplace transform.
00:21:09.000 --> 00:21:15.000
That was just a generic thing
when you had to do something
00:21:14.000 --> 00:21:20.000
like this.
But, the answer is that it's
00:21:18.000 --> 00:21:24.000
hard to go from here to here,
but trivial to go from here to
00:21:23.000 --> 00:21:29.000
here.
This solution step is the
00:21:25.000 --> 00:21:31.000
easiest step of all.
This is not very hard.
00:21:29.000 --> 00:21:35.000
It's easy, in fact.
This is easy and
00:21:33.000 --> 00:21:39.000
straightforward.
This is trivial,
00:21:36.000 --> 00:21:42.000
essentially,
yeah, trivial.
00:21:38.000 --> 00:21:44.000
But, this step is the hard
step.
00:21:41.000 --> 00:21:47.000
This is where you have to use
partial fractions,
00:21:45.000 --> 00:21:51.000
look up things in the table to
get back there so that most of
00:21:50.000 --> 00:21:56.000
the work of the procedure isn't
going from here to here.
00:21:55.000 --> 00:22:01.000
Going from here to there is a
breeze.
00:22:00.000 --> 00:22:06.000
Okay, now, in order to
implement this,
00:22:02.000 --> 00:22:08.000
what is it we have to do?
Well, the basic thing is I have
00:22:05.000 --> 00:22:11.000
to explain to you,
you already know at least a
00:22:08.000 --> 00:22:14.000
little bit, a reasonable amount
of technique for taking that
00:22:12.000 --> 00:22:18.000
step if you went to recitation
yesterday and practiced a little
00:22:16.000 --> 00:22:22.000
bit.
This part, I assure you,
00:22:17.000 --> 00:22:23.000
is nothing.
So, all I have to do now is
00:22:20.000 --> 00:22:26.000
explain to you how to take the
Laplace transform of the
00:22:23.000 --> 00:22:29.000
differential equation.
And, that really means,
00:22:26.000 --> 00:22:32.000
how do you take the Laplace
transform of a derivative?
00:22:31.000 --> 00:22:37.000
So, that's our problem.
What I want to form,
00:22:35.000 --> 00:22:41.000
in other words,
is a formula for the Laplace
00:22:39.000 --> 00:22:45.000
transform f prime of t.
00:22:43.000 --> 00:22:49.000
Now, in terms of what?
Well, since f is an arbitrary
00:22:48.000 --> 00:22:54.000
function, the only thing I could
hope for is somehow to express
00:22:54.000 --> 00:23:00.000
the Laplace transform of the
derivative in terms of the
00:23:00.000 --> 00:23:06.000
Laplace transform of the
original function.
00:23:06.000 --> 00:23:12.000
So, that's what I'm aiming for.
Okay, where are we going to
00:23:10.000 --> 00:23:16.000
start?
Well, starting is easy because
00:23:12.000 --> 00:23:18.000
we know nothing.
If you don't know anything,
00:23:15.000 --> 00:23:21.000
then there's no place to start
but the definition.
00:23:18.000 --> 00:23:24.000
Since I know nothing whatever
about the function f of t,
00:23:22.000 --> 00:23:28.000
and I want to
calculate the Laplace transform,
00:23:26.000 --> 00:23:32.000
I'd better start with the
definition.
00:23:30.000 --> 00:23:36.000
Whatever this is,
it's the integral from zero to
00:23:33.000 --> 00:23:39.000
infinity of e to the minus s t
times f prime of t dt.
00:23:39.000 --> 00:23:45.000
Now, what am I looking for?
I'm looking for somehow to
00:23:43.000 --> 00:23:49.000
transform this so that what
appears here is not
00:23:47.000 --> 00:23:53.000
f prime of t, which I'm
clueless about, but f of t
00:23:50.000 --> 00:23:56.000
because if this were f of t,
this expression would be the
00:23:54.000 --> 00:24:00.000
Laplace transform of f of t.
00:23:56.000 --> 00:24:02.000
And, I'm assuming I know that.
So, the question is how do I
00:24:02.000 --> 00:24:08.000
take this and somehow do
something clever to it that
00:24:05.000 --> 00:24:11.000
turns this into f of t
instead of f prime of t?
00:24:10.000 --> 00:24:16.000
Now, to first the question that
00:24:13.000 --> 00:24:19.000
way, I hope I would get 100%
response on what to do.
00:24:16.000 --> 00:24:22.000
But, I'll go for 1%.
So, what should I do?
00:24:20.000 --> 00:24:26.000
I want to change that, so that
instead of f prime of t,
00:24:23.000 --> 00:24:29.000
f of t
appears there instead.
00:24:27.000 --> 00:24:33.000
What should I do?
Integrate by parts,
00:24:31.000 --> 00:24:37.000
the most fundamental procedure
in advanced analysis.
00:24:35.000 --> 00:24:41.000
Everything important and
interesting depends on
00:24:39.000 --> 00:24:45.000
integration by parts.
And, when you consider that
00:24:43.000 --> 00:24:49.000
integration by parts is nothing
more than just the formula for
00:24:48.000 --> 00:24:54.000
the derivative of a product read
backwards, it's amazing.
00:24:53.000 --> 00:24:59.000
It never fails to amaze me,
but it's okay.
00:24:56.000 --> 00:25:02.000
That's what mathematics are so
great.
00:24:59.000 --> 00:25:05.000
Okay, so let's use integration
by parts.
00:25:04.000 --> 00:25:10.000
Integration by parts:
okay, so, we have to decide,
00:25:06.000 --> 00:25:12.000
of course, there's no doubt
that this is the factor we want
00:25:10.000 --> 00:25:16.000
to integrate,
which means we have to be
00:25:12.000 --> 00:25:18.000
willing to differentiate this
factor.
00:25:14.000 --> 00:25:20.000
But that will be okay because
it looks practically,
00:25:17.000 --> 00:25:23.000
like any exponential,
it looks practically the same
00:25:20.000 --> 00:25:26.000
after you've differentiated it.
So, let's do the work.
00:25:23.000 --> 00:25:29.000
First step is you don't do the
differentiation.
00:25:26.000 --> 00:25:32.000
You only do the integration.
So, the first step is e to the
00:25:29.000 --> 00:25:35.000
negative s t.
And, the integral of f prime of
00:25:34.000 --> 00:25:40.000
t is just f of t.
00:25:36.000 --> 00:25:42.000
And, that's to be evaluated
between the limits zero and
00:25:40.000 --> 00:25:46.000
infinity.
And then, minus,
00:25:42.000 --> 00:25:48.000
again, before you forget it,
put down that minus sign.
00:25:45.000 --> 00:25:51.000
The integral between the limits
of what you get by doing both
00:25:49.000 --> 00:25:55.000
operations, both the
differentiation and the
00:25:52.000 --> 00:25:58.000
integration.
So, the differentiation will be
00:25:55.000 --> 00:26:01.000
by using the chain rule.
Remember, I'm differentiating
00:25:59.000 --> 00:26:05.000
with respect to t.
The variable is t here,
00:26:03.000 --> 00:26:09.000
not s.
s is just a parameter.
00:26:06.000 --> 00:26:12.000
It's just a constant,
a variable constant,
00:26:09.000 --> 00:26:15.000
if you get my meaning.
That's not an oxymoron.
00:26:13.000 --> 00:26:19.000
A variable constant:
a parameter is a variable
00:26:16.000 --> 00:26:22.000
constant, variable because you
can manipulate the little slider
00:26:21.000 --> 00:26:27.000
and make a change its value,
right?
00:26:24.000 --> 00:26:30.000
That's why it's variable.
It's not a variable.
00:26:28.000 --> 00:26:34.000
It's variable,
if you get the distinction.
00:26:33.000 --> 00:26:39.000
Okay, well, I mean,
it becomes a variable
00:26:35.000 --> 00:26:41.000
[LAUGHTER].
But right now,
00:26:37.000 --> 00:26:43.000
it's not a variable.
It's just sitting there in the
00:26:41.000 --> 00:26:47.000
integral.
All right, so,
00:26:43.000 --> 00:26:49.000
minus s, e to the negative s t,
f of t dt.
00:26:47.000 --> 00:26:53.000
Now, this part's easy.
00:26:49.000 --> 00:26:55.000
The interesting thing is this
expression.
00:26:52.000 --> 00:26:58.000
So, and the most interesting
thing is I have to evaluate it
00:26:56.000 --> 00:27:02.000
at infinity.
Now, of course,
00:26:58.000 --> 00:27:04.000
that means take the limit as
you go towards,
00:27:01.000 --> 00:27:07.000
as you let t goes to infinity.
Now, so what I'm interested in
00:27:07.000 --> 00:27:13.000
knowing is what's the limit of
that expression?
00:27:10.000 --> 00:27:16.000
I'll write it as f of t divided
by e to the s t.
00:27:13.000 --> 00:27:19.000
Remember, s is a positive
00:27:16.000 --> 00:27:22.000
number.
s t goes to infinity,
00:27:18.000 --> 00:27:24.000
and I want to know what the
limit of that is.
00:27:21.000 --> 00:27:27.000
Well, the limit is what it is.
But really, if that limit isn't
00:27:25.000 --> 00:27:31.000
zero, I'm in deep trouble since
the whole process is out of
00:27:29.000 --> 00:27:35.000
control.
What will make that limit zero?
00:27:33.000 --> 00:27:39.000
Well, that f of t
should not grow faster than e to
00:27:37.000 --> 00:27:43.000
the s t if s is a big
enough number.
00:27:41.000 --> 00:27:47.000
And now, that's just what will
happen if f of t is of
00:27:45.000 --> 00:27:51.000
exponential type.
It's for this step right here
00:27:48.000 --> 00:27:54.000
that is the most crucial place
at which we need to know that f
00:27:53.000 --> 00:27:59.000
of t is of exponential type.
So, that limit is zero since f
00:27:57.000 --> 00:28:03.000
of t is of exponential type,
in other words,
00:28:01.000 --> 00:28:07.000
that the value,
the absolute value of f of t,
00:28:04.000 --> 00:28:10.000
becomes less than,
let's say, put in the c if you
00:28:09.000 --> 00:28:15.000
want, but it's not very
important, c e to the k t
00:28:13.000 --> 00:28:19.000
efor all values of t.
And, therefore,
00:28:18.000 --> 00:28:24.000
this will go to zero as soon as
s becomes bigger than that k.
00:28:22.000 --> 00:28:28.000
In other words,
00:28:23.000 --> 00:28:29.000
if f of t isn't
growing any faster than e to the
00:28:27.000 --> 00:28:33.000
k t ,
then as soon as s is a number,
00:28:30.000 --> 00:28:36.000
that parameter has the value
bigger than k,
00:28:33.000 --> 00:28:39.000
this ratio is going to go to
zero because the denominator
00:28:37.000 --> 00:28:43.000
will always be bigger than the
numerator, and getting bigger
00:28:41.000 --> 00:28:47.000
faster.
So, this goes to zero if s is
00:28:45.000 --> 00:28:51.000
bigger than k.
At the upper limit,
00:28:48.000 --> 00:28:54.000
therefore, this is zero.
Again, assuming that s is
00:28:52.000 --> 00:28:58.000
bigger than that k,
the k of the exponential type,
00:28:57.000 --> 00:29:03.000
how about at the lower limit?
We're used to seeing zero
00:29:01.000 --> 00:29:07.000
there, but we're not going to
get zero.
00:29:04.000 --> 00:29:10.000
If I plug in t equals zero,
this factor becomes one.
00:29:09.000 --> 00:29:15.000
And, what happens to that one?
f of zero.
00:29:13.000 --> 00:29:19.000
You mean, I'm going to have to
know what f of zero is before I
00:29:18.000 --> 00:29:24.000
can take the Laplace transform
of this derivative?
00:29:22.000 --> 00:29:28.000
The answer is yes,
and that's why you have to have
00:29:26.000 --> 00:29:32.000
an initial value problem.
You have to know in advance
00:29:30.000 --> 00:29:36.000
what the value of the function
that you are looking for is at
00:29:34.000 --> 00:29:40.000
zero because it enters into the
formula.
00:29:37.000 --> 00:29:43.000
I didn't make up these rules;
I'm just following them.
00:29:40.000 --> 00:29:46.000
So, what's the rest?
The two negatives cancel,
00:29:43.000 --> 00:29:49.000
and you get plus s.
It's just a parameter,
00:29:47.000 --> 00:29:53.000
so I can pull it out of the
integral.
00:29:49.000 --> 00:29:55.000
I'm integrating with respect to
t, and what's left is,
00:29:53.000 --> 00:29:59.000
well, what is left?
If I take out minus s,
00:29:56.000 --> 00:30:02.000
combine it there,
I get what's left is just the
00:29:59.000 --> 00:30:05.000
Laplace transform of the
function I started with.
00:30:04.000 --> 00:30:10.000
So, it's F of s.
And, that's the magic formula
00:30:10.000 --> 00:30:16.000
for the Laplace transform of the
derivative.
00:30:15.000 --> 00:30:21.000
So, it's worth putting up on
our little list.
00:30:20.000 --> 00:30:26.000
So, f prime of t,
assuming it's of exponential
00:30:26.000 --> 00:30:32.000
type, has as its Laplace
transform, well,
00:30:31.000 --> 00:30:37.000
what is it?
Let's put down the major part
00:30:35.000 --> 00:30:41.000
of it is s times whatever the
Laplace transform of the
00:30:39.000 --> 00:30:45.000
original function,
F of t,
00:30:41.000 --> 00:30:47.000
was.
So, I take the original Laplace
00:30:44.000 --> 00:30:50.000
transform.
When I multiply it by s,
00:30:46.000 --> 00:30:52.000
that corresponds to taking the
derivative.
00:30:49.000 --> 00:30:55.000
But there's also that little
extra piece.
00:30:51.000 --> 00:30:57.000
I must know the value of the
starting value of the function.
00:30:55.000 --> 00:31:01.000
This is the formula you'll used
to take a Laplace transform of
00:31:00.000 --> 00:31:06.000
the differential equation.
Now, but you see I'm not done
00:31:05.000 --> 00:31:11.000
yet because that will take care
of the term a y prime.
00:31:09.000 --> 00:31:15.000
But, I don't know what the
Laplace transform of the second
00:31:13.000 --> 00:31:19.000
derivative is.
Okay, so, we need a formula for
00:31:16.000 --> 00:31:22.000
the Laplace transform of a
second derivative as well as the
00:31:20.000 --> 00:31:26.000
first.
Now, the hack method is to say,
00:31:23.000 --> 00:31:29.000
secondary, all right.
I've got to do this.
00:31:25.000 --> 00:31:31.000
I'll second derivative here,
second derivative here,
00:31:29.000 --> 00:31:35.000
what do I do with that?
Ah-ha, I integrate by parts
00:31:33.000 --> 00:31:39.000
twice.
Yes, you can do that.
00:31:34.000 --> 00:31:40.000
But that's a hack method.
And, of course,
00:31:40.000 --> 00:31:46.000
I know you're too smart to do
that.
00:31:45.000 --> 00:31:51.000
What you would do instead is--
How are we going to fill that
00:31:53.000 --> 00:31:59.000
in?
Well, a second derivative is
00:31:58.000 --> 00:32:04.000
also a first derivative.
A second derivative is the
00:32:07.000 --> 00:32:13.000
first derivative of the first
derivative.
00:32:14.000 --> 00:32:20.000
Okay, now, we'll just call this
glop, something.
00:32:22.000 --> 00:32:28.000
So, it's glop prime.
What is the formula for the
00:32:31.000 --> 00:32:37.000
Laplace transform of glop prime?
It is, well,
00:32:39.000 --> 00:32:45.000
I have my formula.
It is the glop prime.
00:32:43.000 --> 00:32:49.000
The formula for it is s times
the Laplace transform of glop,
00:32:50.000 --> 00:32:56.000
okay, glop.
Well, glop is f prime of t.
00:32:55.000 --> 00:33:01.000
I'm not done yet,
00:32:58.000 --> 00:33:04.000
minus glop evaluated at zero.
What's glop evaluated at zero?
00:33:05.000 --> 00:33:11.000
Well, f prime of zero.
00:33:10.000 --> 00:33:16.000
Now, I don't want the formula
in that form,
00:33:13.000 --> 00:33:19.000
but I have to have it in that
form because I know what the
00:33:17.000 --> 00:33:23.000
Laplace transform of f prime of
t is.
00:33:20.000 --> 00:33:26.000
I just calculated that.
So, this is equal to s times
00:33:24.000 --> 00:33:30.000
the Laplace transform of f prime
of t, which is s times F of s,
00:33:28.000 --> 00:33:34.000
capital F of s,
minus f of zero.
00:33:31.000 --> 00:33:37.000
All that bracket stuff
00:33:34.000 --> 00:33:40.000
corresponds to this guy.
And, don't forget the stuff
00:33:38.000 --> 00:33:44.000
that's tagging along,
minus f prime of zero.
00:33:42.000 --> 00:33:48.000
And now, put that all together.
00:33:45.000 --> 00:33:51.000
What is it going to be?
Well, there's the principal
00:33:49.000 --> 00:33:55.000
term which consists of s squared
multiplied by F of s.
00:33:54.000 --> 00:34:00.000
That's the main part of it.
00:33:56.000 --> 00:34:02.000
And, the rest is the sort of
fellow travelers.
00:34:00.000 --> 00:34:06.000
So, we have minus s times f of
zero,
00:34:04.000 --> 00:34:10.000
little term tagging along.
This is a constant times s.
00:34:10.000 --> 00:34:16.000
And then, we've got another
one, still another constant.
00:34:14.000 --> 00:34:20.000
So, what we have is to
calculate the Laplace transform
00:34:18.000 --> 00:34:24.000
of the second derivative,
I need to know both f of zero
00:34:22.000 --> 00:34:28.000
and f prime of zero,
exactly the initial
00:34:27.000 --> 00:34:33.000
conditions that the problem was
given for the initial value
00:34:31.000 --> 00:34:37.000
problem.
But, notice,
00:34:33.000 --> 00:34:39.000
there's a principal part of it.
That's the s squared F of s.
00:34:37.000 --> 00:34:43.000
That's the guts of it,
00:34:39.000 --> 00:34:45.000
so to speak.
The rest of it,
00:34:41.000 --> 00:34:47.000
you know, you might hope that
these two numbers are zero.
00:34:44.000 --> 00:34:50.000
It could happen,
and often it is made to happen
00:34:47.000 --> 00:34:53.000
and problems to simplify them.
And I case, you don't have to
00:34:51.000 --> 00:34:57.000
worry; they're not there.
But, if they are there,
00:34:54.000 --> 00:35:00.000
you must put them in or you get
the wrong answer.
00:34:56.000 --> 00:35:02.000
So, that's the list of
formulas.
00:35:00.000 --> 00:35:06.000
So, those formulas on the top
board and these two extra ones,
00:35:06.000 --> 00:35:12.000
those are the things you will
be working with on Friday.
00:35:12.000 --> 00:35:18.000
But I stress,
the Laplace transform won't be
00:35:17.000 --> 00:35:23.000
a big part of the exam.
The exam, of course,
00:35:22.000 --> 00:35:28.000
doesn't exist,
let's say a maximum of 20%,
00:35:27.000 --> 00:35:33.000
maybe 15.
I don't know,
00:35:29.000 --> 00:35:35.000
give or take a few points.
Yeah, what's a point or two?
00:35:37.000 --> 00:35:43.000
Okay, let's solve,
yeah, we have time.
00:35:41.000 --> 00:35:47.000
We have time to solve a
problem.
00:35:44.000 --> 00:35:50.000
Let's solve a problem.
See, I can't touch that.
00:35:49.000 --> 00:35:55.000
It's untouchable.
Okay, this, we've got to keep.
00:36:14.000 --> 00:36:20.000
Problem? Okay.
00:36:39.000 --> 00:36:45.000
Okay, now you know how to solve
this problem by operators.
00:36:43.000 --> 00:36:49.000
Let me just briefly remind you
of the basic steps.
00:36:47.000 --> 00:36:53.000
You have to do two separate
tasks.
00:36:50.000 --> 00:36:56.000
You have to first solve the
homogeneous equation,
00:36:54.000 --> 00:37:00.000
putting a zero there.
That's the first thing you
00:36:58.000 --> 00:37:04.000
learned to do.
That's easy.
00:37:00.000 --> 00:37:06.000
You could almost do that in
your head now.
00:37:05.000 --> 00:37:11.000
You get the characteristic
polynomial, get its roots,
00:37:08.000 --> 00:37:14.000
get the two functions,
e to the t and e to the
00:37:11.000 --> 00:37:17.000
negative t,
which are the solutions.
00:37:14.000 --> 00:37:20.000
You make up c1 times one,
and c2 times the other.
00:37:18.000 --> 00:37:24.000
That's the complementary
function that solves the
00:37:21.000 --> 00:37:27.000
homogeneous problem.
And then you have to find a
00:37:24.000 --> 00:37:30.000
particular solution.
Can you see what would happen
00:37:27.000 --> 00:37:33.000
if you try to find the
particular solution?
00:37:30.000 --> 00:37:36.000
The number here is negative
one, right?
00:37:34.000 --> 00:37:40.000
Negative one is a root of the
characteristic polynomial,
00:37:37.000 --> 00:37:43.000
so you've got to use that extra
formula.
00:37:40.000 --> 00:37:46.000
It's okay.
That's why I gave it to you.
00:37:42.000 --> 00:37:48.000
You've used the exponential
input theorem with the extra
00:37:46.000 --> 00:37:52.000
formula.
Then, you will get the
00:37:48.000 --> 00:37:54.000
particular solution.
And now, you have to make the
00:37:51.000 --> 00:37:57.000
general solution.
The particular solution plus
00:37:54.000 --> 00:38:00.000
the complementary function,
and now you are ready to put in
00:37:58.000 --> 00:38:04.000
the initial conditions.
At the very end,
00:38:02.000 --> 00:38:08.000
when you've got the whole
general solution,
00:38:05.000 --> 00:38:11.000
now you put in,
not before, you put in the
00:38:07.000 --> 00:38:13.000
initial conditions.
You have to calculate the
00:38:11.000 --> 00:38:17.000
derivative of that thing and
substitute this.
00:38:14.000 --> 00:38:20.000
You take it as it stands to
substitute this.
00:38:17.000 --> 00:38:23.000
You get a pair of simultaneous
equations for c1 and c2.
00:38:21.000 --> 00:38:27.000
You solve them:
answer.
00:38:22.000 --> 00:38:28.000
It's a rather elaborate
procedure, which has at least
00:38:26.000 --> 00:38:32.000
three or four separate steps,
all of which,
00:38:29.000 --> 00:38:35.000
of course, must be done
correctly.
00:38:33.000 --> 00:38:39.000
Now, the Laplace transform,
instead, feeds the entire
00:38:37.000 --> 00:38:43.000
problem into the Laplace
transform machine.
00:38:40.000 --> 00:38:46.000
You follow that little blue
pattern, and you come out with
00:38:44.000 --> 00:38:50.000
the answer.
So, let's do the Laplace
00:38:47.000 --> 00:38:53.000
transform way.
Okay, so, the first step is to
00:38:51.000 --> 00:38:57.000
say, if here's my unknown
function, y of t,
00:38:55.000 --> 00:39:01.000
it obeys this law,
and here are its starting
00:38:58.000 --> 00:39:04.000
values, a bit of its derivative.
What I'm going to take is the
00:39:03.000 --> 00:39:09.000
Laplace transform of this
equation.
00:39:06.000 --> 00:39:12.000
In other words,
I'll take the Laplace transform
00:39:09.000 --> 00:39:15.000
of this side,
and this side also.
00:39:11.000 --> 00:39:17.000
And, they must be equal because
if they were equal to start
00:39:15.000 --> 00:39:21.000
with, the Laplace transforms
also have to be equal.
00:39:18.000 --> 00:39:24.000
Okay, so let's take the Laplace
transform of this equation.
00:39:22.000 --> 00:39:28.000
Okay, first ID the Laplace
transform of the second
00:39:25.000 --> 00:39:31.000
derivative.
Okay, that's going to be,
00:39:27.000 --> 00:39:33.000
don't forget the principal
terms.
00:39:31.000 --> 00:39:37.000
There is some people who get so
hypnotized by this.
00:39:34.000 --> 00:39:40.000
I just know I'm going to forget
this.
00:39:36.000 --> 00:39:42.000
So, they read it.
Then they forget this.
00:39:38.000 --> 00:39:44.000
But that's everything.
That's the important part.
00:39:41.000 --> 00:39:47.000
Okay, so it's s times,
I'm calling the Laplace
00:39:44.000 --> 00:39:50.000
transform not capital F but
capital Y because my original
00:39:48.000 --> 00:39:54.000
function is called little y.
So, it's s squared Y.
00:39:51.000 --> 00:39:57.000
It's Y of s,
but I'm not going to put that,
00:39:54.000 --> 00:40:00.000
the of s in because it just
makes the thing look more
00:39:58.000 --> 00:40:04.000
complicated.
And now, before you forget,
00:40:02.000 --> 00:40:08.000
you have to put in the rest.
So, minus s times the value at
00:40:06.000 --> 00:40:12.000
zero, which is one,
minus the value of the
00:40:10.000 --> 00:40:16.000
derivative.
But, that's zero.
00:40:12.000 --> 00:40:18.000
So, this is not too hard a
problem.
00:40:15.000 --> 00:40:21.000
So, minus s minus zero,
so I don't have to put that
00:40:19.000 --> 00:40:25.000
in.
So, all this is the Laplace
00:40:22.000 --> 00:40:28.000
transform of y double prime.
00:40:25.000 --> 00:40:31.000
And now, minus the Laplace
transform of y,
00:40:29.000 --> 00:40:35.000
well, that's just capital Y.
What's that equal to?
00:40:35.000 --> 00:40:41.000
The Laplace transform of the
right-hand side.
00:40:40.000 --> 00:40:46.000
Okay, look up the formula.
It is e to the negative t,
00:40:45.000 --> 00:40:51.000
a is minus one,
so, it's one over s minus minus
00:40:51.000 --> 00:40:57.000
one;
so, it is s plus one.
00:40:57.000 --> 00:41:03.000
This is that.
Okay, the next thing we have to
00:41:02.000 --> 00:41:08.000
do is solve for Y.
That doesn't look too hard.
00:41:05.000 --> 00:41:11.000
Solve it for y.
Okay, the best thing to do is
00:41:08.000 --> 00:41:14.000
put s squared,
group all the Y terms together
00:41:11.000 --> 00:41:17.000
unless you're really quite a
good calculator.
00:41:14.000 --> 00:41:20.000
Maybe make one extra line out
of it.
00:41:17.000 --> 00:41:23.000
Yeah, definitely do this.
And then, the extra garbage I
00:41:21.000 --> 00:41:27.000
refer to as the garbage,
this stuff, and this stuff,
00:41:25.000 --> 00:41:31.000
the stuff, the linear
polynomials which are tagging
00:41:28.000 --> 00:41:34.000
along move to the right-hand
side because they don't involve
00:41:32.000 --> 00:41:38.000
capital Y.
So, this we will move to the
00:41:37.000 --> 00:41:43.000
other side.
And so, that's equal to (one
00:41:40.000 --> 00:41:46.000
over (s plus one)) plus s.
00:41:44.000 --> 00:41:50.000
Now, you have a basic choice.
About half the time,
00:41:48.000 --> 00:41:54.000
it's a good idea to combine
these terms.
00:41:51.000 --> 00:41:57.000
The other half of the time,
it's not a good idea to combine
00:41:56.000 --> 00:42:02.000
those terms.
So, how do we know whether to
00:41:59.000 --> 00:42:05.000
do it or not to do it?
Experience, which you will get
00:42:03.000 --> 00:42:09.000
by solving many,
many problems.
00:42:06.000 --> 00:42:12.000
Okay, I'm going to combine them
because I think it's a good
00:42:11.000 --> 00:42:17.000
thing to do here.
So, what is that?
00:42:15.000 --> 00:42:21.000
That's s squared plus s plus
one.
00:42:20.000 --> 00:42:26.000
So, it's s squared plus s plus
one divided by s plus one,
00:42:25.000 --> 00:42:31.000
okay?
I'm still not done because now
00:42:28.000 --> 00:42:34.000
we have to know,
what's Y?
00:42:31.000 --> 00:42:37.000
All right, now we have to
think.
00:42:35.000 --> 00:42:41.000
What we're going to do is get Y
in this form.
00:42:38.000 --> 00:42:44.000
But, I want it in the form in
which it's most suited for using
00:42:42.000 --> 00:42:48.000
partial fractions.
In other words,
00:42:44.000 --> 00:42:50.000
I want the denominator as
factored as I possibly can be.
00:42:48.000 --> 00:42:54.000
Okay, well, the numerator is
going to be just what it was.
00:42:52.000 --> 00:42:58.000
How should I write the
denominator?
00:42:55.000 --> 00:43:01.000
Well, the denominator is going
to have the factor s plus one
00:42:59.000 --> 00:43:05.000
in it from here.
But after I divide through,
00:43:04.000 --> 00:43:10.000
the other factor will be s
squared minus one,
00:43:09.000 --> 00:43:15.000
right?
But, s squared minus one is s
00:43:12.000 --> 00:43:18.000
minus one times s plus one.
00:43:17.000 --> 00:43:23.000
So, I have to divide this by s
squared minus one.
00:43:23.000 --> 00:43:29.000
Factored, it's this.
So, the end result is there are
00:43:27.000 --> 00:43:33.000
two of these and one of the
other.
00:43:32.000 --> 00:43:38.000
And now, it's ready to be used.
It's better to be a partial
00:43:36.000 --> 00:43:42.000
fraction.
So, the final step is to use a
00:43:40.000 --> 00:43:46.000
partial fraction's decomposition
on this so that you can
00:43:44.000 --> 00:43:50.000
calculate its inverse Laplace
transform.
00:43:48.000 --> 00:43:54.000
So, let's do that.
Okay, (s squared plus s plus
00:43:51.000 --> 00:43:57.000
one) divided by that thing,
(s plus one) squared times (s
00:43:56.000 --> 00:44:02.000
minus one) equals s plus
00:44:01.000 --> 00:44:07.000
one squared plus s
plus one plus s minus one.
00:44:07.000 --> 00:44:13.000
In the top will be constants,
00:44:12.000 --> 00:44:18.000
just constants.
Let's do it this way first,
00:44:15.000 --> 00:44:21.000
and I'll say at the very end,
something else.
00:44:18.000 --> 00:44:24.000
Maybe now.
Many of you are upset.
00:44:21.000 --> 00:44:27.000
Some of you are upset.
I know this for a fact because
00:44:25.000 --> 00:44:31.000
in high school,
or wherever you learned to do
00:44:28.000 --> 00:44:34.000
this before, there weren't two
terms here.
00:44:33.000 --> 00:44:39.000
There was just one term,
s plus one squared.
00:44:36.000 --> 00:44:42.000
If you do it that way,
00:44:39.000 --> 00:44:45.000
then it's all right.
Then, it's all right,
00:44:42.000 --> 00:44:48.000
but I don't recommend it.
In that case,
00:44:45.000 --> 00:44:51.000
the numerators will not be
constants.
00:44:48.000 --> 00:44:54.000
But, if you just have that,
then because this is a
00:44:52.000 --> 00:44:58.000
quadratic polynomial all by
itself.
00:44:54.000 --> 00:45:00.000
You've got to have a linear
polynomial, a s plus b
00:44:59.000 --> 00:45:05.000
in the top, see?
00:45:02.000 --> 00:45:08.000
So, you must have a s plus b
here,
00:45:04.000 --> 00:45:10.000
as I'm sure you remember if
that's the way you learned to do
00:45:08.000 --> 00:45:14.000
it. But, to do cover-up,
00:45:09.000 --> 00:45:15.000
the best way as much as
possible to separate out the
00:45:12.000 --> 00:45:18.000
terms.
If this were a cubic term,
00:45:14.000 --> 00:45:20.000
God forbid, s plus one cubed,
00:45:16.000 --> 00:45:22.000
then you'd have to have s plus
one cubed,
00:45:20.000 --> 00:45:26.000
s plus one squared.
00:45:23.000 --> 00:45:29.000
Okay, I won't give you anything
bigger than quadratic.
00:45:26.000 --> 00:45:32.000
[LAUGHTER]
You can trust me.
00:45:29.000 --> 00:45:35.000
Okay, now, what can we find by
the cover up method?
00:45:33.000 --> 00:45:39.000
Well, surely this.
Cover up the s minus one,
00:45:37.000 --> 00:45:43.000
put s equals one,
and I get three divided by two
00:45:42.000 --> 00:45:48.000
squared, four.
So, this is three quarters.
00:45:45.000 --> 00:45:51.000
Now, in this,
00:45:48.000 --> 00:45:54.000
you can always find the highest
power by cover-up because,
00:45:52.000 --> 00:45:58.000
cover it up,
put s equals negative one,
00:45:56.000 --> 00:46:02.000
and you get one minus one
plus one.
00:46:02.000 --> 00:46:08.000
So, one up there,
negative one here makes
00:46:04.000 --> 00:46:10.000
negative two here.
So, one over negative two.
00:46:07.000 --> 00:46:13.000
So, it's minus one half.
00:46:09.000 --> 00:46:15.000
Now, this you cannot determine
00:46:12.000 --> 00:46:18.000
by cover-up because you'd want
to cover-up just one of these s
00:46:16.000 --> 00:46:22.000
plus ones.
But then you can't put s equals
00:46:19.000 --> 00:46:25.000
negative one because
you get infinity.
00:46:22.000 --> 00:46:28.000
You get zero there,
makes infinity.
00:46:24.000 --> 00:46:30.000
So, this must be determined
some other way,
00:46:27.000 --> 00:46:33.000
either by undetermined
coefficients,
00:46:29.000 --> 00:46:35.000
or if you've just got one
thing, for heaven's sake,
00:46:32.000 --> 00:46:38.000
just make a substitution.
See, this is supposed to be
00:46:37.000 --> 00:46:43.000
true.
This is an algebraic identity,
00:46:40.000 --> 00:46:46.000
true for all values of the
variable, and therefore,
00:46:43.000 --> 00:46:49.000
it ought to be true when s
equals zero,
00:46:47.000 --> 00:46:53.000
for instance.
Why zero?
00:46:48.000 --> 00:46:54.000
Well, because I haven't used it
yet.
00:46:51.000 --> 00:46:57.000
I used negative one and
positive one,
00:46:53.000 --> 00:46:59.000
but I didn't use zero.
Okay, let's use zero.
00:46:56.000 --> 00:47:02.000
Put s equals zero.
What do we get?
00:47:00.000 --> 00:47:06.000
Well, on the left-hand side,
I get one divided by one
00:47:03.000 --> 00:47:09.000
squared, negative.
So, I get minus one on the left
00:47:06.000 --> 00:47:12.000
hand side equals,
what do I get on the right?
00:47:09.000 --> 00:47:15.000
Put s equals zero,
you get negative one half.
00:47:12.000 --> 00:47:18.000
Well, this is the number I'm
00:47:15.000 --> 00:47:21.000
trying to find.
So, let's write that simply as
00:47:18.000 --> 00:47:24.000
plus c, putting s equals zero.
s equals zero here gives me
00:47:21.000 --> 00:47:27.000
negative three quarters.
00:47:23.000 --> 00:47:29.000
Okay, what's c?
This is minus a half,
00:47:26.000 --> 00:47:32.000
minus three quarters,
is minus five quarters.
00:47:29.000 --> 00:47:35.000
Put it on the other side,
00:47:32.000 --> 00:47:38.000
minus one plus five quarters is
plus one quarter.
00:47:35.000 --> 00:47:41.000
So, c equals one quarter.
00:47:39.000 --> 00:47:45.000
And now, we are ready to do the
00:47:42.000 --> 00:47:48.000
final step.
Take the inverse Laplace
00:47:44.000 --> 00:47:50.000
transform.
You see what I said when I said
00:47:47.000 --> 00:47:53.000
that all the work is in this
last step?
00:47:49.000 --> 00:47:55.000
Just look how much of the work,
how much of the board is
00:47:53.000 --> 00:47:59.000
devoted to the first two steps,
and how much is going to be
00:47:57.000 --> 00:48:03.000
devoted to the last step?
Okay, so we get e to the
00:48:01.000 --> 00:48:07.000
inverse Laplace transform.
Well, the first term is the
00:48:05.000 --> 00:48:11.000
hardest.
Let's let that go for the
00:48:08.000 --> 00:48:14.000
moment.
So, I leave a space for it,
00:48:10.000 --> 00:48:16.000
and then we will have one
quarter.
00:48:13.000 --> 00:48:19.000
Well, one over s plus one is,
00:48:16.000 --> 00:48:22.000
that's just the exponential
formula.
00:48:19.000 --> 00:48:25.000
One over s plus one would be e
to the negative t,e to the minus
00:48:24.000 --> 00:48:30.000
one times t.
So, it's one quarter e to the
00:48:28.000 --> 00:48:34.000
minus one times t.
00:48:32.000 --> 00:48:38.000
And, how about the next thing
would be three quarters times,
00:48:36.000 --> 00:48:42.000
well, here it's negative one,
so that's e to the plus t.
00:48:41.000 --> 00:48:47.000
Notice how those signs work.
00:48:44.000 --> 00:48:50.000
And, that just leaves us the
Laplace transform of this thing.
00:48:49.000 --> 00:48:55.000
Now, you look at it and you
say, this Laplace transform
00:48:54.000 --> 00:49:00.000
happened in two steps.
I took something and I got,
00:48:58.000 --> 00:49:04.000
essentially,
one over s squared.
00:49:03.000 --> 00:49:09.000
And then, I changed s to s plus
one.
00:49:08.000 --> 00:49:14.000
All right, what gives one over
s squared?
00:49:13.000 --> 00:49:19.000
The Laplace transform of what
is one over s squared?
00:49:18.000 --> 00:49:24.000
t, you say to yourself,
one over s to some power is
00:49:23.000 --> 00:49:29.000
essentially some power of t.
And then, you look at the
00:49:28.000 --> 00:49:34.000
formula.
Notice at the top is one
00:49:31.000 --> 00:49:37.000
factorial, which is one,
of course.
00:49:34.000 --> 00:49:40.000
Okay, now, then how do I
convert this to one over s plus
00:49:39.000 --> 00:49:45.000
one squared?
That's the exponential shift
00:49:44.000 --> 00:49:50.000
formula.
If you know what the Laplace
00:49:47.000 --> 00:49:53.000
transform, so the first formula
in the middle of the board on
00:49:51.000 --> 00:49:57.000
the top, there,
if you know what,
00:49:54.000 --> 00:50:00.000
change s to s plus one,
corresponds to
00:49:58.000 --> 00:50:04.000
multiplying by e to the t.
00:50:03.000 --> 00:50:09.000
So, it is t times e to the
negative t.
00:50:06.000 --> 00:50:12.000
Sorry, that corresponds to
this.
00:50:08.000 --> 00:50:14.000
So, this is the exponential
shift formula.
00:50:11.000 --> 00:50:17.000
If t goes to one over s squared,
then t e to the
00:50:15.000 --> 00:50:21.000
minus t goes to one
over s plus one squared.
00:50:20.000 --> 00:50:26.000
Okay, but there's a constant
00:50:22.000 --> 00:50:28.000
out front.
So, it's minus one half t e to
00:50:25.000 --> 00:50:31.000
the negative t.
00:50:27.000 --> 00:50:33.000
Now, tell me,
what parts of this solution,
00:50:30.000 --> 00:50:36.000
oh boy, we're over time.
But, notice,
00:50:32.000 --> 00:50:38.000
this is what would have been
the particular solution,
00:50:36.000 --> 00:50:42.000
(y)p before,
and this is the stuff that
00:50:39.000 --> 00:50:45.000
occurs in the complementary
function, but already the
00:50:42.000 --> 00:50:48.000
appropriate constants have been
supplied for the coefficients.
00:50:49.000 --> 00:50:55.000
You don't have to calculate
them separately.
00:50:52.000 --> 00:50:58.000
They were built into the
method.
00:50:54.000 --> 00:51:00.000
Okay, good luck on Friday,
and see you there.