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PROFESSOR: Well,
OK, Professor Frey
00:00:24.580 --> 00:00:27.840
invited me to give the
two lectures this week
00:00:27.840 --> 00:00:35.610
on first order equations, like
that one, first order dy dt.
00:00:35.610 --> 00:00:39.560
And the lectures next week will
be on second order equation.
00:00:39.560 --> 00:00:45.380
So we're looking for, you could
say, formulas for the solution.
00:00:45.380 --> 00:00:48.460
We'll get as far as we
can with formulas, then
00:00:48.460 --> 00:00:49.500
numerical methods.
00:00:49.500 --> 00:00:54.520
Graphical methods take over
in more complicated problems.
00:00:54.520 --> 00:00:57.560
This is a model problem.
00:00:57.560 --> 00:00:58.140
It's linear.
00:01:01.320 --> 00:01:05.010
I chose it to have
constant coefficient a,
00:01:05.010 --> 00:01:08.760
and let me check the units.
00:01:08.760 --> 00:01:11.990
Always good to see the
units in a problem.
00:01:11.990 --> 00:01:17.780
So let me think of this
y, as the money in a bank,
00:01:17.780 --> 00:01:26.930
or bank balance, so y as in
dollars, and t, time, in years.
00:01:26.930 --> 00:01:35.070
So we're looking at the ups
and downs of bank balance y.
00:01:35.070 --> 00:01:43.300
The rate of change, so the
units then are dollars per year.
00:01:43.300 --> 00:01:47.010
So every term in the equation
has to have the right units.
00:01:47.010 --> 00:01:51.600
So y is in dollars,
so the interest rate a
00:01:51.600 --> 00:01:56.750
is percent per
year, say 6% a year.
00:01:56.750 --> 00:02:02.400
So a could be 6%-- that's
dimensionless-- per year,
00:02:02.400 --> 00:02:10.430
or half a percent per
month if we change.
00:02:10.430 --> 00:02:12.830
So if we change
units, the constant a
00:02:12.830 --> 00:02:14.580
would change from 6 to a half.
00:02:14.580 --> 00:02:17.170
But let's stay with 6.
00:02:17.170 --> 00:02:24.240
And then q of t represents
deposits and withdrawals,
00:02:24.240 --> 00:02:27.760
so that's in dollars
per year again.
00:02:27.760 --> 00:02:28.890
Has to be.
00:02:28.890 --> 00:02:34.050
So that's continuous.
00:02:34.050 --> 00:02:38.330
We think of the deposits
and the interest
00:02:38.330 --> 00:02:44.840
as being computed continuously
as time goes forward.
00:02:44.840 --> 00:02:48.190
So if that's a constant-- and
I'll take that case first,
00:02:48.190 --> 00:02:51.700
q equal 1-- that would
mean that we're putting in,
00:02:51.700 --> 00:02:57.330
depositing $1 per year,
continuously through the year.
00:02:57.330 --> 00:03:00.750
So that's the model that comes
from a differential equation.
00:03:00.750 --> 00:03:05.980
A difference equation would
give us finite time steps.
00:03:05.980 --> 00:03:10.050
So I'm looking for the solution.
00:03:10.050 --> 00:03:15.210
And with constant
coefficients, linear, we're
00:03:15.210 --> 00:03:18.610
going to get a formula
for the solution.
00:03:18.610 --> 00:03:21.750
I could actually deal
with variable interest
00:03:21.750 --> 00:03:26.120
rate for this one
first order equation,
00:03:26.120 --> 00:03:27.750
but the formula becomes messy.
00:03:27.750 --> 00:03:29.760
But you can still do it.
00:03:29.760 --> 00:03:32.210
After that point, for a
second order equations
00:03:32.210 --> 00:03:36.210
like oscillation,
or for a system
00:03:36.210 --> 00:03:40.480
of several equations
coupled together,
00:03:40.480 --> 00:03:44.690
constant coefficients is
where you can get formulas.
00:03:44.690 --> 00:03:47.320
So let's go with that case.
00:03:47.320 --> 00:03:49.400
So how to solve that equation?
00:03:52.030 --> 00:03:57.810
Let me take first of all, a
constant, constant source.
00:03:57.810 --> 00:04:01.260
So I think of q as
the source term.
00:04:01.260 --> 00:04:08.080
To get one nice formula, let me
take this example, ay plus 1,
00:04:08.080 --> 00:04:09.361
let's say.
00:04:09.361 --> 00:04:12.240
How do you find y
of t to solve that?
00:04:12.240 --> 00:04:15.960
And you start with some
initial condition y of 0.
00:04:15.960 --> 00:04:21.130
That's the opening deposit
that you make at time 0.
00:04:21.130 --> 00:04:23.410
How to solve that equation?
00:04:23.410 --> 00:04:32.380
Well, we're looking
for a solution.
00:04:32.380 --> 00:04:36.240
And solutions to linear
equations have two parts.
00:04:36.240 --> 00:04:40.290
So the same will happen
in linear algebra.
00:04:40.290 --> 00:04:43.660
One part is a solution
to that equation,
00:04:43.660 --> 00:04:47.550
so we're just looking
for one, any one,
00:04:47.550 --> 00:04:49.940
and we'll call it a
particular solution.
00:04:49.940 --> 00:04:56.350
And the associated null
equation, dy dt equal ay.
00:04:59.240 --> 00:05:04.350
So this is an equation
with q equals 0.
00:05:04.350 --> 00:05:07.260
That's why it's called null.
00:05:07.260 --> 00:05:10.630
And it's also
called homogeneous.
00:05:10.630 --> 00:05:14.390
So more textbooks use that
long word homogeneous,
00:05:14.390 --> 00:05:18.030
but I use the word null
because it's shorter
00:05:18.030 --> 00:05:23.410
and because it's the same
word in linear algebra.
00:05:26.970 --> 00:05:30.620
So let me call yn
the null solution,
00:05:30.620 --> 00:05:32.530
the general null solution.
00:05:32.530 --> 00:05:37.270
And y, I'm looking here for
a particular solution yp,
00:05:37.270 --> 00:05:41.710
and I'm going to-- here's
the key for linear equations.
00:05:41.710 --> 00:05:47.665
Let me take that off and
focus on those two equations.
00:05:50.300 --> 00:05:54.390
How does solving the null
equation, which is easy to do,
00:05:54.390 --> 00:05:56.080
help me?
00:05:56.080 --> 00:06:02.870
Why can I, as I plan
to do, add in yn to yp?
00:06:02.870 --> 00:06:04.910
I just add the two equations.
00:06:04.910 --> 00:06:08.490
Can I just add
those two equations?
00:06:08.490 --> 00:06:17.090
I get the derivative of yp
plus yn on the left side.
00:06:17.090 --> 00:06:22.110
And I have a times yp plus yn.
00:06:22.110 --> 00:06:27.190
And that is a critical moment
there when we use linearity.
00:06:27.190 --> 00:06:33.410
I had a yp a yn, and I
could put them together.
00:06:33.410 --> 00:06:38.070
If it was y squared, yp
squared and yn squared
00:06:38.070 --> 00:06:42.580
would not be the same
as yp plus yn squared.
00:06:42.580 --> 00:06:48.490
It's the linearity that
comes, and then I add the 1.
00:06:48.490 --> 00:06:51.100
So what do I see from this?
00:06:51.100 --> 00:06:56.840
I see that yp plus yn
also solves my equation.
00:06:56.840 --> 00:07:04.600
So the whole family of
solutions is 1 yp plus any yn.
00:07:04.600 --> 00:07:06.480
And why do I say any yn?
00:07:06.480 --> 00:07:11.470
Because when I find
one, I find more.
00:07:11.470 --> 00:07:14.930
The solutions to
this equation are
00:07:14.930 --> 00:07:20.940
yn could be e to the at,
because the derivative of e
00:07:20.940 --> 00:07:23.870
to the at does bring
down a factor a.
00:07:23.870 --> 00:07:29.940
But you see, I've left space
for any multiple of e to the at.
00:07:29.940 --> 00:07:33.580
This is where that long
word homogeneous comes from.
00:07:33.580 --> 00:07:37.650
It's homogeneous means I can
multiply by any constant,
00:07:37.650 --> 00:07:39.630
and I still solve the equation.
00:07:39.630 --> 00:07:43.710
And of course, the
key again is linear.
00:07:43.710 --> 00:07:49.420
So now I have-- well, you could
say I've done half the job.
00:07:49.420 --> 00:07:53.210
I've found yn, the general yn.
00:07:53.210 --> 00:07:57.410
And now I just have
to find one yp,
00:07:57.410 --> 00:08:01.150
one solution to the equation.
00:08:01.150 --> 00:08:08.100
And with this source
term, a constant,
00:08:08.100 --> 00:08:11.690
there's a nice way to
find that solution.
00:08:11.690 --> 00:08:14.210
Look for a constant solution.
00:08:14.210 --> 00:08:16.970
So certain right
hand sides, and those
00:08:16.970 --> 00:08:19.450
are the like the
special functions
00:08:19.450 --> 00:08:23.080
for the special source terms
for differential equation,
00:08:23.080 --> 00:08:24.780
certain right hand
sides-- and I'm just
00:08:24.780 --> 00:08:27.750
going to go down a
list of them today.
00:08:27.750 --> 00:08:29.610
The next one on the
list-- can I tell you
00:08:29.610 --> 00:08:32.320
what the next one
on the list will be?
00:08:32.320 --> 00:08:35.659
y prime equal ay.
00:08:35.659 --> 00:08:41.070
I use prime for--
well, I'll write dy dt,
00:08:41.070 --> 00:08:43.010
but often I'll write y prime.
00:08:43.010 --> 00:08:46.230
dy dt equal ay plus
an exponential.
00:08:49.200 --> 00:08:51.440
That'll be number two.
00:08:51.440 --> 00:08:55.290
So I'm just preparing
the way for number two.
00:08:55.290 --> 00:08:59.240
Well, actually number
one, this example
00:08:59.240 --> 00:09:02.580
is the same as that exponential
example with exponent
00:09:02.580 --> 00:09:06.230
s equal 0, right?
00:09:06.230 --> 00:09:09.320
If s is 0, then I
have a constant.
00:09:09.320 --> 00:09:11.840
So this is a special
case of that one.
00:09:11.840 --> 00:09:15.080
This is the most
important source term
00:09:15.080 --> 00:09:18.450
in the whole subject.
00:09:18.450 --> 00:09:23.810
But here we go
with a constant 1.
00:09:23.810 --> 00:09:25.480
So we've got yn.
00:09:25.480 --> 00:09:26.380
And what's yp?
00:09:31.870 --> 00:09:33.550
I just looked to see.
00:09:33.550 --> 00:09:35.500
Can I think of one?
00:09:35.500 --> 00:09:38.630
And with these
special functions,
00:09:38.630 --> 00:09:43.200
you can often find a
solution of the same form
00:09:43.200 --> 00:09:45.810
as the source term.
00:09:45.810 --> 00:09:48.830
And in this case,
that means a constant.
00:09:48.830 --> 00:09:52.120
So if yp is a constant,
this will be 0.
00:09:52.120 --> 00:09:54.100
So I just want to
pick the constant that
00:09:54.100 --> 00:09:55.770
makes this thing 0.
00:09:55.770 --> 00:10:03.160
And of course, their right hand
side is 0 when yp is minus 1
00:10:03.160 --> 00:10:04.780
over a.
00:10:04.780 --> 00:10:06.920
So I've got it.
00:10:06.920 --> 00:10:09.590
We've solved that
equation, except we
00:10:09.590 --> 00:10:12.080
didn't match the
initial condition yet.
00:10:12.080 --> 00:10:14.570
Let me if you take
that final step.
00:10:14.570 --> 00:10:22.220
So the general y is any
multiple, any null solution,
00:10:22.220 --> 00:10:27.260
plus any one particular
solution, that one.
00:10:27.260 --> 00:10:33.650
And we want to match it
to y of 0 at t equals 0.
00:10:33.650 --> 00:10:36.600
So I want to take that solution.
00:10:36.600 --> 00:10:38.530
I want to find that
constant, here.
00:10:38.530 --> 00:10:42.190
That's the only remaining
step is find that constant.
00:10:42.190 --> 00:10:44.430
You've done it in the homework.
00:10:44.430 --> 00:10:51.600
So at t equals 0,
y of 0 is-- at t
00:10:51.600 --> 00:10:55.880
equals 0, this is C. This
is the minus 1 over a.
00:10:55.880 --> 00:10:59.210
So I learn what the C has to be.
00:10:59.210 --> 00:11:00.870
And that's the final step.
00:11:00.870 --> 00:11:04.490
C is bring the 1 over
a onto that side,
00:11:04.490 --> 00:11:18.130
so C will be y of 0 e to the
at minus 1 over a e to at.
00:11:18.130 --> 00:11:22.310
And here we had
a minus 1 over a.
00:11:22.310 --> 00:11:28.590
Well, it'll be plus
1 over a e to the at.
00:11:28.590 --> 00:11:35.190
So now I've just put in the C,
y of 0 plus 1 over a. y of 0
00:11:35.190 --> 00:11:37.880
plus 1 over a has gone
in for C. And now I
00:11:37.880 --> 00:11:40.140
have to subtract this 1 over a.
00:11:40.140 --> 00:11:43.670
Here, I see a 1 over a,
so I can do it neatly.
00:11:48.990 --> 00:11:50.310
Got a solution.
00:11:50.310 --> 00:11:51.790
We can check it, of course.
00:11:51.790 --> 00:11:58.900
At t equals 0, this
disappears, and this is y of 0.
00:11:58.900 --> 00:12:00.380
And it has the form.
00:12:00.380 --> 00:12:06.030
It's a multiple of e to the
at and a particular solution.
00:12:06.030 --> 00:12:06.960
So that's a good one.
00:12:11.190 --> 00:12:16.810
Notice that to get the
initial condition right,
00:12:16.810 --> 00:12:20.440
I couldn't take C to be y of
0 to get the initial condition
00:12:20.440 --> 00:12:21.240
right.
00:12:21.240 --> 00:12:22.750
To get the initial
condition right,
00:12:22.750 --> 00:12:28.580
I had to get that, this
minus 1 over a in there.
00:12:28.580 --> 00:12:29.400
Good for that one?
00:12:32.160 --> 00:12:36.230
Let me move to the
next one, exponentials.
00:12:40.220 --> 00:12:48.220
So again, we know that the
null equation with no source
00:12:48.220 --> 00:12:52.240
has this solution e to the at.
00:12:52.240 --> 00:12:57.410
And I'm going to suppose
that the a in e to the
00:12:57.410 --> 00:13:02.794
at in the null solution
is different from the s
00:13:02.794 --> 00:13:07.070
in the source function,
which will come up
00:13:07.070 --> 00:13:09.200
in the particular solution.
00:13:09.200 --> 00:13:11.910
So you're going to
see either the st
00:13:11.910 --> 00:13:14.530
in the particular
solution and an e to the
00:13:14.530 --> 00:13:17.230
at in the null solution.
00:13:17.230 --> 00:13:24.260
And in the case when s equals
a, that's called resonance,
00:13:24.260 --> 00:13:28.720
the two exponents are the
same, and the formula changes
00:13:28.720 --> 00:13:30.700
a little.
00:13:30.700 --> 00:13:33.850
Let's leave that case for later.
00:13:33.850 --> 00:13:36.590
How do I solve this?
00:13:36.590 --> 00:13:38.690
I'm looking for a
particular solution
00:13:38.690 --> 00:13:42.140
because I know the
null solutions.
00:13:42.140 --> 00:13:44.690
How am I going to get
a particular solution
00:13:44.690 --> 00:13:46.170
of this equation?
00:13:46.170 --> 00:13:50.970
Fundamental observation,
the key point
00:13:50.970 --> 00:13:56.140
is it's going to be a
multiple of e to the st.
00:13:56.140 --> 00:14:02.470
If an exponential goes in, then
that will be an exponential.
00:14:02.470 --> 00:14:04.600
Its derivative will
be an exponential.
00:14:04.600 --> 00:14:08.920
I'll have e to the
st's everywhere.
00:14:08.920 --> 00:14:12.910
And I can get the number right.
00:14:12.910 --> 00:14:15.950
So I'm looking for y try.
00:14:15.950 --> 00:14:24.180
So I'll put try, knowing it's
going to work, as some number
00:14:24.180 --> 00:14:31.310
times e to the st.
So this would be
00:14:31.310 --> 00:14:33.580
like the exponential response.
00:14:33.580 --> 00:14:36.370
Response, do you know
that word response?
00:14:36.370 --> 00:14:39.820
So response is the solution.
00:14:39.820 --> 00:14:44.510
The input is q, and
the response is Y.
00:14:44.510 --> 00:14:47.036
And here, the input
is e to the st,
00:14:47.036 --> 00:14:53.550
and the response is a multiple
of e to the st. So plug it in.
00:14:53.550 --> 00:14:58.940
The timed derivative
will be Y. Taking
00:14:58.940 --> 00:15:04.000
the derivative will bring down
a 1. e to the st equals aY.
00:15:04.000 --> 00:15:12.090
A aY e to the st plus 1 e to
the st. Just what we hoped.
00:15:15.350 --> 00:15:18.980
The beauty of
exponentials is that when
00:15:18.980 --> 00:15:21.710
you take their derivatives,
you just have more exponential.
00:15:21.710 --> 00:15:23.770
That's the key thing.
00:15:23.770 --> 00:15:26.580
That's why exponential is
the most important function
00:15:26.580 --> 00:15:30.610
in this course, absolutely
the most important function.
00:15:30.610 --> 00:15:33.360
So it happened here.
00:15:33.360 --> 00:15:36.120
I can cancel e to the
st, because every term
00:15:36.120 --> 00:15:37.430
has one of them.
00:15:37.430 --> 00:15:41.070
So I'm seeing that--
what am I getting for Y?
00:15:41.070 --> 00:15:44.080
Getting a very important
number for Y. So
00:15:44.080 --> 00:15:48.050
I bring aY onto
this side with sY.
00:15:48.050 --> 00:15:49.710
On this side I just have a 1.
00:15:52.540 --> 00:15:55.560
Maybe it's worth putting
on its own board.
00:15:55.560 --> 00:16:06.080
Y is, so Ys aY comes with a
minus, and the 1, 1 over--
00:16:06.080 --> 00:16:08.760
so Y was multiplied
by s minus a.
00:16:12.130 --> 00:16:17.610
That's the right quantity to
get a particular solution.
00:16:17.610 --> 00:16:21.490
And that 1 over s
minus a, you see
00:16:21.490 --> 00:16:24.490
why I wanted s to
be different from a.
00:16:24.490 --> 00:16:28.630
I If s equaled a in that
case, in that possibility
00:16:28.630 --> 00:16:32.620
of resonance when the two
exponents are the same,
00:16:32.620 --> 00:16:35.630
we would have 1 over 0, and we'd
have to look somewhere else.
00:16:39.690 --> 00:16:43.160
The name for that-- this has to
have a name because it shows up
00:16:43.160 --> 00:16:43.890
all the time.
00:16:43.890 --> 00:16:46.220
The exponential
response function,
00:16:46.220 --> 00:16:47.660
you could call it that.
00:16:47.660 --> 00:16:49.995
Most people would call
it the transfer function.
00:16:57.210 --> 00:17:00.130
So any constant coefficient
linear equation's
00:17:00.130 --> 00:17:03.610
going to have a transfer
function, easy to find.
00:17:03.610 --> 00:17:06.520
Everything easy, that's
what I'm emphasizing, here.
00:17:06.520 --> 00:17:08.960
Everything's straightforward.
00:17:08.960 --> 00:17:12.540
That transfer function
tells you what
00:17:12.540 --> 00:17:14.780
multiplies the exponential.
00:17:14.780 --> 00:17:18.200
So the source was here.
00:17:20.720 --> 00:17:28.640
And the response is here, the
response factor, you could say,
00:17:28.640 --> 00:17:29.800
the transfer function.
00:17:29.800 --> 00:17:32.510
Multiply by 1 over s minus a.
00:17:32.510 --> 00:17:37.600
So if s is close to a,
if the input is almost
00:17:37.600 --> 00:17:45.960
at the same exponent as the
natural, as the null solution,
00:17:45.960 --> 00:17:50.290
then we're going to
get a big response.
00:17:50.290 --> 00:17:52.180
So that's good.
00:17:52.180 --> 00:17:55.490
For a constant coefficient
problem second order,
00:17:55.490 --> 00:17:58.560
other problems we can find
that response function.
00:17:58.560 --> 00:18:00.520
It's the key function.
00:18:00.520 --> 00:18:05.640
It's the function
if we have, or if we
00:18:05.640 --> 00:18:09.580
were to look at Laplace
transforms, that
00:18:09.580 --> 00:18:11.540
would be the key.
00:18:11.540 --> 00:18:13.690
When you take
Laplace transforms,
00:18:13.690 --> 00:18:16.800
the transfer function shows up.
00:18:16.800 --> 00:18:19.150
Then when you take inverse
Laplace transforms,
00:18:19.150 --> 00:18:23.245
you have to find what function
has that Laplace transform.
00:18:26.670 --> 00:18:30.660
So did we get the-- we
got the final answer then.
00:18:30.660 --> 00:18:36.980
Let me put it here. y is e
to the st times this factor.
00:18:36.980 --> 00:18:39.255
So I divide by s minus a.
00:18:39.255 --> 00:18:41.210
A nice solution.
00:18:46.160 --> 00:18:49.220
Let me also anticipate
something more.
00:18:51.960 --> 00:18:59.510
An important case for e to
the st is e to the i omega t.
00:18:59.510 --> 00:19:03.580
e to the st, we think about
as exponential growth,
00:19:03.580 --> 00:19:05.380
exponential decay.
00:19:05.380 --> 00:19:10.200
But that's for positive
s and negative s.
00:19:10.200 --> 00:19:16.570
And all important in
applications is oscillation.
00:19:16.570 --> 00:19:25.740
So coming, let me say,
coming is either late today
00:19:25.740 --> 00:19:31.980
or early Wednesday will
be s equal i omega,
00:19:31.980 --> 00:19:41.610
so where the source term
is e to the i omega t.
00:19:41.610 --> 00:19:45.960
And alternating, so this
is electrical engineers
00:19:45.960 --> 00:19:50.950
would meet it constantly from
alternating voltage source,
00:19:50.950 --> 00:19:56.610
alternating current source,
AC, with frequency omega,
00:19:56.610 --> 00:19:58.590
60 cycles per
second, for example.
00:20:02.400 --> 00:20:04.285
Why don't I just
deal with this now?
00:20:06.900 --> 00:20:11.440
Because it involves
complex numbers.
00:20:11.440 --> 00:20:17.810
And we've got to take a little
step back and prepare for that.
00:20:17.810 --> 00:20:22.180
But when we do it,
we'll get not only e
00:20:22.180 --> 00:20:25.910
to the i omega t,
which I brought out,
00:20:25.910 --> 00:20:28.650
but also, it's real part.
00:20:28.650 --> 00:20:31.750
You remember the great
formula with complex numbers,
00:20:31.750 --> 00:20:34.340
Euler's formula,
that e to the i omega
00:20:34.340 --> 00:20:39.340
t is a combination of cosine
omega t, the real part,
00:20:39.340 --> 00:20:44.150
and then the imaginary
part is sine omega t.
00:20:44.150 --> 00:20:52.510
So this is looking
like a complex problem.
00:20:52.510 --> 00:20:58.700
But it actually solves two
real problems, cosine and sine.
00:20:58.700 --> 00:21:02.870
Cosine and sine will be on our
short list of great functions
00:21:02.870 --> 00:21:04.480
that we can deal with.
00:21:04.480 --> 00:21:10.430
But to deal with them neatly,
we need a little thought
00:21:10.430 --> 00:21:11.790
about complex numbers.
00:21:11.790 --> 00:21:15.330
So OK if I leave
e to the i omega
00:21:15.330 --> 00:21:18.490
t for the end of the list, here?
00:21:21.440 --> 00:21:24.600
So I'm ready for another
one, another source term.
00:21:24.600 --> 00:21:29.830
And I'm going to pick
the step function.
00:21:29.830 --> 00:21:41.190
So the next example is going to
be dy dt equals ay plus a step.
00:21:41.190 --> 00:21:48.680
Well, suppose I
put H of t there.
00:21:48.680 --> 00:21:50.205
Suppose I put H of t.
00:21:52.820 --> 00:21:55.240
And I ask you for the
solution to that guy.
00:21:57.820 --> 00:22:02.120
So that step function,
its graph is here.
00:22:02.120 --> 00:22:06.450
It's 0 for negative time,
and it's 1 for positive time.
00:22:09.520 --> 00:22:12.330
So we've already solved
that problem, right?
00:22:12.330 --> 00:22:16.130
Where did I solve this equation?
00:22:16.130 --> 00:22:18.740
This equation is
already on that board.
00:22:21.410 --> 00:22:22.190
Because why?
00:22:26.120 --> 00:22:30.470
Because H of t is
for t positive.
00:22:30.470 --> 00:22:32.630
That's the only
place we're looking.
00:22:32.630 --> 00:22:36.380
This whole problem, we're
not looking at negative t.
00:22:36.380 --> 00:22:39.980
We're only looking
at t from 0 forward.
00:22:39.980 --> 00:22:46.070
And what is H of
t from 0 forward?
00:22:46.070 --> 00:22:46.600
It's 1.
00:22:46.600 --> 00:22:48.120
It's a constant.
00:22:48.120 --> 00:22:55.130
So that problem, as it stands,
is identical to that problem.
00:22:55.130 --> 00:22:58.910
Same thing, we have a 1.
00:22:58.910 --> 00:23:01.020
No need to solve that again.
00:23:01.020 --> 00:23:09.970
The real example is when
this function jumps up
00:23:09.970 --> 00:23:20.422
at some later time T. Now I have
the function is H of t minus T.
00:23:20.422 --> 00:23:27.980
Do you see that, why the step
function that jumps at time T
00:23:27.980 --> 00:23:30.340
has that formula?
00:23:30.340 --> 00:23:34.140
Because for little
t before that time,
00:23:34.140 --> 00:23:39.120
in here, this is--
what's the deal?
00:23:39.120 --> 00:23:42.720
If little t is
smaller than big T,
00:23:42.720 --> 00:23:50.880
then t minus T is
negative, right?
00:23:50.880 --> 00:23:55.440
If t is in here, then
t minus capital T
00:23:55.440 --> 00:23:56.950
is going to be a
negative number.
00:23:56.950 --> 00:24:00.680
And H of a negative number is 0.
00:24:00.680 --> 00:24:04.980
But for t greater than capital
T, this is a positive number.
00:24:04.980 --> 00:24:07.460
And H of a positive number is 1.
00:24:07.460 --> 00:24:11.340
Do you see how if you
want to shift a graph,
00:24:11.340 --> 00:24:13.480
if you want the graph
to shift, if you want
00:24:13.480 --> 00:24:19.570
to move the starting time,
then algebraically, the way
00:24:19.570 --> 00:24:24.118
you do it is to change t to
t minus the starting time.
00:24:26.522 --> 00:24:27.730
And that's what I want to do.
00:24:33.420 --> 00:24:40.010
So physically, what's
happening with this equation?
00:24:40.010 --> 00:24:42.780
So it starts with
y of 0 as before.
00:24:42.780 --> 00:24:46.910
Let's think of a bank balance
and then other things, too.
00:24:46.910 --> 00:24:53.870
If it's a bank balance, we put
in a certain amount, y of 0.
00:24:53.870 --> 00:24:54.860
We hope.
00:24:54.860 --> 00:24:57.710
And that grew.
00:24:57.710 --> 00:25:00.900
And then starting
at time, capital T,
00:25:00.900 --> 00:25:04.850
this switch turns on.
00:25:04.850 --> 00:25:09.570
Actually, physically,
step function
00:25:09.570 --> 00:25:15.300
is really often describing a
switch that's turned on, now.
00:25:15.300 --> 00:25:20.190
This source term act
begins to act at that time.
00:25:20.190 --> 00:25:21.590
And it acts at 1.
00:25:24.240 --> 00:25:29.320
So at time capital T we start
putting money into our account.
00:25:29.320 --> 00:25:30.920
Or taking it out, of course.
00:25:30.920 --> 00:25:37.740
If this with a minus sign,
I'd be putting money in.
00:25:37.740 --> 00:25:40.650
Sorry, I would start with
some money in, y of 0.
00:25:44.290 --> 00:25:46.120
I would start with money in.
00:25:46.120 --> 00:25:49.080
Yeah, actually, tell
me what's the solution
00:25:49.080 --> 00:25:55.560
to this equation that
starts from y of 0?
00:25:55.560 --> 00:26:00.750
What's the solution up until
the switch is turned on?
00:26:00.750 --> 00:26:05.300
What's the solution before
this switch happens,
00:26:05.300 --> 00:26:08.640
this solution while
this is still 0?
00:26:08.640 --> 00:26:11.870
So let's put that part
of the answer down.
00:26:11.870 --> 00:26:17.275
This is for t smaller
than T. What's the answer?
00:26:22.410 --> 00:26:23.620
This is all common sense.
00:26:23.620 --> 00:26:28.010
It's coming fast, so I'm
asking these questions.
00:26:28.010 --> 00:26:32.730
And when I asked that question,
it's a sort of indication
00:26:32.730 --> 00:26:36.180
that you can really
see the answer.
00:26:36.180 --> 00:26:38.660
You don't need to go back
to the textbook for that.
00:26:38.660 --> 00:26:40.070
What have we got here?
00:26:40.070 --> 00:26:40.570
Yeah?
00:26:40.570 --> 00:26:42.486
AUDIENCE: Is it the null
solution [INAUDIBLE]?
00:26:42.486 --> 00:26:45.010
PROFESSOR: It'll be this guy.
00:26:45.010 --> 00:26:47.450
Yeah, the particular
solution will be 0.
00:26:47.450 --> 00:26:52.850
Right, the particular solution
is 0 before this is on.
00:26:52.850 --> 00:26:56.020
I'm sorry, the
null solution is 0,
00:26:56.020 --> 00:26:58.980
and the particular solution,
well, the particular solution
00:26:58.980 --> 00:27:01.640
is a guy that starts right.
00:27:01.640 --> 00:27:03.230
I don't know.
00:27:03.230 --> 00:27:07.370
Those names were not important.
00:27:07.370 --> 00:27:09.960
And then the question
is-- so it's just
00:27:09.960 --> 00:27:11.790
our initial deposit growing.
00:27:15.920 --> 00:27:20.350
Now, all I ask, what
about after time T?
00:27:20.350 --> 00:27:22.151
What about after time T?
00:27:25.540 --> 00:27:31.450
For t after time T, and
hopefully, equal time T,
00:27:31.450 --> 00:27:33.470
what do you think
y of t will be?
00:27:39.230 --> 00:27:43.120
Again, we want to
separate in our minds
00:27:43.120 --> 00:27:48.810
the stuff that's starting
from the initial condition
00:27:48.810 --> 00:27:54.410
from the stuff that's piling
up because of the source.
00:27:54.410 --> 00:27:59.500
So one part will be that guy.
00:28:02.730 --> 00:28:04.730
I haven't given the
complete answer.
00:28:04.730 --> 00:28:09.380
But this is continuing to grow.
00:28:09.380 --> 00:28:15.560
And because it's linear, we're
always using this neat fact
00:28:15.560 --> 00:28:16.920
that our equation is linear.
00:28:16.920 --> 00:28:19.530
We can watch things
separately, and then
00:28:19.530 --> 00:28:21.140
just add them together.
00:28:21.140 --> 00:28:23.770
So I plan to add
this part, which
00:28:23.770 --> 00:28:29.520
comes from initial
condition to a part
00:28:29.520 --> 00:28:35.880
that-- maybe we can guess it--
that's coming from the source.
00:28:35.880 --> 00:28:40.100
And how do we have any
chance to guess it?
00:28:40.100 --> 00:28:45.770
Only because that particular
source, once it's turned on,
00:28:45.770 --> 00:28:48.800
jumps to a constant
1, and we've solved
00:28:48.800 --> 00:28:51.710
the equation for a constant 1.
00:28:51.710 --> 00:28:52.910
Let me go back here.
00:28:56.030 --> 00:29:04.208
I think our answer
to this question--
00:29:04.208 --> 00:29:09.090
so this is like just first
practice with a step function,
00:29:09.090 --> 00:29:12.750
to get the hang of
a step function.
00:29:12.750 --> 00:29:17.100
So I'm seeing this same y of
0 e to the at in every case,
00:29:17.100 --> 00:29:20.970
because that's what happens
to the initial deposit.
00:29:20.970 --> 00:29:23.720
I'll say grow, assuming
the bank's paying
00:29:23.720 --> 00:29:25.900
a positive interest rate.
00:29:25.900 --> 00:29:29.940
And now, where did
this term comes from?
00:29:29.940 --> 00:29:31.395
What did that term represent?
00:29:33.459 --> 00:29:35.000
AUDIENCE: The money
that [INAUDIBLE].
00:29:35.000 --> 00:29:36.390
PROFESSOR: The money that, yeah?
00:29:36.390 --> 00:29:38.015
AUDIENCE: They had
each of [INAUDIBLE].
00:29:38.015 --> 00:29:40.770
PROFESSOR: The money
that came in and grew.
00:29:40.770 --> 00:29:45.090
It came in, and then
it grew by itself,
00:29:45.090 --> 00:29:47.150
grew separately from
that these guys.
00:29:47.150 --> 00:29:50.100
So the initial condition
is growing along.
00:29:50.100 --> 00:29:52.650
And the money we put
in starts growing.
00:29:52.650 --> 00:29:54.810
Now, the point is what?
00:29:54.810 --> 00:30:00.776
That over here, it's going
to look just like that.
00:30:00.776 --> 00:30:03.870
So I'm going to have a 1 over a.
00:30:03.870 --> 00:30:06.690
And I'm going to have
something like that.
00:30:06.690 --> 00:30:11.520
But can you just guess
what's going to go in there?
00:30:11.520 --> 00:30:14.920
When I write it down,
it'll make sense.
00:30:14.920 --> 00:30:20.410
So this term is representing
what we have at time little t,
00:30:20.410 --> 00:30:24.080
later on, from the
deposits we made,
00:30:24.080 --> 00:30:28.130
not the initial one, but
the source, the continuing
00:30:28.130 --> 00:30:29.020
deposits.
00:30:29.020 --> 00:30:30.560
And let me write it.
00:30:30.560 --> 00:30:39.250
It's going to be a 1 over a
e to the a something minus 1.
00:30:39.250 --> 00:30:42.500
It's going to look
just like that guy.
00:30:42.500 --> 00:30:45.910
When I say that guy, let
me point to it again--
00:30:45.910 --> 00:30:48.440
e to the at minus 1.
00:30:48.440 --> 00:30:51.810
But it's not quite
e to the at minus 1.
00:30:51.810 --> 00:30:52.520
What is it?
00:30:52.520 --> 00:30:53.770
AUDIENCE: t minus [INAUDIBLE].
00:30:53.770 --> 00:31:03.560
PROFESSOR: t minus
capital T, because it
00:31:03.560 --> 00:31:05.000
didn't start until that time.
00:31:08.900 --> 00:31:17.540
So I'm going to leave that as,
like, reasonable, sensible.
00:31:20.550 --> 00:31:24.600
Think about a step function
that's turned on a capital time
00:31:24.600 --> 00:31:27.200
T. Then it grows from that time.
00:31:27.200 --> 00:31:29.860
Of course, mentally, I
never write down a formula
00:31:29.860 --> 00:31:35.900
like that without
checking at t equal to T,
00:31:35.900 --> 00:31:39.040
because that's the one important
point, at t equal capital
00:31:39.040 --> 00:31:43.940
T. What is this at
t equal capital T?
00:31:43.940 --> 00:31:45.350
It's 0.
00:31:45.350 --> 00:31:49.680
At t equal capital T, this is
e to the 0, which is 1 minus 1
00:31:49.680 --> 00:31:51.520
altogether 0.
00:31:51.520 --> 00:31:54.000
And is that the right answer?
00:31:54.000 --> 00:31:58.020
At t equal capital T is 0,
should I have nothing here?
00:32:01.020 --> 00:32:01.520
Yes?
00:32:01.520 --> 00:32:02.020
No?
00:32:02.020 --> 00:32:04.190
Give me a head shake.
00:32:04.190 --> 00:32:07.350
Should I have nothing
at t equal capital T?
00:32:07.350 --> 00:32:08.410
I've got nothing.
00:32:08.410 --> 00:32:12.120
e to the 0 minus
1, that's nothing?
00:32:12.120 --> 00:32:14.420
Yes, yes that's the right thing.
00:32:14.420 --> 00:32:21.770
Because at capital T, the
source has just turned on,
00:32:21.770 --> 00:32:25.170
hasn't had time to
build up anything,
00:32:25.170 --> 00:32:27.080
just that was the
instant it turned on.
00:32:30.430 --> 00:32:32.900
So that's a step function.
00:32:32.900 --> 00:32:35.890
A step function is a
little bit of a stretch
00:32:35.890 --> 00:32:40.420
from an ordinary
function, but not as much
00:32:40.420 --> 00:32:44.860
of a stretch as its derivative.
00:32:44.860 --> 00:32:50.190
In a way, this is like the
highlight for today, coming up,
00:32:50.190 --> 00:32:55.300
to deal with not only a step
function, but a delta function.
00:33:01.490 --> 00:33:03.710
I guess every author
and every teacher
00:33:03.710 --> 00:33:07.840
has to think am I going
to let this delta function
00:33:07.840 --> 00:33:11.930
into my course or into the book?
00:33:11.930 --> 00:33:16.750
And my answer is yes.
00:33:16.750 --> 00:33:17.650
You have to do it.
00:33:17.650 --> 00:33:19.050
You should do it.
00:33:19.050 --> 00:33:24.800
Delta functions are--
they're not true functions.
00:33:24.800 --> 00:33:27.480
As we'll see, no
true function can
00:33:27.480 --> 00:33:29.600
do what a delta function does.
00:33:29.600 --> 00:33:35.170
But it's such an
intuitive, fantastic model
00:33:35.170 --> 00:33:39.760
of things happening over
a very, very short time.
00:33:39.760 --> 00:33:42.440
We just make that
short time into 0.
00:33:42.440 --> 00:33:45.400
So we're saying with
the delta function,
00:33:45.400 --> 00:33:53.110
we're going to say that
something can happen in 0 time.
00:33:55.620 --> 00:33:58.020
Something can happen in 0 time.
00:33:58.020 --> 00:34:03.950
It's a model of, you know,
when a bat hits a ball.
00:34:03.950 --> 00:34:06.520
There's a very short time.
00:34:06.520 --> 00:34:08.610
Or a golf club hits a golf ball.
00:34:08.610 --> 00:34:11.170
There's a very
short time interval
00:34:11.170 --> 00:34:12.194
when they're in contact.
00:34:14.969 --> 00:34:20.170
We're modeling that by 0
time, but still, the ball
00:34:20.170 --> 00:34:23.050
gets an impulse.
00:34:23.050 --> 00:34:28.940
Normally, for 0 time, if you're
doing things continuously,
00:34:28.940 --> 00:34:31.790
what you do over 0
time is no importance.
00:34:31.790 --> 00:34:34.389
But we're not doing things
continuously, at all.
00:34:34.389 --> 00:34:36.960
So here we go.
00:34:36.960 --> 00:34:40.929
You've seen this guy, I think.
00:34:40.929 --> 00:34:44.449
But if you haven't,
here's the time to see it.
00:34:44.449 --> 00:34:48.360
So the delta function
is the derivative
00:34:48.360 --> 00:34:53.219
of-- so I've written three
important functions up here.
00:34:53.219 --> 00:34:57.210
Let me start with
a continuous one.
00:34:57.210 --> 00:35:05.090
That function, the ramp
is 0, and then the ramp
00:35:05.090 --> 00:35:06.810
suddenly ramps up to t.
00:35:11.160 --> 00:35:13.070
Take its derivative.
00:35:13.070 --> 00:35:16.570
So the derivative, the
slope of the ramp function
00:35:16.570 --> 00:35:18.340
is certainly 0 there.
00:35:18.340 --> 00:35:20.220
And here, the slope is 1.
00:35:20.220 --> 00:35:23.760
So the slope jumped from 0 to 1.
00:35:23.760 --> 00:35:31.190
The slope of the ramp
function is the step function.
00:35:31.190 --> 00:35:34.640
Derivative of ramp equals step.
00:35:34.640 --> 00:35:36.340
Why don't I write
those words down?
00:35:40.350 --> 00:35:45.790
Derivative of ramp equals step.
00:35:50.040 --> 00:35:53.345
So there is already
the step function.
00:35:56.332 --> 00:36:00.240
In pure calculus,
the step function
00:36:00.240 --> 00:36:04.190
has already got a
little question mark.
00:36:04.190 --> 00:36:10.410
Because at that point, the
derivative in a calculus course
00:36:10.410 --> 00:36:13.260
doesn't exist,
strictly doesn't exist,
00:36:13.260 --> 00:36:15.500
because we get a
different answer
00:36:15.500 --> 00:36:21.540
0 on the left side from the
answer, 1 on the right side.
00:36:21.540 --> 00:36:23.670
We just go with that.
00:36:23.670 --> 00:36:25.600
I'm not going to
worry about what
00:36:25.600 --> 00:36:28.780
is its value at that point.
00:36:28.780 --> 00:36:33.570
It's 0 up for t negative,
and it's 1 for t positive.
00:36:33.570 --> 00:36:36.720
And often, I'll take it
1 for t equals 0, also.
00:36:36.720 --> 00:36:39.590
Usually, I will.
00:36:39.590 --> 00:36:43.460
That's the small problem.
00:36:43.460 --> 00:36:48.020
Now, the bigger problem is the
derivative of the-- so this
00:36:48.020 --> 00:36:52.910
is now the derivative
of the step function.
00:36:52.910 --> 00:36:55.300
So what's the derivative
of this step function?
00:36:55.300 --> 00:36:58.630
Well, the derivative along
there is certainly 0.
00:36:58.630 --> 00:37:02.050
The derivative along
here is certainly 0.
00:37:02.050 --> 00:37:07.250
But the derivative,
when that jumped,
00:37:07.250 --> 00:37:10.930
the derivative, the
slope was infinite.
00:37:10.930 --> 00:37:12.500
That line is vertical.
00:37:12.500 --> 00:37:14.090
Its slope is infinite.
00:37:14.090 --> 00:37:20.520
So at that one point, you have
an affinity, here, delta of 0.
00:37:20.520 --> 00:37:22.510
You could say delta
of 0 is infinite.
00:37:26.390 --> 00:37:29.640
But you haven't
said much, there.
00:37:29.640 --> 00:37:34.130
Infinite is too vague.
00:37:34.130 --> 00:37:37.280
Actually, I wouldn't
know if you gave
00:37:37.280 --> 00:37:39.410
me infinite or 2 times infinite.
00:37:39.410 --> 00:37:41.420
I couldn't tell the difference.
00:37:41.420 --> 00:37:48.240
So I'll put it in quotes,
because it sort of gives us
00:37:48.240 --> 00:37:48.850
comfort.
00:37:48.850 --> 00:37:51.390
But it doesn't mean much.
00:37:51.390 --> 00:37:53.060
What does mean much?
00:37:53.060 --> 00:37:54.470
Somehow that's important.
00:37:58.530 --> 00:38:00.790
Can I tell you how to
work with delta functions,
00:38:00.790 --> 00:38:02.340
how to think about
delta functions?
00:38:05.660 --> 00:38:08.590
It's the right way to
think about delta function.
00:38:08.590 --> 00:38:11.990
So here's some comment
on delta function.
00:38:18.990 --> 00:38:22.370
Giving the values of the
function, 0, and infinity,
00:38:22.370 --> 00:38:26.690
and 0, is not the best.
00:38:26.690 --> 00:38:30.050
What you can do with
a delta function
00:38:30.050 --> 00:38:33.420
is you can integrate it.
00:38:33.420 --> 00:38:38.380
You can define the
function by integrals.
00:38:38.380 --> 00:38:40.340
Integrals of things are nice.
00:38:40.340 --> 00:38:44.200
Do you think in your mind
when you take derivatives,
00:38:44.200 --> 00:38:48.450
as we did going left to right,
we were taking derivatives.
00:38:48.450 --> 00:38:51.650
The function was getting crazy.
00:38:51.650 --> 00:38:57.570
When we go right to left, take
integrals, those are smoothing.
00:38:57.570 --> 00:39:00.160
Integrals make
functions smoother.
00:39:00.160 --> 00:39:03.090
They cancel noise.
00:39:03.090 --> 00:39:05.210
They smooth the function out.
00:39:05.210 --> 00:39:09.320
So what we can do is to take the
integral of the delta function.
00:39:11.920 --> 00:39:16.900
We could take it from
any negative number
00:39:16.900 --> 00:39:19.890
to any positive number.
00:39:19.890 --> 00:39:23.390
And what answer would we get?
00:39:23.390 --> 00:39:27.800
What would be the right,
well, the one thing people
00:39:27.800 --> 00:39:29.990
know about the
delta function is--
00:39:29.990 --> 00:39:33.750
and actually, it's the
key thing-- the integral
00:39:33.750 --> 00:39:36.900
of the delta function.
00:39:36.900 --> 00:39:40.910
Again, I'm integrating
the delta function
00:39:40.910 --> 00:39:45.150
from some negative number
up to some positive number.
00:39:45.150 --> 00:39:48.930
And it doesn't matter where n
is, because the function is 0
00:39:48.930 --> 00:39:49.780
there and there.
00:39:49.780 --> 00:39:53.240
But what's the answer here?
00:39:53.240 --> 00:39:54.430
Put me out of my misery.
00:39:54.430 --> 00:39:57.440
Just tell me the number
I'm looking for, here,
00:39:57.440 --> 00:39:58.980
the integral of
the delta function.
00:39:58.980 --> 00:40:00.504
Or maybe you haven't met it.
00:40:00.504 --> 00:40:01.420
AUDIENCE: [INAUDIBLE].
00:40:01.420 --> 00:40:03.640
PROFESSOR: It's?
00:40:03.640 --> 00:40:07.510
It's the one good
number you could guess.
00:40:07.510 --> 00:40:09.160
It's 1.
00:40:09.160 --> 00:40:12.280
Now, why is it 1?
00:40:12.280 --> 00:40:16.380
Because if the delta function
is the derivative of the step
00:40:16.380 --> 00:40:23.470
function, this should be the
step function evaluated between
00:40:23.470 --> 00:40:29.000
N and P. This should be
the step function, , here,
00:40:29.000 --> 00:40:30.960
minus the step function, there
00:40:30.960 --> 00:40:34.472
And what is the step function?
00:40:34.472 --> 00:40:35.680
You have to keep it straight.
00:40:35.680 --> 00:40:37.420
Am I talking about
the delta function?
00:40:37.420 --> 00:40:42.700
No, right now, I've
integrated it to get H of t.
00:40:42.700 --> 00:40:46.410
So this is H of P at
the positive side,
00:40:46.410 --> 00:40:52.640
minus H of N. That's
what integration's about.
00:40:52.640 --> 00:40:54.960
And what do I get?
00:40:54.960 --> 00:41:00.440
1, because H of P, the
step function here, H is 1.
00:41:00.440 --> 00:41:02.670
And here, it's 0, so I get 1.
00:41:08.310 --> 00:41:11.220
Good, that's the
thing that everybody
00:41:11.220 --> 00:41:13.690
remembers about
the delta function.
00:41:13.690 --> 00:41:18.860
And now I can make sense out of
2 delta function, 2 delta of t.
00:41:18.860 --> 00:41:19.900
That could be my source.
00:41:22.580 --> 00:41:26.930
So if 2 delta of
t was my source,
00:41:26.930 --> 00:41:29.240
what's the graph
of 2 delta of t?
00:41:29.240 --> 00:41:33.820
Again, it's 0 infinite 0.
00:41:33.820 --> 00:41:38.470
You really can't tell from
the infinity what's up,
00:41:38.470 --> 00:41:43.820
but what would be the
integral of 2 delta of t,
00:41:43.820 --> 00:41:47.230
the integral of 2 delta
of t or some other?
00:41:47.230 --> 00:41:49.340
Well, let me put in the 2, here?
00:41:49.340 --> 00:41:53.290
What's the integral of 2
delta of t, would be 2H of t.
00:41:53.290 --> 00:41:53.840
Keep going.
00:41:53.840 --> 00:41:54.870
What do I get here?
00:41:54.870 --> 00:41:55.400
AUDIENCE: 2.
00:41:55.400 --> 00:41:57.880
PROFESSOR: It would
be 2 of these guys, 2
00:41:57.880 --> 00:41:59.851
of these, 2 of these, 2.
00:41:59.851 --> 00:42:00.350
All right?
00:42:03.990 --> 00:42:09.120
So we made sense out of the
strength of the impulse,
00:42:09.120 --> 00:42:13.260
how hard the bat hit the ball.
00:42:13.260 --> 00:42:16.320
But of course, we
need units in there.
00:42:16.320 --> 00:42:18.700
We have to have units.
00:42:18.700 --> 00:42:25.650
And here, the value
for that unit was 2.
00:42:25.650 --> 00:42:29.780
Now, I'm going to-- because
this is really worth
00:42:29.780 --> 00:42:32.990
doing with delta functions.
00:42:32.990 --> 00:42:37.290
I didn't ask at the start
have you seen them before.
00:42:37.290 --> 00:42:40.720
But they are worth seeing.
00:42:40.720 --> 00:42:42.470
And they just take
a little practice.
00:42:42.470 --> 00:42:45.500
But then in the
end, delta functions
00:42:45.500 --> 00:42:49.950
are way easier to work with than
some complicated function that
00:42:49.950 --> 00:42:54.880
attempts to model this.
00:42:54.880 --> 00:43:01.320
We could model that by some
Gaussian curve or something.
00:43:01.320 --> 00:43:05.330
All the integrations would
become impossible right away.
00:43:05.330 --> 00:43:12.040
We could model it by
a step function up
00:43:12.040 --> 00:43:13.890
and a step function down.
00:43:13.890 --> 00:43:17.920
Then the integrations
would be possible.
00:43:17.920 --> 00:43:22.360
But still, we have
this finite width.
00:43:22.360 --> 00:43:25.230
I could let that
width go to 0 and let
00:43:25.230 --> 00:43:27.444
the height go to infinity.
00:43:27.444 --> 00:43:28.360
And what would happen?
00:43:28.360 --> 00:43:30.590
I'd get the delta function.
00:43:30.590 --> 00:43:33.930
So that's one way to create a
delta function, if you like.
00:43:33.930 --> 00:43:36.570
If you're OK with
step functions,
00:43:36.570 --> 00:43:41.000
then one way to create delta
is to take a big step up, step
00:43:41.000 --> 00:43:45.320
down, and then let the
size of the step grow
00:43:45.320 --> 00:43:47.960
and the width of
the steps shrink.
00:43:47.960 --> 00:43:52.990
Keep the area 1, because
area is integral.
00:43:52.990 --> 00:43:56.360
So I keep this,
that little width,
00:43:56.360 --> 00:43:58.580
times this big
height equal to 1.
00:43:58.580 --> 00:44:02.110
And in the end, I get delta.
00:44:02.110 --> 00:44:08.610
Now again, my point is
that delta functions,
00:44:08.610 --> 00:44:10.190
that you really understand them.
00:44:10.190 --> 00:44:13.320
What you can legitimately do
with them is integrate them.
00:44:13.320 --> 00:44:19.090
But now in later problems,
we might have not a 1 or a 2,
00:44:19.090 --> 00:44:27.570
but a function in here, like
cosine t, or e to the t,
00:44:27.570 --> 00:44:30.380
or q of t.
00:44:30.380 --> 00:44:32.560
Can I practice with those?
00:44:32.560 --> 00:44:35.100
Can I put in a function f of t?
00:44:35.100 --> 00:44:37.640
I didn't leave enough
space to write f of t,
00:44:37.640 --> 00:44:47.090
so I'm going to put it in
here. f of t delta of t dt.
00:44:47.090 --> 00:44:50.873
And I'm going to go
for the answer, there.
00:44:57.780 --> 00:45:01.390
My question is what
does that equal?
00:45:01.390 --> 00:45:03.900
You see what the question is?
00:45:03.900 --> 00:45:08.100
I got my delta function,
which I only just met.
00:45:08.100 --> 00:45:11.970
And I'm multiplying it by
some ordinary function.
00:45:11.970 --> 00:45:15.040
f of t gives us no problems.
00:45:15.040 --> 00:45:16.250
Think of cosine t.
00:45:16.250 --> 00:45:19.140
Think of e to the t.
00:45:19.140 --> 00:45:23.697
What do you think is the
right answer for that?
00:45:23.697 --> 00:45:25.280
What do you think
is the right answer?
00:45:25.280 --> 00:45:28.350
And this tells you
what the delta function
00:45:28.350 --> 00:45:29.960
is when you see this.
00:45:35.450 --> 00:45:39.083
What do I need to know about
f of t to get an answer, here?
00:45:42.540 --> 00:45:48.030
Do I need to know what f
is at t equals minus 1?
00:45:48.030 --> 00:45:51.100
You could see from
the way my voice asked
00:45:51.100 --> 00:45:55.220
that question that
the answer is no.
00:45:55.220 --> 00:45:59.830
Why do I not care
what f is at minus 1?
00:45:59.830 --> 00:46:00.330
Yeah?
00:46:00.330 --> 00:46:01.310
AUDIENCE: Because you're
multiplying by [INAUDIBLE].
00:46:01.310 --> 00:46:02.726
PROFESSOR: Because
I'm multiplying
00:46:02.726 --> 00:46:04.940
by somebody that's 0.
00:46:04.940 --> 00:46:09.890
And similarly, at f equal minus
1/2, or at f equal plus 1/3,
00:46:09.890 --> 00:46:11.730
all those f's make
no difference,
00:46:11.730 --> 00:46:17.250
because they're
all multiplying 0.
00:46:17.250 --> 00:46:18.720
What does make a difference?
00:46:18.720 --> 00:46:24.680
What's the key information
about f that does
00:46:24.680 --> 00:46:27.220
come into the answer?
00:46:27.220 --> 00:46:28.910
f at?
00:46:28.910 --> 00:46:32.154
At just at that one point, f at?
00:46:32.154 --> 00:46:33.420
AUDIENCE: [INAUDIBLE]
00:46:33.420 --> 00:46:37.350
PROFESSOR: 0, f at
0 is the action.
00:46:37.350 --> 00:46:38.996
The impulse is happening.
00:46:38.996 --> 00:46:40.120
The bat's hitting the ball.
00:46:43.980 --> 00:46:46.700
So we're modeling
rocket launching, here.
00:46:46.700 --> 00:46:52.640
We're launching in 0 seconds
instead of a finite time.
00:46:52.640 --> 00:46:57.020
So in other words,
well, I don't know
00:46:57.020 --> 00:47:02.130
how to put this answer down
other than just to write it.
00:47:02.130 --> 00:47:04.910
I guess I'm hoping
you're with me
00:47:04.910 --> 00:47:06.930
in seeing that
what it should be.
00:47:10.760 --> 00:47:12.440
Can I just write it?
00:47:12.440 --> 00:47:15.690
All that matters
is what f is at t
00:47:15.690 --> 00:47:19.670
equals 0, because that's
where all the action is.
00:47:19.670 --> 00:47:24.170
And that f of 0,
if f of 0 was the 2
00:47:24.170 --> 00:47:28.280
that I had there a little while
ago, then the answer will be 2.
00:47:28.280 --> 00:47:34.880
If f of 0 is a 1, if the
answer is f of 0 times
00:47:34.880 --> 00:47:38.430
1-- and I won't write times 1.
00:47:38.430 --> 00:47:39.180
That's ridiculous.
00:47:43.880 --> 00:47:46.520
Now we can integrate
delta functions, not just
00:47:46.520 --> 00:47:50.390
a single integral of
delta, but integral
00:47:50.390 --> 00:47:53.800
of a function, a nice
function times delta.
00:47:53.800 --> 00:47:55.330
And we get f of 0.
00:47:55.330 --> 00:48:00.110
So can I just, while we're on
the subject of delta functions,
00:48:00.110 --> 00:48:02.020
ask you a few examples?
00:48:02.020 --> 00:48:11.635
What is the integral of
e to the t delta of t dt?
00:48:11.635 --> 00:48:12.431
AUDIENCE: It's 1.
00:48:12.431 --> 00:48:13.680
PROFESSOR: Yeah, say it again?
00:48:13.680 --> 00:48:14.430
AUDIENCE: It's 1.
00:48:14.430 --> 00:48:15.440
PROFESSOR: It's 1.
00:48:15.440 --> 00:48:16.920
It's 1, right.
00:48:16.920 --> 00:48:21.340
Because e to the t, at the only
point we care about, t equal 0
00:48:21.340 --> 00:48:23.740
is 1.
00:48:23.740 --> 00:48:26.310
And what if I change
that to sine t?
00:48:30.480 --> 00:48:34.630
Suppose I integrate
sine t times delta of t?
00:48:34.630 --> 00:48:37.580
What do I get now?
00:48:37.580 --> 00:48:39.030
I get?
00:48:39.030 --> 00:48:39.530
AUDIENCE: 0.
00:48:39.530 --> 00:48:40.550
PROFESSOR: 0, right.
00:48:40.550 --> 00:48:43.080
And actually, that's
totally reasonable.
00:48:43.080 --> 00:48:47.600
This is a function, which is
yeah, it's an odd function.
00:48:51.190 --> 00:48:57.010
Anyway, sine, if I switch t to
negative t, it goes negative.
00:48:57.010 --> 00:48:59.310
0 is the right answer.
00:48:59.310 --> 00:49:00.670
Let me ask you this one.
00:49:00.670 --> 00:49:02.730
What about delta of t squared?
00:49:07.190 --> 00:49:11.456
Because if we're up for a delta
function, we might square it.
00:49:13.980 --> 00:49:16.320
Now we've got a
high-powered function,
00:49:16.320 --> 00:49:20.480
because squaring this
crazy function delta of t
00:49:20.480 --> 00:49:23.170
gives us something truly crazy.
00:49:23.170 --> 00:49:27.604
And what answer would
you expect for that?
00:49:27.604 --> 00:49:28.463
AUDIENCE: 1.
00:49:28.463 --> 00:49:29.713
PROFESSOR: Would you expect 1?
00:49:33.790 --> 00:49:35.990
So this is like?
00:49:35.990 --> 00:49:38.350
I'm just getting
intuition working, here,
00:49:38.350 --> 00:49:40.801
for delta functions.
00:49:40.801 --> 00:49:41.550
What do you think?
00:49:41.550 --> 00:49:44.505
I'm looking at the energy
when I square something.
00:49:49.140 --> 00:49:50.747
OK, so we had a guess of 1.
00:49:50.747 --> 00:49:51.705
Is there another guess?
00:49:54.530 --> 00:49:55.030
Yeah?
00:49:55.030 --> 00:49:55.780
AUDIENCE: A third?
00:49:55.780 --> 00:49:56.562
PROFESSOR: Sorry?
00:49:56.562 --> 00:49:57.460
AUDIENCE: 1/3.
00:49:57.460 --> 00:50:01.130
PROFESSOR: 1/3, that's
our second guess.
00:50:01.130 --> 00:50:10.090
I'm open for other
guesses before I-- OK, we
00:50:10.090 --> 00:50:14.770
have a rule here for f of t.
00:50:14.770 --> 00:50:20.120
And now what is the f of t that
I'm asking about in this case?
00:50:20.120 --> 00:50:22.150
It's delta of t, right?
00:50:22.150 --> 00:50:27.510
If f of t is delta of t,
then that would match this.
00:50:27.510 --> 00:50:30.886
And therefore, the
answer should match.
00:50:30.886 --> 00:50:32.510
Do you see what I'm
shooting for, yeah?
00:50:32.510 --> 00:50:33.140
AUDIENCE: It'd be infinity?
00:50:33.140 --> 00:50:34.405
PROFESSOR: It'd be infinity.
00:50:34.405 --> 00:50:35.970
It would be infinity.
00:50:35.970 --> 00:50:44.650
That's delta of t squared
is that's an infinite energy
00:50:44.650 --> 00:50:45.150
function.
00:50:45.150 --> 00:50:47.980
You never meet it, actually.
00:50:47.980 --> 00:50:50.570
I apologize, so so
write it down there.
00:50:50.570 --> 00:50:53.600
I could erase it right
away because you basically
00:50:53.600 --> 00:50:55.000
never see it.
00:50:55.000 --> 00:50:57.260
It's infinite energy.
00:50:57.260 --> 00:50:59.220
Well, I think you'd see it.
00:50:59.220 --> 00:51:03.060
I mean, we're really going back
to the days of Norbert Wiener.
00:51:03.060 --> 00:51:05.060
When I came to the
math department,
00:51:05.060 --> 00:51:09.210
Norbert Wiener was still
here, still alive, still
00:51:09.210 --> 00:51:17.100
walking the hallway by touching
the wall and counting offices.
00:51:17.100 --> 00:51:23.310
And hard to talk to, because
he always had a lot to say.
00:51:23.310 --> 00:51:26.930
And you got kind of
allowed to listen.
00:51:26.930 --> 00:51:32.250
So anyway, Wiener
was among the first
00:51:32.250 --> 00:51:37.450
to really use delta
functions, successfully
00:51:37.450 --> 00:51:38.380
use delta functions.
00:51:38.380 --> 00:51:42.320
Anyway, this is the big one.
00:51:42.320 --> 00:51:43.178
This is the big one.
00:51:47.600 --> 00:51:49.820
Now, so what's all that about?
00:51:49.820 --> 00:51:55.820
I guess I was trying
to prepare by talking
00:51:55.820 --> 00:52:01.730
about this function
prepare for the equation
00:52:01.730 --> 00:52:04.390
when that's the source.
00:52:04.390 --> 00:52:09.532
So dy equal ay plus
a delta function.
00:52:13.900 --> 00:52:19.840
Let me bring that delta
function in at time T.
00:52:19.840 --> 00:52:22.460
So how do you interpret
that equation?
00:52:22.460 --> 00:52:24.440
So like part of this
morning's lecture
00:52:24.440 --> 00:52:30.450
is to get a first
handle on an impulse.
00:52:30.450 --> 00:52:33.430
So let me write that
word impulse, here.
00:52:36.940 --> 00:52:39.110
Where am I going to write it?
00:52:39.110 --> 00:52:45.490
So delta is an impulse.
00:52:45.490 --> 00:52:48.200
That's our ordinary
English word for something
00:52:48.200 --> 00:52:49.920
that happens fast.
00:52:49.920 --> 00:52:53.660
And y of t is the
impulse response.
00:53:05.260 --> 00:53:09.130
And this is the most important.
00:53:13.110 --> 00:53:16.020
Well, I said e to the st
was the most important.
00:53:16.020 --> 00:53:18.830
How can I have two most
important examples?
00:53:18.830 --> 00:53:21.670
Well, they're a tie, let's say.
00:53:21.670 --> 00:53:26.260
e to the st is the most
important ordinary function.
00:53:26.260 --> 00:53:30.610
It's the key to
the whole course.
00:53:30.610 --> 00:53:36.450
Delta of t, the impulse,
is the important one
00:53:36.450 --> 00:53:40.030
because if I can solve
it for a delta function,
00:53:40.030 --> 00:53:42.690
I can solve it for anything.
00:53:42.690 --> 00:53:48.080
Let's see if we can solve
it for a delta function,
00:53:48.080 --> 00:53:53.390
a delta function, an impulse
that starts at time T. Again,
00:53:53.390 --> 00:53:57.620
I'm just going to start
writing down the solution
00:53:57.620 --> 00:54:01.890
and ask for your help
what to write next.
00:54:01.890 --> 00:54:06.550
So what do you expect as a
first term in the solution?
00:54:06.550 --> 00:54:08.980
So I'm starting
again from y of 0.
00:54:13.780 --> 00:54:18.070
Let's see if we can
solve it by common sense.
00:54:22.530 --> 00:54:25.850
So how do I start
the solution to this?
00:54:28.410 --> 00:54:31.070
Everybody sees what
this equation is saying.
00:54:31.070 --> 00:54:36.260
I have an initial deposit of
y of 0 that starts growing.
00:54:36.260 --> 00:54:39.910
And then at time capital
T I make a deposit.
00:54:39.910 --> 00:54:45.980
At that moment, at that
instant, I make a deposit of 1.
00:54:45.980 --> 00:54:49.640
That's an instant deposit of 1.
00:54:49.640 --> 00:54:51.530
Which is, of course,
what I do in reality.
00:54:51.530 --> 00:54:53.250
I take $1 to the bank.
00:54:53.250 --> 00:54:54.420
They've got it now.
00:54:54.420 --> 00:54:59.290
At time T, I give them that $1,
and it starts earning interest.
00:54:59.290 --> 00:55:01.820
So what about y of t?
00:55:01.820 --> 00:55:02.570
What do you think?
00:55:02.570 --> 00:55:06.355
What's the first term
coming from y of 0?
00:55:09.750 --> 00:55:11.930
So the term coming
from y of 0 will
00:55:11.930 --> 00:55:15.160
be y of 0 to start
with, e to at.
00:55:20.060 --> 00:55:23.460
That takes care of the y of 0.
00:55:23.460 --> 00:55:24.580
Now, I need something.
00:55:27.640 --> 00:55:34.310
It's like this, plus
I need something
00:55:34.310 --> 00:55:43.010
that accounts for what
this deposit brings.
00:55:43.010 --> 00:55:47.940
So up until time
T, what do I put?
00:55:47.940 --> 00:55:54.550
So this is for t smaller
than T and t bigger than T.
00:55:54.550 --> 00:55:58.080
So what goes there?
00:55:58.080 --> 00:56:08.330
For t smaller than
T, what's the benefit
00:56:08.330 --> 00:56:11.320
from the delta function?
00:56:11.320 --> 00:56:15.530
0, didn't happen yet.
00:56:15.530 --> 00:56:18.810
For t bigger than T,
what's the benefit
00:56:18.810 --> 00:56:20.360
from the delta function?
00:56:20.360 --> 00:56:22.460
AUDIENCE: [INAUDIBLE].
00:56:22.460 --> 00:56:26.470
PROFESSOR: For t bigger
than T, well, that's right.
00:56:26.470 --> 00:56:28.170
OK, but now I've
made that deposit
00:56:28.170 --> 00:56:34.570
at time capital T.
Whatever's going
00:56:34.570 --> 00:56:40.800
there is whatever I'm
getting from that deposit.
00:56:40.800 --> 00:56:45.850
At time capital
T, I gave them $1,
00:56:45.850 --> 00:56:49.140
and they start paying
interest on it.
00:56:49.140 --> 00:56:50.748
What's going to go there?
00:56:56.120 --> 00:57:01.050
So if I gave them $1 at that
initial time, so that $1
00:57:01.050 --> 00:57:03.910
would have been part of y of 0.
00:57:03.910 --> 00:57:07.160
What did I get at a later time?
00:57:07.160 --> 00:57:09.390
e to the at.
00:57:09.390 --> 00:57:11.860
Now I'm waiting.
00:57:11.860 --> 00:57:15.740
I'm giving them the
dollar at time capital T,
00:57:15.740 --> 00:57:18.220
and it starts growing.
00:57:18.220 --> 00:57:20.930
So what do I have
at a later time,
00:57:20.930 --> 00:57:24.240
for t later than capital T?
00:57:24.240 --> 00:57:26.040
What has that $1 grown into?
00:57:28.690 --> 00:57:36.350
e to the a times the--
right, it's critical.
00:57:36.350 --> 00:57:37.760
It's the elapsed time.
00:57:37.760 --> 00:57:39.830
It's the time since the deposit.
00:57:39.830 --> 00:57:41.240
Is that right?
00:57:41.240 --> 00:57:42.414
So what do I put here?
00:57:42.414 --> 00:57:43.580
AUDIENCE: t minus capital T?
00:57:43.580 --> 00:57:45.570
PROFESSOR: t minus
capital T, good.
00:57:48.690 --> 00:57:56.780
Apologies to bug you about
this, but the only way
00:57:56.780 --> 00:58:03.310
to learn this stuff from a
lecture is to be part of it.
00:58:03.310 --> 00:58:07.920
So I constantly ask you, instead
of just writing down a formula.
00:58:07.920 --> 00:58:10.560
I think that looks good.
00:58:10.560 --> 00:58:18.630
So suddenly, what does this
amount to at t equal capital T?
00:58:18.630 --> 00:58:21.490
Maybe I should allow
t equal capital T.
00:58:21.490 --> 00:58:24.010
At t equal capital T,
what do I have here?
00:58:24.010 --> 00:58:24.840
AUDIENCE: 1.
00:58:24.840 --> 00:58:25.830
PROFESSOR: 1.
00:58:25.830 --> 00:58:27.700
That's my $1.
00:58:27.700 --> 00:58:31.160
At t equal capital
T, we've got $1.
00:58:31.160 --> 00:58:32.220
And later it's grown.
00:58:34.900 --> 00:58:37.210
So we have now solved.
00:58:37.210 --> 00:58:38.930
We have found the
impulse response.
00:58:41.490 --> 00:58:43.290
We have found the
impulse response
00:58:43.290 --> 00:58:52.820
when the impulse happened at
capital T. That was good going.
00:58:52.820 --> 00:59:02.220
Now, I've given you
my list of examples
00:59:02.220 --> 00:59:08.220
with the pause on
the sine and cosine.
00:59:08.220 --> 00:59:15.040
I pause on the sine and
cosine because one way
00:59:15.040 --> 00:59:18.690
to think about sine and cosine
is to get into complex numbers.
00:59:18.690 --> 00:59:23.618
And that's really for next time.
00:59:27.610 --> 00:59:31.170
But apart from that, we've
done all the examples,
00:59:31.170 --> 00:59:33.500
so are we ready?
00:59:33.500 --> 00:59:38.280
Oh yeah, I'm going to try for
the big thing, the big formula.
00:59:38.280 --> 00:59:41.870
So this is the key
result of section 1.4,
00:59:41.870 --> 00:59:44.470
the solution to this equation.
00:59:44.470 --> 00:59:46.555
So I'm going back to
the original equation.
00:59:51.410 --> 00:59:57.020
And just see if we can write
down a formula for the answer.
00:59:57.020 --> 00:59:59.460
So let me write
the equation again.
00:59:59.460 --> 01:00:04.280
dy dt is ay plus some source.
01:00:08.280 --> 01:00:12.106
I think we can write down
a formula that looks right.
01:00:16.210 --> 01:00:19.030
And we could then actually
plug it in and see, yeah,
01:00:19.030 --> 01:00:21.010
it is right.
01:00:21.010 --> 01:00:23.710
So what's going to
go into this formula?
01:00:23.710 --> 01:00:28.870
We got enough examples, so now
let's go for the whole thing.
01:00:28.870 --> 01:00:38.710
So y of t, first of
all, comes whatever
01:00:38.710 --> 01:00:40.562
depends on the
initial condition.
01:00:44.210 --> 01:00:48.280
So how much do we have
at a later time when
01:00:48.280 --> 01:00:52.480
our initial deposit was y of 0?
01:00:52.480 --> 01:00:56.430
So that's the one we've
seen in every example.
01:00:56.430 --> 01:01:01.560
Every one of these things
has this term growing out
01:01:01.560 --> 01:01:03.150
of y of 0.
01:01:03.150 --> 01:01:06.190
So let me put that in again.
01:01:06.190 --> 01:01:13.460
So the part that grows out of
y of 0 is y of 0 e to the at.
01:01:13.460 --> 01:01:14.380
That's OK.
01:01:17.280 --> 01:01:19.170
So that's what the initial.
01:01:19.170 --> 01:01:24.730
So our money is coming from two
sources, this initial deposit,
01:01:24.730 --> 01:01:31.800
which was easy, and this
continuous, over time deposit,
01:01:31.800 --> 01:01:33.730
q of t.
01:01:33.730 --> 01:01:37.510
And I have to ask
you about that.
01:01:37.510 --> 01:01:41.420
That's going to be like
the particular solution,
01:01:41.420 --> 01:01:45.470
the particular solution that
comes from the source term.
01:01:45.470 --> 01:01:50.370
This is the solution it comes
from the initial condition.
01:01:50.370 --> 01:01:52.420
So what do you think
this thing looks like?
01:01:52.420 --> 01:01:57.470
I just think once we see
it, we can say, yeah,
01:01:57.470 --> 01:01:58.290
that makes sense.
01:02:01.880 --> 01:02:05.070
So now I'm saying what?
01:02:05.070 --> 01:02:09.780
If we've deposited q of
t in varying amounts,
01:02:09.780 --> 01:02:12.950
maybe a constant for a while,
maybe a ramp for awhile,
01:02:12.950 --> 01:02:23.310
maybe whatever, a step, how am
I going to think about this?
01:02:23.310 --> 01:02:27.430
So at every time t
equal to s, so I'm
01:02:27.430 --> 01:02:32.490
using little t for
the time I've reached.
01:02:32.490 --> 01:02:33.660
Right?
01:02:33.660 --> 01:02:37.120
Here's t starting at 0.
01:02:37.120 --> 01:02:42.930
Now, let me use s for
a time part way along.
01:02:42.930 --> 01:02:47.000
So part way along, I input.
01:02:47.000 --> 01:02:50.565
I deposit q of s.
01:02:50.565 --> 01:02:54.370
I deposit it at time s.
01:02:54.370 --> 01:02:56.380
And then what does it do?
01:02:56.380 --> 01:02:59.200
That money is in the
bank with everybody else.
01:02:59.200 --> 01:03:02.170
It grows along with
everything else.
01:03:02.170 --> 01:03:05.290
So what's the growth factor?
01:03:05.290 --> 01:03:07.190
What's the growth factor?
01:03:07.190 --> 01:03:11.250
This is the amount I
deposited at time s.
01:03:11.250 --> 01:03:15.510
And how much has
it grown at time t?
01:03:15.510 --> 01:03:20.890
This is the key question,
and you can answer it.
01:03:20.890 --> 01:03:22.380
It went in a time s.
01:03:22.380 --> 01:03:23.460
I'm looking at time t.
01:03:23.460 --> 01:03:24.210
What's the factor?
01:03:24.210 --> 01:03:26.540
AUDIENCE: Is it e
to the a t minus s.
01:03:26.540 --> 01:03:28.670
PROFESSOR: e to the a t minus s.
01:03:39.090 --> 01:03:45.730
So that's the contribution
to our balance at time t
01:03:45.730 --> 01:03:48.370
from our input at time s.
01:03:48.370 --> 01:03:51.800
But now, I've been
inputting all the way along.
01:03:51.800 --> 01:03:55.380
s is running all the
way from here to here.
01:03:55.380 --> 01:03:57.660
So finish my formula.
01:03:57.660 --> 01:04:00.082
Put me out of my misery.
01:04:00.082 --> 01:04:01.690
Or it's not misery, actually.
01:04:01.690 --> 01:04:05.020
Its success at this moment.
01:04:05.020 --> 01:04:08.470
What do I do now?
01:04:08.470 --> 01:04:09.095
I?
01:04:09.095 --> 01:04:09.928
AUDIENCE: Integrate.
01:04:09.928 --> 01:04:11.360
PROFESSOR: I integrate, exactly.
01:04:11.360 --> 01:04:12.720
I integrate.
01:04:12.720 --> 01:04:13.970
I integrate.
01:04:13.970 --> 01:04:16.600
So all these deposits went in.
01:04:16.600 --> 01:04:19.670
They grew that amount
in the remaining time.
01:04:19.670 --> 01:04:26.680
And I integrate from 0
up to the current time t.
01:04:26.680 --> 01:04:29.340
So you see that formula?
01:04:29.340 --> 01:04:30.260
Have a look at it.
01:04:34.490 --> 01:04:37.800
This is a general formula, and
every one of those examples
01:04:37.800 --> 01:04:39.830
could be found
from that formula.
01:04:43.180 --> 01:04:47.690
If q of s was 1, that was
our very first example.
01:04:47.690 --> 01:04:50.270
We could do that integration.
01:04:50.270 --> 01:04:57.170
If q of s was e to the--
anyway, we could do every one.
01:04:57.170 --> 01:05:02.350
I just want you to see that
that formula makes sense.
01:05:02.350 --> 01:05:06.520
Again, this is what grew out
of the initial condition.
01:05:06.520 --> 01:05:10.700
This is what grew out of
the deposit at time s.
01:05:10.700 --> 01:05:13.510
And the whole point of
calculus, the whole point
01:05:13.510 --> 01:05:17.880
of learning [? 1801 ?],
the integral equation
01:05:17.880 --> 01:05:23.720
part, the integrals part,
is integrals just add up.
01:05:23.720 --> 01:05:28.950
This term just adds up
all the later deposits,
01:05:28.950 --> 01:05:34.280
times the growth factor
in the remaining time.
01:05:34.280 --> 01:05:40.720
And as I say, if I took q of s
equal 1-- the examples I gave
01:05:40.720 --> 01:05:43.570
are really the examples where
you can do the integral.
01:05:46.240 --> 01:05:50.830
If q of s is e to the i omega
s, I can do that integral.
01:05:50.830 --> 01:05:56.390
Actually, it's not hard to do
because e to the at doesn't
01:05:56.390 --> 01:05:57.210
depend on s.
01:05:57.210 --> 01:06:01.980
I can bring an e to the
at out in this case.
01:06:01.980 --> 01:06:03.870
That formula is just
worth thinking about.
01:06:03.870 --> 01:06:05.400
It's worth understanding.
01:06:05.400 --> 01:06:08.690
I didn't, like, derive it.
01:06:08.690 --> 01:06:11.580
And the book does, of course.
01:06:11.580 --> 01:06:13.575
There's something called
an integrating factor.
01:06:16.650 --> 01:06:19.200
You can get at this
formula systematically.
01:06:19.200 --> 01:06:25.460
I'd rather get at it
and understand it.
01:06:25.460 --> 01:06:27.800
I'm more interested
in understanding
01:06:27.800 --> 01:06:32.380
what the meaning of that
formula is than the algebra.
01:06:32.380 --> 01:06:35.910
Algebra is just a
goal to understand,
01:06:35.910 --> 01:06:39.020
and that's what I
shot for directly.
01:06:39.020 --> 01:06:41.750
And as I say, that
the book also,
01:06:41.750 --> 01:06:46.490
early section of the book,
uses practice in calculus.
01:06:46.490 --> 01:06:49.930
Substitute that in
to the equation.
01:06:49.930 --> 01:06:52.120
Figure out what is dy dt.
01:06:52.120 --> 01:06:56.470
And check that it works.
01:06:56.470 --> 01:07:05.080
It's worth actually
looking at that end of what
01:07:05.080 --> 01:07:07.290
you need to know from
calculus It's is.
01:07:07.290 --> 01:07:10.530
You should be able
to plug that in for y
01:07:10.530 --> 01:07:14.090
and see that solves
the equation.
01:07:14.090 --> 01:07:22.930
Right, now I have enough
time to do cosine omega t.
01:07:22.930 --> 01:07:27.710
But I don't have enough time
to do it the complex way.
01:07:27.710 --> 01:07:36.280
So let me do as a final
example, the equation.
01:07:36.280 --> 01:07:37.120
Let me just think.
01:07:37.120 --> 01:07:39.630
I don't know if I have
enough space here.
01:07:39.630 --> 01:07:43.960
I'm now going to do dy dt--
can I call that y prime to save
01:07:43.960 --> 01:07:53.160
a little space-- equal
ay plus cosine of t.
01:07:53.160 --> 01:07:54.721
I'll take omega to be 1.
01:07:58.650 --> 01:08:02.682
Now, how could we
solve that one?
01:08:05.410 --> 01:08:09.690
I'm going to solve it
without complex numbers,
01:08:09.690 --> 01:08:12.960
just to see how easy
or hard that is.
01:08:12.960 --> 01:08:17.130
And you'll see,
actually, it's easy.
01:08:17.130 --> 01:08:20.050
But complex numbers
will tell us more.
01:08:20.050 --> 01:08:25.279
So it's easy, but
not totally easy.
01:08:25.279 --> 01:08:29.010
So what did I do in
the earlier example
01:08:29.010 --> 01:08:32.970
if the right hand side
was a 1, a constant?
01:08:32.970 --> 01:08:35.500
I look for the solution
to be a constant.
01:08:35.500 --> 01:08:38.240
If the right hand side
was an exponential,
01:08:38.240 --> 01:08:41.010
I look for the solution
to be an exponential.
01:08:41.010 --> 01:08:46.180
Now, my right hand side, my
source term, is a cosine.
01:08:46.180 --> 01:08:49.479
So what form of the solution
am I going to look for?
01:08:52.020 --> 01:08:54.329
I naturally think,
OK, look for a cosine.
01:08:57.859 --> 01:09:04.090
We could try y equals
some number M cosine t.
01:09:07.899 --> 01:09:14.109
Now, you have to see what
goes wrong and how to fix it.
01:09:14.109 --> 01:09:17.640
So if I plug that in,
looking for M the same way
01:09:17.640 --> 01:09:20.684
I look for capital Y
earlier, I plug this in,
01:09:20.684 --> 01:09:24.439
and I get aM cosine t cosine t.
01:09:24.439 --> 01:09:27.240
But what do I get for y prime?
01:09:27.240 --> 01:09:28.020
Sine t.
01:09:30.910 --> 01:09:32.160
And I can't match.
01:09:32.160 --> 01:09:33.979
I can make it work.
01:09:33.979 --> 01:09:38.050
I can't make a sine there
magic a cosine here.
01:09:38.050 --> 01:09:40.510
So what's the solution?
01:09:40.510 --> 01:09:43.620
How do I fix it?
01:09:43.620 --> 01:09:47.529
I better allow my
solution to include
01:09:47.529 --> 01:09:51.563
some sine plus N sine t.
01:09:56.110 --> 01:10:02.240
So that's the problem with
doing it, keeping things real.
01:10:02.240 --> 01:10:06.750
I'll push this
through, no problem.
01:10:06.750 --> 01:10:09.180
But cosine by itself won't work.
01:10:09.180 --> 01:10:13.820
I need to have sines there,
because derivatives bring out
01:10:13.820 --> 01:10:14.700
sines.
01:10:14.700 --> 01:10:18.570
So I have a combination
of cosine and sine.
01:10:18.570 --> 01:10:20.933
I have a combination
of cosine and sine.
01:10:24.140 --> 01:10:27.140
So the complex method
will work in one shot
01:10:27.140 --> 01:10:36.410
because e to the i omega t is a
combination of cosine and sine.
01:10:36.410 --> 01:10:39.560
Or another way to say it
is when I see cosine here,
01:10:39.560 --> 01:10:41.410
that's got two exponentials.
01:10:41.410 --> 01:10:45.030
That's got e to the it
and e to the-- anyway.
01:10:45.030 --> 01:10:47.180
Let's go for the real one.
01:10:47.180 --> 01:10:48.790
So I'm going to plug
that into there.
01:10:52.860 --> 01:10:56.560
So I'll get sines
and cosines, right?
01:10:56.560 --> 01:10:59.180
When I plug this
into there, I'll
01:10:59.180 --> 01:11:01.480
have some sines
and some cosines,
01:11:01.480 --> 01:11:03.980
and I'll just match
the two separately.
01:11:03.980 --> 01:11:06.190
So I'm going to
get two equations.
01:11:06.190 --> 01:11:11.370
First of all, let me say
what's the cosine equation?
01:11:11.370 --> 01:11:14.130
And then what's
the sine equation?
01:11:14.130 --> 01:11:18.190
So when I match cosine
terms, what do I have?
01:11:18.190 --> 01:11:21.300
What cosine terms do I
get out of y prime, here?
01:11:23.970 --> 01:11:26.290
The derivative.
01:11:26.290 --> 01:11:28.470
Well, the derivative
of cosine is a sine.
01:11:28.470 --> 01:11:30.560
That that's not a cosine term.
01:11:30.560 --> 01:11:32.780
The derivative of
sine is cosine.
01:11:32.780 --> 01:11:38.000
I think I get, if I
just match cosines,
01:11:38.000 --> 01:11:39.740
I think I get an N cosine.
01:11:39.740 --> 01:11:44.940
N cosine t equal ay.
01:11:44.940 --> 01:11:49.400
How many cosines do I
have from that term?
01:11:49.400 --> 01:11:52.040
ay has an M cosine t.
01:11:52.040 --> 01:11:55.832
I think I have an aM,
and here I've got 1.
01:11:59.990 --> 01:12:07.070
That was a natural
step, but new to us.
01:12:07.070 --> 01:12:09.940
I'm matching the cosines.
01:12:09.940 --> 01:12:15.440
I have on the left side, with
this form of the solution,
01:12:15.440 --> 01:12:18.020
the derivative will
have an N cosine t.
01:12:18.020 --> 01:12:22.850
So I had N cosines, aM
cosines, and 1 cosine.
01:12:22.850 --> 01:12:24.760
Now, what if I match signs?
01:12:24.760 --> 01:12:26.500
What happens there?
01:12:26.500 --> 01:12:29.900
We're pushing more
than an hour, so
01:12:29.900 --> 01:12:34.020
hang on for another five
minutes, and we're there.
01:12:34.020 --> 01:12:38.310
Now, what happens if
I match sines, sine t?
01:12:38.310 --> 01:12:40.510
How do I get sine t in y prime?
01:12:44.740 --> 01:12:47.734
So take the derivative of
that, and what do you have?
01:12:47.734 --> 01:12:48.900
AUDIENCE: Minus [INAUDIBLE].
01:12:48.900 --> 01:12:51.190
PROFESSOR: Minus M sine t.
01:12:51.190 --> 01:12:55.170
That tells me how many
sine t's are in there.
01:12:55.170 --> 01:12:58.690
And on the right
hand side, a times y,
01:12:58.690 --> 01:13:01.379
how many sine t's
do I have from that?
01:13:01.379 --> 01:13:02.420
AUDIENCE: You have N t's.
01:13:02.420 --> 01:13:04.740
PROFESSOR: N, good thinking.
01:13:04.740 --> 01:13:09.210
And what about from this term?
01:13:09.210 --> 01:13:11.800
None, no sine there.
01:13:11.800 --> 01:13:18.940
So I have two equations by
matching the cosines and sines.
01:13:18.940 --> 01:13:22.360
Once you see it, you
could do it again.
01:13:22.360 --> 01:13:25.240
And we can solve
those equations,
01:13:25.240 --> 01:13:30.290
two ordinary, very simple
equations for M and N.
01:13:30.290 --> 01:13:32.180
Let's see if I make space.
01:13:32.180 --> 01:13:38.210
Why don't I do it here,
so you can see it.
01:13:41.260 --> 01:13:43.780
So how do I solve
those two equations?
01:13:43.780 --> 01:13:50.070
Well, this equation gives me--
easy-- gives me M as minus aN.
01:13:50.070 --> 01:13:51.930
So I'll just put that in for N.
01:13:51.930 --> 01:13:58.086
So I have N equals aM.
01:13:58.086 --> 01:14:00.360
But M is minus aN.
01:14:00.360 --> 01:14:06.264
I think I've got minus
a squared N plus that 1.
01:14:11.990 --> 01:14:17.570
All I did was solve the
equation, just by common sense.
01:14:17.570 --> 01:14:21.860
You could say by linear
algebra, but linear algebra's
01:14:21.860 --> 01:14:24.310
got a little more
to it than this.
01:14:24.310 --> 01:14:29.720
So now I know M, and now I know
N. So now I know the answer.
01:14:29.720 --> 01:14:38.944
y is M, so M is minus aN.
01:14:41.590 --> 01:14:44.690
Oh, well, I have to figure
out what N is, here.
01:14:44.690 --> 01:14:46.510
What is N?
01:14:46.510 --> 01:14:50.570
This is giving me N, but
I better figure it out.
01:14:50.570 --> 01:14:53.280
What is N from that
first equation?
01:14:53.280 --> 01:14:56.000
And then I'll plug in.
01:14:56.000 --> 01:14:57.843
And then I'm quit.
01:14:57.843 --> 01:14:58.769
AUDIENCE: [INAUDIBLE].
01:14:58.769 --> 01:15:01.084
PROFESSOR: 1 over, yeah.
01:15:01.084 --> 01:15:02.480
AUDIENCE: 1 plus a squared.
01:15:02.480 --> 01:15:05.530
PROFESSOR: 1 plus
a squared, good.
01:15:05.530 --> 01:15:07.530
Because that term goes
over there, and we have 1
01:15:07.530 --> 01:15:08.650
plus a squared.
01:15:08.650 --> 01:15:13.980
So now y is M cosine t.
01:15:13.980 --> 01:15:16.110
So M is minus aN.
01:15:16.110 --> 01:15:24.990
So minus aN is 1 over 1
plus a squared cosine t.
01:15:24.990 --> 01:15:26.580
Is that right?
01:15:26.580 --> 01:15:28.940
That was the cosines.
01:15:28.940 --> 01:15:35.330
And we had N sine t.
01:15:35.330 --> 01:15:38.710
But N is just 1-- I think
I just add the sine t.
01:15:38.710 --> 01:15:40.284
Have I got it?
01:15:40.284 --> 01:15:41.760
I think so.
01:15:46.190 --> 01:15:51.520
Here is the N sine t,
and here is the M cos t.
01:15:55.365 --> 01:15:56.415
It was just algebra.
01:16:01.054 --> 01:16:02.470
Typical of these
problems, there's
01:16:02.470 --> 01:16:05.970
a little thinking and
then some algebra.
01:16:05.970 --> 01:16:10.350
The thinking led us to this.
01:16:10.350 --> 01:16:14.420
The thinking led us to the fact
we needed cosines in there,
01:16:14.420 --> 01:16:16.650
as well as cosines.
01:16:16.650 --> 01:16:18.590
But then once we did
it, then the thinking
01:16:18.590 --> 01:16:23.275
said, OK, separately match the
cosine terms and the sine term.
01:16:23.275 --> 01:16:24.690
And then do the algebra.
01:16:27.260 --> 01:16:38.230
Now, I just want to
do this with complex.
01:16:38.230 --> 01:16:44.260
So y prime equals
ay plus e to the it.
01:16:49.464 --> 01:16:51.660
To get an idea, you see the two.
01:16:51.660 --> 01:16:54.000
And then I have
to talk about it.
01:16:54.000 --> 01:16:56.820
You see, I'm only going
to go part way with this
01:16:56.820 --> 01:17:00.810
and then save it for Wednesday.
01:17:00.810 --> 01:17:06.470
But if I see this, what
solution do I assume?
01:17:06.470 --> 01:17:16.304
This is like an e to the st. I
assume y is some Y e to the it.
01:17:16.304 --> 01:17:19.340
See, I don't have cosines
and sines anymore.
01:17:19.340 --> 01:17:20.680
I have e to the it.
01:17:20.680 --> 01:17:22.960
And if I take the
derivative of e to the it,
01:17:22.960 --> 01:17:25.180
I'm still in the
e to the it world.
01:17:25.180 --> 01:17:25.870
So I do this.
01:17:28.710 --> 01:17:29.790
I plug it in.
01:17:29.790 --> 01:17:32.730
Uh-huh, let me leave
that for Wednesday.
01:17:32.730 --> 01:17:36.390
We have to have some
excitement for Wednesday.
01:17:36.390 --> 01:17:41.220
So we'll get a complex
answer, and then we'll
01:17:41.220 --> 01:17:43.570
take the real part to
solve that problem.
01:17:46.370 --> 01:17:49.880
So we've got two steps,
one way or the other way.
01:17:49.880 --> 01:17:54.300
Here, we had two steps because
we had to let sines sneak in.
01:17:54.300 --> 01:17:58.920
Here, we have two steps
because I could solve it,
01:17:58.920 --> 01:18:02.160
and you could solve
that right away.
01:18:02.160 --> 01:18:05.370
But then you have to
take the real part.
01:18:05.370 --> 01:18:06.990
I'll leave that.
01:18:06.990 --> 01:18:08.905
Is there questions?
01:18:08.905 --> 01:18:12.606
Do you want me to recap
quickly what we've done.
01:18:12.606 --> 01:18:15.450
AUDIENCE: Yes.
01:18:15.450 --> 01:18:17.510
PROFESSOR: I try to
leave on the board
01:18:17.510 --> 01:18:21.500
enough to make a recap possible.
01:18:21.500 --> 01:18:25.020
Everything was
about that equation.
01:18:25.020 --> 01:18:28.480
We have only
solved-- I shouldn't
01:18:28.480 --> 01:18:33.490
say only-- we have solved the
constant coefficient, model
01:18:33.490 --> 01:18:35.770
constant coefficient,
first order equation.
01:18:42.480 --> 01:18:44.165
Wednesday comes
nonlinear equation.
01:18:46.760 --> 01:18:49.330
This one today was
strictly linear.
01:18:49.330 --> 01:18:51.470
So what did we do?
01:18:51.470 --> 01:18:57.670
We solved this equation,
first of all, for q equal 1;
01:18:57.670 --> 01:19:02.220
secondly, for q equal
e to the st; thirdly,
01:19:02.220 --> 01:19:13.680
for q equal a step; fourthly
for q equal-- where is it?
01:19:13.680 --> 01:19:14.790
Where is that delta of t?
01:19:14.790 --> 01:19:17.430
Maybe it's here.
01:19:17.430 --> 01:19:19.450
Ah, it got erased.
01:19:19.450 --> 01:19:28.610
So the fourth guy was y prime
equal ay plus delta of t,
01:19:28.610 --> 01:19:34.730
or delta of t minus
capital T. So those
01:19:34.730 --> 01:19:36.010
were our four examples.
01:19:38.850 --> 01:19:41.280
And then what did we finally do?
01:19:44.500 --> 01:19:48.200
So if we're recapping,
compressing,
01:19:48.200 --> 01:19:51.170
we're compressing
everything into two minutes.
01:19:51.170 --> 01:19:57.010
We solved those four
examples, and then we
01:19:57.010 --> 01:20:00.100
solved the general problem.
01:20:00.100 --> 01:20:01.810
And when we solved
the general problem,
01:20:01.810 --> 01:20:06.920
that gave us this integral,
which my whole goal was
01:20:06.920 --> 01:20:11.120
that you should understand that
this should seem right to you.
01:20:11.120 --> 01:20:15.000
This is adding up
the value at time t
01:20:15.000 --> 01:20:20.350
from all the inputs
at different times s.
01:20:20.350 --> 01:20:24.850
So to add them up, we
integrate from 0 to t.
01:20:24.850 --> 01:20:27.880
And finally, we
returned to the question
01:20:27.880 --> 01:20:31.000
of cos t, all
important question.
01:20:31.000 --> 01:20:33.990
But awkward question,
because we needed to let sine
01:20:33.990 --> 01:20:36.140
t in there too.