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PROFESSOR: Monday's lecture
was all linear equations.
00:00:26.020 --> 00:00:28.070
And I thought I
would start today
00:00:28.070 --> 00:00:32.330
with nonlinear equations,
still first order.
00:00:32.330 --> 00:00:36.620
And we can't deal with
every nonlinear equation.
00:00:36.620 --> 00:00:38.990
That's too much to ask.
00:00:38.990 --> 00:00:43.020
These are going
to be made easier
00:00:43.020 --> 00:00:45.910
by a property
called "separable."
00:00:45.910 --> 00:00:54.280
So these will be separable
nonlinear equations.
00:00:54.280 --> 00:00:57.950
And let me start with
a couple of examples
00:00:57.950 --> 00:01:01.360
and then you'll
see the whole idea.
00:01:01.360 --> 00:01:10.570
So one example would be the
simplest nonlinear equation
00:01:10.570 --> 00:01:13.980
I can think of,
with a y squared.
00:01:13.980 --> 00:01:17.180
So how to get there?
00:01:19.910 --> 00:01:21.470
Here's the trick.
00:01:21.470 --> 00:01:24.340
This is the separable idea.
00:01:24.340 --> 00:01:26.750
You're going to
see it in one shot.
00:01:26.750 --> 00:01:30.230
We can separate, put
the Ys on one side
00:01:30.230 --> 00:01:31.730
and the Ds on the other.
00:01:31.730 --> 00:01:39.210
So I write this as dy
over y squared equal dt.
00:01:39.210 --> 00:01:41.820
I put the dt up and
brought the y squared down.
00:01:41.820 --> 00:01:44.100
So now they're
separated, in a kind
00:01:44.100 --> 00:01:48.210
of hookie way with
infinitesimals.
00:01:48.210 --> 00:01:51.540
But I'll makes sense out
of that by integrating.
00:01:51.540 --> 00:01:53.610
I'll integrate both sides.
00:01:53.610 --> 00:01:58.790
I'll integrate time from 0 to t.
00:01:58.790 --> 00:02:05.570
And I have an initial
condition, y of 0, always.
00:02:05.570 --> 00:02:11.700
And since this one is about y,
when t starts at 0, this guy
00:02:11.700 --> 00:02:16.550
starts at y of 0, up
to, this ends at t,
00:02:16.550 --> 00:02:20.810
so this ends at y of t.
00:02:20.810 --> 00:02:21.850
OK.
00:02:21.850 --> 00:02:25.570
Now the point is, also
the problem was nonlinear,
00:02:25.570 --> 00:02:29.740
we've got two separate
ordinary integrals to do.
00:02:29.740 --> 00:02:32.330
And we can do them.
00:02:32.330 --> 00:02:36.010
We can certainly do
the right hand side.
00:02:36.010 --> 00:02:37.790
I get t.
00:02:37.790 --> 00:02:40.350
And on the left hand
side, what do I get?
00:02:40.350 --> 00:02:43.280
Well, that maybe I better
leave a little space
00:02:43.280 --> 00:02:45.480
to figure out this one.
00:02:48.220 --> 00:02:51.610
But the point is we can
integrate 1 over y squared.
00:02:51.610 --> 00:02:54.960
And I guess we get
minus 1 over y.
00:02:54.960 --> 00:03:02.460
So I get minus 1 over y
between y of 0 and y of t.
00:03:02.460 --> 00:03:06.770
In other words, I'm
getting let's see,
00:03:06.770 --> 00:03:11.430
so what the right, the
derivative of the integral of 1
00:03:11.430 --> 00:03:14.980
over y squared is minus
1 over y because I always
00:03:14.980 --> 00:03:18.700
check the derivative
gives me that back.
00:03:18.700 --> 00:03:21.680
So now I'm ready to
plug-in those limits.
00:03:21.680 --> 00:03:26.330
So I'll do the
bottom limit first
00:03:26.330 --> 00:03:30.160
because it comes with a minus
sign, canceling that minus, 1
00:03:30.160 --> 00:03:35.590
over y of 0 minus 1 over y of t.
00:03:35.590 --> 00:03:36.740
Got it.
00:03:36.740 --> 00:03:40.640
And that equals the other
integral, which is just t.
00:03:43.740 --> 00:03:51.240
So that's the answer as it
comes directly from integration.
00:03:51.240 --> 00:03:53.510
And we can do more.
00:03:53.510 --> 00:04:01.130
You can see that finding the
solution when these things are
00:04:01.130 --> 00:04:04.930
separable has boiled
down to two integrals.
00:04:04.930 --> 00:04:16.485
And we could have a
function of t here, too.
00:04:16.485 --> 00:04:19.269
And that would be
allowed, a function of t
00:04:19.269 --> 00:04:21.339
multiplying this
guy, because then I
00:04:21.339 --> 00:04:24.430
would leave the function
of t on that side.
00:04:24.430 --> 00:04:27.020
And I would have
to integrate that.
00:04:27.020 --> 00:04:28.240
And I would bring the y.
00:04:28.240 --> 00:04:30.620
You see, I've just
separated the y.
00:04:30.620 --> 00:04:37.000
In general, these
equations look like dy, dt
00:04:37.000 --> 00:04:43.240
is some function of t divided
by some function of y.
00:04:43.240 --> 00:04:46.560
Maybe the book
calls the top one g,
00:04:46.560 --> 00:04:51.440
I think, and the bottom one f.
00:04:51.440 --> 00:04:56.050
And everybody in this room
sees that I can put the f of y
00:04:56.050 --> 00:04:57.290
up there.
00:04:57.290 --> 00:05:00.000
I can put the dt up there.
00:05:00.000 --> 00:05:01.310
And I've separated it.
00:05:01.310 --> 00:05:02.030
OK.
00:05:02.030 --> 00:05:04.030
So that's sort of the
general situation.
00:05:05.340 --> 00:05:10.290
This is a kind of nice example,
nice example, dy, dt equals
00:05:10.290 --> 00:05:11.700
y squared.
00:05:11.700 --> 00:05:14.200
Can we just play with
this a little bit?
00:05:14.200 --> 00:05:19.900
Let me take y of 0 to be 1,
just to make the numbers easy.
00:05:19.900 --> 00:05:24.140
So if y of 0 is 1,
then I have, I'll
00:05:24.140 --> 00:05:26.940
just keep going a little bit.
00:05:26.940 --> 00:05:29.090
You do have to keep
going a little bit
00:05:29.090 --> 00:05:32.600
because when you finish
the integral right there,
00:05:32.600 --> 00:05:35.370
you haven't got y equal.
00:05:35.370 --> 00:05:38.600
You've got some equation
that involves y,
00:05:38.600 --> 00:05:40.880
but you have to solve for y.
00:05:40.880 --> 00:05:45.070
So I have to solve
that equation for y.
00:05:45.070 --> 00:05:46.620
Let me just do it.
00:05:46.620 --> 00:05:48.250
So how would I solve it?
00:05:48.250 --> 00:05:50.670
And let me take y of 0 to be 1.
00:05:50.670 --> 00:05:56.680
So now, if I just write
it below, I'm at 1 minus 1
00:05:56.680 --> 00:05:59.960
over y equals t.
00:05:59.960 --> 00:06:02.590
Good?
00:06:02.590 --> 00:06:09.850
So I'm going to put the 1 over
y of t on that side and the t
00:06:09.850 --> 00:06:10.680
on that side.
00:06:10.680 --> 00:06:16.120
So if I just continue here,
I've got 1 over y of t
00:06:16.120 --> 00:06:23.050
on this side and, do I have
1 minus t on that side?
00:06:23.050 --> 00:06:23.990
Yeah.
00:06:23.990 --> 00:06:25.795
Looking good.
00:06:25.795 --> 00:06:33.430
So solution starting from
y of 0 equal 1 is y of t
00:06:33.430 --> 00:06:35.950
equal 1 over 1 minus t.
00:06:38.870 --> 00:06:41.640
You could do that.
00:06:41.640 --> 00:06:44.230
You could do that.
00:06:44.230 --> 00:06:49.610
And I can always, like, mentally
I check the algebra at t
00:06:49.610 --> 00:06:50.300
equals 0.
00:06:50.300 --> 00:06:52.430
That gives me the answer, 1.
00:06:52.430 --> 00:06:56.320
But let's step back and
look at that answer.
00:06:56.320 --> 00:06:58.330
I mean, that's part of
differential equations
00:06:58.330 --> 00:07:03.390
is to do some algebra, if
possible, and get to a formula.
00:07:03.390 --> 00:07:05.940
But if we don't think
about the formula,
00:07:05.940 --> 00:07:07.240
we haven't learned anything.
00:07:07.240 --> 00:07:08.310
Right there, yes.
00:07:08.310 --> 00:07:09.720
Good.
00:07:09.720 --> 00:07:14.650
So what happens?
00:07:14.650 --> 00:07:24.060
I want to compare with
the linear case that
00:07:24.060 --> 00:07:26.870
was like e to the t.
00:07:26.870 --> 00:07:30.480
This was y prime equal y, right?
00:07:30.480 --> 00:07:32.030
And that led to e to the t.
00:07:32.030 --> 00:07:37.370
Y prime equals y squared
leads to that one.
00:07:37.370 --> 00:07:40.260
So first observation.
00:07:40.260 --> 00:07:44.010
I haven't got exponentials
anymore in that solution.
00:07:44.010 --> 00:07:46.920
Exponentials are
just like perfection
00:07:46.920 --> 00:07:49.170
for linear equations.
00:07:49.170 --> 00:07:52.760
For nonlinear equations,
we get other functions.
00:07:52.760 --> 00:07:56.345
Professor Fry had a
hyperbolic tangent function
00:07:56.345 --> 00:07:58.010
in his first lecture.
00:07:58.010 --> 00:07:59.550
Other things happen.
00:07:59.550 --> 00:08:00.160
OK.
00:08:00.160 --> 00:08:06.890
Now, how do those
compare if I graph those?
00:08:06.890 --> 00:08:09.140
It's just like, why not try?
00:08:09.140 --> 00:08:12.510
So they both started at 1.
00:08:12.510 --> 00:08:16.610
And e to the t went
up exponentially.
00:08:16.610 --> 00:08:17.480
E to the t.
00:08:20.520 --> 00:08:23.810
And, I don't know,
we use exponential.
00:08:23.810 --> 00:08:26.815
In our minds, we think,
that's pretty fast growth.
00:08:26.815 --> 00:08:31.690
! mean, that's the common
expression, grew exponentially.
00:08:31.690 --> 00:08:35.330
But here, this guy
is going to grow
00:08:35.330 --> 00:08:39.850
faster because y is going
to be bigger than 1.
00:08:39.850 --> 00:08:44.220
So y squared is going
to be bigger than y.
00:08:44.220 --> 00:08:46.800
That one's going to grow faster.
00:08:46.800 --> 00:08:49.030
Faster than exponential.
00:08:49.030 --> 00:08:51.290
This has the exponential growth.
00:08:51.290 --> 00:08:53.070
Pretty fast.
00:08:53.070 --> 00:08:56.690
Polynomial, of course,
some parabola or something
00:08:56.690 --> 00:09:01.590
would be hanging way down
here, left behind in the dust.
00:09:01.590 --> 00:09:04.080
But this 1 over
1 minus t, that's
00:09:04.080 --> 00:09:05.540
going to grow really fast.
00:09:05.540 --> 00:09:11.080
And what's more, it's
going to go to infinity.
00:09:11.080 --> 00:09:16.435
So that y prime equal
y squared, the solution
00:09:16.435 --> 00:09:21.770
to that doesn't
just-- e to the t
00:09:21.770 --> 00:09:26.720
goes to infinity
at time infinity.
00:09:26.720 --> 00:09:30.270
At any finite time,
we get an answer.
00:09:30.270 --> 00:09:37.970
Eventually, at t equal infinity,
it's gone above every bound.
00:09:37.970 --> 00:09:43.360
But this one, 1 over 1 minus
t is what I want to graph now.
00:09:43.360 --> 00:09:50.410
I believe that that takes
off and at a certain point,
00:09:50.410 --> 00:09:54.310
capital T, it's
going to infinity.
00:09:54.310 --> 00:09:55.240
It's blown up.
00:09:55.240 --> 00:09:58.570
So it's blow up in finite time.
00:09:58.570 --> 00:10:00.000
Blow up in finite time.
00:10:00.000 --> 00:10:01.500
And what is that time?
00:10:01.500 --> 00:10:07.010
What's the time at which the
y prime equal y squared has
00:10:07.010 --> 00:10:10.220
taken off, gone off the charts?
00:10:10.220 --> 00:10:11.360
T equal--
00:10:11.360 --> 00:10:11.860
AUDIENCE: 1.
00:10:11.860 --> 00:10:13.220
PROFESSOR: --1.
00:10:13.220 --> 00:10:17.470
Because when t reaches
1, I have 1 over 0,
00:10:17.470 --> 00:10:22.600
and I'm dividing by 0,
and so that's the blow up.
00:10:22.600 --> 00:10:25.590
Finite time blow up.
00:10:25.590 --> 00:10:27.060
OK.
00:10:27.060 --> 00:10:31.540
So this can happen for
some nonlinear equations.
00:10:31.540 --> 00:10:33.610
It wouldn't happen
for a linear equation.
00:10:33.610 --> 00:10:37.390
For a linear equation,
exponentials are in control.
00:10:37.390 --> 00:10:39.410
OK.
00:10:39.410 --> 00:10:40.780
So that's one nice example.
00:10:40.780 --> 00:10:43.030
Oh, another nice thing
about that example.
00:10:47.190 --> 00:10:51.120
Well, I say nice if you're
OK with infinite series.
00:10:51.120 --> 00:10:53.050
I just want to compare.
00:10:53.050 --> 00:10:59.140
The book mentions the
infinite series for these guys
00:10:59.140 --> 00:11:03.110
because that's an old way to
solve differential equations is
00:11:03.110 --> 00:11:05.040
term-by-term in an
infinite series.
00:11:06.010 --> 00:11:08.730
It's sort of fun to
see the two series.
00:11:08.730 --> 00:11:15.620
Well, because they're the two
most important series in math.
00:11:15.620 --> 00:11:18.870
Actually, they're the two series
that everybody should know.
00:11:18.870 --> 00:11:23.040
The power series,
Taylor series--
00:11:23.040 --> 00:11:26.720
whatever word you want to
give it for those two guys.
00:11:26.720 --> 00:11:28.700
So let me do them.
00:11:28.700 --> 00:11:35.010
E to the t, I'll put that one
first, and 1 over 1 minus t.
00:11:35.010 --> 00:11:39.290
These are the great
series of math.
00:11:39.290 --> 00:11:45.160
Shall I just write them down
and sort of talk through them?
00:11:45.160 --> 00:11:49.540
Because this is not a lecture
on infinite series by any means.
00:11:49.540 --> 00:11:52.470
But having these
two in front of us,
00:11:52.470 --> 00:11:55.020
coming from these two
beautiful equations,
00:11:55.020 --> 00:11:58.010
y prime equal y squared
and y prime equal y,
00:11:58.010 --> 00:12:05.370
I can't resist seeing what
they look like this way.
00:12:05.370 --> 00:12:08.410
So e to the t, do you
remember e to the t?
00:12:08.410 --> 00:12:11.200
It starts at 1.
00:12:11.200 --> 00:12:14.800
What's the slope of e to the t?
00:12:14.800 --> 00:12:16.370
At t equals 0.
00:12:16.370 --> 00:12:18.720
So I'm doing everything--
this series is
00:12:18.720 --> 00:12:20.750
going to be, both
of the series are
00:12:20.750 --> 00:12:26.190
going to be, around t equals 0.
00:12:26.190 --> 00:12:31.450
That's my, like, starting point.
00:12:31.450 --> 00:12:39.100
So this e to the t
thing has a tangent.
00:12:39.100 --> 00:12:41.160
It has a slope there.
00:12:41.160 --> 00:12:45.455
And what's the slope of
e to the t at t equals 0?
00:12:45.455 --> 00:12:45.954
AUDIENCE: 1.
00:12:45.954 --> 00:12:47.760
PROFESSOR: 1.
00:12:47.760 --> 00:12:49.590
It's derivative.
00:12:49.590 --> 00:12:52.240
The derivative of e to
the t is e to the t.
00:12:52.240 --> 00:12:53.750
The slope is 1.
00:12:53.750 --> 00:12:58.880
So that tangent line
has coefficient 1.
00:12:58.880 --> 00:13:01.760
That's how it starts.
00:13:01.760 --> 00:13:03.620
That's the linear approximation.
00:13:03.620 --> 00:13:07.350
That's the heart of
calculus, is this.
00:13:07.350 --> 00:13:09.980
But we're going to go better.
00:13:09.980 --> 00:13:13.140
We're going to
get the next term.
00:13:13.140 --> 00:13:15.940
So what's the next term?
00:13:15.940 --> 00:13:17.750
That gave us the tangent line.
00:13:18.740 --> 00:13:22.170
Now I'm going to move
to the tangent parabola.
00:13:22.170 --> 00:13:26.130
So the parabola has
got another is still
00:13:26.130 --> 00:13:29.360
going to be below
the real thing.
00:13:29.360 --> 00:13:34.210
Can I squeeze in the
words "line" and "parab,"
00:13:34.210 --> 00:13:37.650
for "parabola?"
00:13:39.040 --> 00:13:42.180
Parabola has bending.
00:13:42.180 --> 00:13:44.680
I'm really explaining
the Taylor series
00:13:44.680 --> 00:13:46.520
in what I hope is
a sensible way.
00:13:49.090 --> 00:13:51.300
Here is the starting point.
00:13:51.300 --> 00:13:53.100
This has the slope.
00:13:53.100 --> 00:13:55.530
The next term has the bending.
00:13:55.530 --> 00:13:59.630
The bending comes
from what derivative?
00:13:59.630 --> 00:14:02.670
What derivative tells
us about bending?
00:14:02.670 --> 00:14:04.041
Second derivative.
00:14:04.041 --> 00:14:06.340
Second derivative tells
us how much it curves.
00:14:08.100 --> 00:14:12.650
Well, the second derivative of
e to the t is still e to the t.
00:14:12.650 --> 00:14:15.300
So the bending is 1.
00:14:15.300 --> 00:14:17.310
The bending is also 1.
00:14:17.310 --> 00:14:20.700
Now that comes in with
a factor of a 1/2.
00:14:23.930 --> 00:14:27.020
There is the tangent parabola.
00:14:27.020 --> 00:14:32.490
And you will see what
these numbers become.
00:14:32.490 --> 00:14:35.580
Let me just go to,
the third derivative
00:14:35.580 --> 00:14:38.720
would be responsible
for the t cubed term.
00:14:38.720 --> 00:14:42.100
And its coefficient would
be 1 over 3 factorial.
00:14:42.100 --> 00:14:45.490
So 2 is the same as 2 factorial.
00:14:45.490 --> 00:14:49.330
3 factorial is 3
times 2 times 16.
00:14:49.330 --> 00:14:59.300
So the numbers here go 1, 6, 24,
120, whatever the next one is.
00:14:59.300 --> 00:15:01.470
720 or something.
00:15:01.470 --> 00:15:03.530
They grow fast.
00:15:03.530 --> 00:15:07.660
So that series always
gives a finite answer.
00:15:07.660 --> 00:15:09.530
It does grow with t.
00:15:09.530 --> 00:15:14.200
But it doesn't spike with t.
00:15:14.200 --> 00:15:17.800
Now compare that famous series.
00:15:17.800 --> 00:15:21.970
And of course, this
is 1 over 1 factorial,
00:15:21.970 --> 00:15:23.860
everything consistent.
00:15:23.860 --> 00:15:28.810
Compare that with the
series for 1 over 1 minus t.
00:15:28.810 --> 00:15:32.540
That's the other famous series
that they learned in algebra.
00:15:32.540 --> 00:15:33.950
I'll just write it.
00:15:33.950 --> 00:15:39.415
That's 1 plus t plus t
squared plus t cubed plus 1
00:15:39.415 --> 00:15:41.180
so on, with coefficient 1.
00:15:43.860 --> 00:15:48.060
This had 1 over n factorials.
00:15:48.060 --> 00:15:50.550
Those make the series converge.
00:15:50.550 --> 00:15:52.980
These don't have
the n factorials.
00:15:52.980 --> 00:15:55.410
This is 1, 1, 1, 1.
00:15:55.410 --> 00:15:57.980
And, well, I could
check that formula.
00:15:57.980 --> 00:16:01.790
But do you remember the
name for that series?
00:16:01.790 --> 00:16:06.690
1 plus t plus t squared
plus t cubed plus so on?
00:16:06.690 --> 00:16:11.900
Algebra is taught differently
in many high schools now.
00:16:11.900 --> 00:16:15.340
And maybe that never got a name.
00:16:15.340 --> 00:16:18.960
I guess I would call it
the Geometric series.
00:16:18.960 --> 00:16:20.630
Geometric series.
00:16:20.630 --> 00:16:23.090
And you see, it's beautiful.
00:16:23.090 --> 00:16:24.680
It's the other important series.
00:16:24.680 --> 00:16:26.650
But it's quite
different from this one
00:16:26.650 --> 00:16:32.255
because, what's the
difference about this series?
00:16:32.255 --> 00:16:32.755
Yeah?
00:16:32.755 --> 00:16:33.976
AUDIENCE: It goes to infinity.
00:16:33.976 --> 00:16:34.760
PROFESSOR: It's a--
00:16:34.760 --> 00:16:36.010
AUDIENCE: It goes to infinity.
00:16:36.010 --> 00:16:37.690
PROFESSOR: It goes to infinity.
00:16:37.690 --> 00:16:38.740
But where?
00:16:38.740 --> 00:16:39.860
At what time?
00:16:39.860 --> 00:16:45.380
At what value of t is this
sum going to fall apart?
00:16:45.380 --> 00:16:47.180
Blow up?
00:16:47.180 --> 00:16:48.670
At t equal 1.
00:16:48.670 --> 00:16:53.390
When have 1 plus 1 plus 1
plus 1, I'm getting infinity.
00:16:53.390 --> 00:16:55.220
So this blows up.
00:16:55.220 --> 00:16:59.830
And of course, we see that it
should because this blows up.
00:16:59.830 --> 00:17:03.265
Left side blows up at t equal
1, the right side blows up a t
00:17:03.265 --> 00:17:04.190
equal 1.
00:17:04.190 --> 00:17:09.200
Where the exponential
series, which
00:17:09.200 --> 00:17:12.950
is the heart of ordinary
differential equations,
00:17:12.950 --> 00:17:18.710
never blows up because of these
big numbers in the denominator.
00:17:18.710 --> 00:17:24.660
OK, I'm good for this first
simple example, y prime equals
00:17:24.660 --> 00:17:26.410
y squared.
00:17:26.410 --> 00:17:30.390
It has so much in it,
it's worth thinking about.
00:17:30.390 --> 00:17:33.522
I'm ready, you OK
for a second example?
00:17:33.522 --> 00:17:41.040
A second important
separable equation.
00:17:41.040 --> 00:17:43.810
I'm going to pick one.
00:17:43.810 --> 00:17:49.870
So I'm going to pick an
equation that starts out
00:17:49.870 --> 00:17:54.110
with our familiar linear growth.
00:17:54.110 --> 00:17:56.670
This could be, you
know, last time
00:17:56.670 --> 00:17:59.380
it was growth of
money in a bank.
00:17:59.380 --> 00:18:02.330
It could be growth
of population.
00:18:02.330 --> 00:18:10.230
The number of, to a sum
first approximation,
00:18:10.230 --> 00:18:14.670
the rate of growth
of the population
00:18:14.670 --> 00:18:18.990
comes from, like,
births minus deaths.
00:18:18.990 --> 00:18:26.550
And with modern medicine,
births are a larger number
00:18:26.550 --> 00:18:27.750
than deaths.
00:18:27.750 --> 00:18:31.030
So a is positive,
and that grows.
00:18:31.030 --> 00:18:35.240
But if we're talking
about the, I mean,
00:18:35.240 --> 00:18:37.900
the United Nations
tries to predict,
00:18:37.900 --> 00:18:40.960
everybody tries to
predict, population
00:18:40.960 --> 00:18:45.980
of the world in future years.
00:18:45.980 --> 00:18:51.470
And so this could be called
the Population Equation.
00:18:51.470 --> 00:18:59.020
But just to leave it as pure
exponential is obviously wrong.
00:18:59.020 --> 00:19:01.190
The world can't grow forever.
00:19:01.190 --> 00:19:05.050
The population
can't grow forever.
00:19:05.050 --> 00:19:10.320
And the, I guess I hope it
doesn't grow like 1 over 1
00:19:10.320 --> 00:19:11.260
minus t.
00:19:11.260 --> 00:19:15.110
So this is at least
a little slower.
00:19:15.110 --> 00:19:21.520
But somehow competition for
space, competition for food,
00:19:21.520 --> 00:19:25.050
for oil, for water-- which
is going to be the big one--
00:19:25.050 --> 00:19:26.770
is in here.
00:19:26.770 --> 00:19:33.740
Competition here, of
people versus people,
00:19:33.740 --> 00:19:38.930
a reasonable term, a
first approximation,
00:19:38.930 --> 00:19:42.830
is a y squared, with a
minus, is a y squared
00:19:42.830 --> 00:19:45.710
and with some coefficient.
00:19:45.710 --> 00:19:47.310
That's a very famous equation.
00:19:51.110 --> 00:19:54.970
A first model of
population is it grows.
00:19:54.970 --> 00:20:01.020
But this is a competition
term, y against y.
00:20:01.020 --> 00:20:07.300
And so, the same
would be true if we
00:20:07.300 --> 00:20:09.900
were talking about epidemics.
00:20:09.900 --> 00:20:12.340
That's a big subject with
ordinary differential
00:20:12.340 --> 00:20:14.850
equations, epidemiology.
00:20:14.850 --> 00:20:17.890
Or say, flu.
00:20:17.890 --> 00:20:21.460
How does flu spread?
00:20:21.460 --> 00:20:22.980
And how does it get cured?
00:20:23.740 --> 00:20:26.810
So partly, people are
getting over the flu.
00:20:26.810 --> 00:20:31.640
But then y against y is
telling us how many infected,
00:20:31.640 --> 00:20:33.120
how many new infections.
00:20:33.120 --> 00:20:35.175
So we would like to
solve that equation.
00:20:37.910 --> 00:20:40.190
And it's separable.
00:20:40.190 --> 00:20:45.480
I can do what I did
before, dy over ay minus by
00:20:45.480 --> 00:20:48.650
squared equal dt.
00:20:48.650 --> 00:20:53.480
And I can integrate,
starting from year of 0.
00:20:53.480 --> 00:20:58.650
Well, why don't we
start from year 2014,
00:20:58.650 --> 00:21:03.460
with the population y at
now-- the present population?
00:21:07.840 --> 00:21:13.930
That would be a model that
the UN would consider using.
00:21:13.930 --> 00:21:17.500
That other people with
very important interest
00:21:17.500 --> 00:21:25.080
in measuring population and
measuring every resource
00:21:25.080 --> 00:21:27.990
would need equations like this.
00:21:27.990 --> 00:21:30.770
And then they would
put on more terms,
00:21:30.770 --> 00:21:36.650
like a term for immigration.
00:21:36.650 --> 00:21:41.155
All sorts, many improvements
have to go into this equation.
00:21:43.910 --> 00:21:45.655
Let me just look
at this as it is.
00:21:48.580 --> 00:21:53.480
Well, I've got two choices here.
00:21:53.480 --> 00:21:59.200
Well, it's this integral
that I'm looking at.
00:21:59.200 --> 00:22:01.220
That is a doable integral.
00:22:01.220 --> 00:22:06.160
It's the type of integral that
we saw in the Rocket problem.
00:22:06.160 --> 00:22:10.560
The Rocket problem was
more constant minus.
00:22:10.560 --> 00:22:15.030
This was a drag term, when
we were looking at rockets.
00:22:15.030 --> 00:22:20.410
And this was a
constant, say, gravity.
00:22:20.410 --> 00:22:24.510
So it was still a second degree.
00:22:24.510 --> 00:22:26.480
Still second degree,
but a little different.
00:22:26.480 --> 00:22:32.410
This has the linear in
second degree terms.
00:22:32.410 --> 00:22:34.350
If you look up that
integral, you'll find it.
00:22:37.310 --> 00:22:40.010
Or there's a systematic
way to do it.
00:22:40.010 --> 00:22:47.320
That's in 1801, I guess,
called partial fractions.
00:22:47.320 --> 00:22:50.710
It's not a lot of fun.
00:22:50.710 --> 00:22:53.190
I don't plan to do it.
00:22:53.190 --> 00:22:54.630
It's in the book.
00:22:54.630 --> 00:22:57.540
Has to be because that's
the way you can integr-- you
00:22:57.540 --> 00:23:02.150
can integrate polynomials
over polynomials
00:23:02.150 --> 00:23:04.770
by partial fractions.
00:23:04.770 --> 00:23:09.160
That's what they're
for, but there's
00:23:09.160 --> 00:23:11.280
a neat way to do this one.
00:23:11.280 --> 00:23:16.600
There's a neat trick
that Bernoulli discovered
00:23:16.600 --> 00:23:22.010
to solve that equation, to
turn it into a linear equation.
00:23:22.010 --> 00:23:24.500
And of course, if we can turn
it into a linear equation,
00:23:24.500 --> 00:23:25.400
we're on our way.
00:23:25.400 --> 00:23:31.480
So the neat trick is
let z be 1 over y.
00:23:35.960 --> 00:23:38.720
You can put this in the
category of lucky accidents,
00:23:38.720 --> 00:23:41.810
if you like.
00:23:41.810 --> 00:23:45.410
So now I want an equation for z.
00:23:45.410 --> 00:23:51.920
So I know that dz, dt if I
take the derivative of that,
00:23:51.920 --> 00:23:54.140
that's y to the minus 1.
00:23:54.140 --> 00:24:01.850
So it's minus 1 y to
the minus 2 dy, dt.
00:24:01.850 --> 00:24:04.700
That's the chain rule.
00:24:04.700 --> 00:24:08.850
Take the derivative of 1 over 1,
you get minus 1 over y squared.
00:24:08.850 --> 00:24:14.100
Multiply by the derivative
of what's inside.
00:24:14.100 --> 00:24:15.331
That's the chain rule.
00:24:15.331 --> 00:24:15.830
OK.
00:24:15.830 --> 00:24:21.120
So I plan to substitute
those in here.
00:24:21.120 --> 00:24:24.430
So dy, dt, let's see.
00:24:24.430 --> 00:24:25.320
Can you see me?
00:24:25.320 --> 00:24:29.090
You can probably do
it better than me.
00:24:29.090 --> 00:24:33.240
So dy, dt is minus.
00:24:33.240 --> 00:24:34.390
I'll bring that up.
00:24:34.390 --> 00:24:39.620
Dy, dt I'm going to
put-- I hope this'll
00:24:39.620 --> 00:24:44.445
work all right-- for dy,
dt, I'm going to put in dz.
00:24:47.300 --> 00:24:52.600
Using this, I'm going to
put minus y squared dz, dt.
00:24:52.600 --> 00:24:54.070
Did that look right?
00:24:54.070 --> 00:24:55.740
I don't think I'm
necessarily doing
00:24:55.740 --> 00:24:57.920
this the most brilliant way.
00:24:57.920 --> 00:25:03.910
But dy, dt-- I put this up here
and I got that-- equals ay.
00:25:06.540 --> 00:25:08.883
So that's a over z.
00:25:08.883 --> 00:25:10.820
Oh, y Is 1 over z.
00:25:10.820 --> 00:25:17.040
So get this, I want all Zs now.
00:25:17.040 --> 00:25:19.540
So that's this part.
00:25:19.540 --> 00:25:23.080
And ay is over z
minus by squared
00:25:23.080 --> 00:25:26.980
is minus b over z squared.
00:25:26.980 --> 00:25:28.755
Would you say OK to that?
00:25:31.310 --> 00:25:34.180
I've got Zs now, instead of Ys.
00:25:34.180 --> 00:25:40.041
I just took every term and
replaced y by 1 over z.
00:25:40.041 --> 00:25:45.130
Y is 1 over z and dy,
dt I can get that way.
00:25:45.130 --> 00:25:45.936
OK.
00:25:45.936 --> 00:25:47.840
Yeah.
00:25:47.840 --> 00:25:49.270
Now what?
00:25:49.270 --> 00:25:52.440
Now look what happens, if I
multiply through by z squared
00:25:52.440 --> 00:25:54.190
or by minus z squared.
00:25:54.190 --> 00:25:56.680
Let me multiply through
by minus z squared.
00:25:56.680 --> 00:25:59.530
I get dz, dt.
00:25:59.530 --> 00:26:04.290
Multiplying by minus z
squared gives me a minus az.
00:26:04.290 --> 00:26:05.910
And what do I get
when I multiply
00:26:05.910 --> 00:26:07.795
this one by minus z squared?
00:26:10.620 --> 00:26:11.600
AUDIENCE: Plus b.
00:26:11.600 --> 00:26:12.690
PROFESSOR: I get plus b.
00:26:16.630 --> 00:26:18.980
Look what happened.
00:26:18.980 --> 00:26:22.160
By this, like, some magic trick.
00:26:25.279 --> 00:26:26.320
You could say, all right.
00:26:26.320 --> 00:26:28.920
That was just a one time shot.
00:26:28.920 --> 00:26:31.560
But it was a good one.
00:26:31.560 --> 00:26:35.730
We ended up with a
linear equation for z.
00:26:35.730 --> 00:26:37.230
A linear equation for z.
00:26:37.230 --> 00:26:41.940
And we solved that
equation last time.
00:26:41.940 --> 00:26:45.550
So let me squeeze in
the solution for z,
00:26:45.550 --> 00:26:47.150
and then elsewhere.
00:26:47.150 --> 00:26:50.900
So what was the
solution for z of t?
00:26:50.900 --> 00:26:56.030
It was some multiple
of, no, yeah.
00:26:56.030 --> 00:26:59.830
This is perfect
review of last time.
00:26:59.830 --> 00:27:03.580
We have a constant times z.
00:27:03.580 --> 00:27:07.250
And so that's going to
go into the exponential.
00:27:07.250 --> 00:27:13.380
This will be the, it's
a minus a, notice.
00:27:13.380 --> 00:27:18.490
That will be the, what
part of the solution
00:27:18.490 --> 00:27:21.150
is that one called?
00:27:21.150 --> 00:27:22.975
That's the null solution.
00:27:22.975 --> 00:27:26.120
The null solution, when b is o.
00:27:26.120 --> 00:27:29.380
And now I add in a
particular solution.
00:27:32.000 --> 00:27:33.330
A particular solution.
00:27:33.330 --> 00:27:39.670
And one good particular
solution is choose the z
00:27:39.670 --> 00:27:41.660
to be a constant.
00:27:41.660 --> 00:27:43.220
Then that'll be 0.
00:27:43.220 --> 00:27:45.320
So I want that to be 0.
00:27:45.320 --> 00:27:49.230
So what constant z makes that 0?
00:27:49.230 --> 00:27:51.520
I think it's b
over a, don't you?
00:27:51.520 --> 00:27:52.517
I think b over a.
00:27:52.517 --> 00:27:53.350
Does that work good?
00:27:57.920 --> 00:28:04.320
That's every null solution
plus one particular solution.
00:28:04.320 --> 00:28:08.850
Let me say now,
and I'll say again,
00:28:08.850 --> 00:28:13.920
looking for solutions which
are steady states, b over
00:28:13.920 --> 00:28:20.630
a-- of this particular solution,
that particular solution made
00:28:20.630 --> 00:28:22.830
this 0.
00:28:22.830 --> 00:28:25.620
So it made this 0.
00:28:25.620 --> 00:28:28.760
So it's a solution that's
not going anywhere.
00:28:28.760 --> 00:28:29.940
It's a constant solution.
00:28:29.940 --> 00:28:34.460
It's a solution that
can live for all time.
00:28:34.460 --> 00:28:35.440
OK.
00:28:35.440 --> 00:28:36.050
B over a.
00:28:39.230 --> 00:28:41.590
Let me put that word
there, steady state.
00:28:46.680 --> 00:28:48.300
OK.
00:28:48.300 --> 00:28:52.520
And now I would want to match
the initial conditions using c.
00:28:56.290 --> 00:28:56.790
Yeah.
00:28:56.790 --> 00:28:58.210
I'd better do that.
00:28:58.210 --> 00:28:59.620
OK.
00:28:59.620 --> 00:29:04.420
And I have to get back to y.
00:29:04.420 --> 00:29:09.200
I have y is 1 over z.
00:29:09.200 --> 00:29:12.065
So I'm going to have to
flip this upside down.
00:29:12.065 --> 00:29:15.440
I'm going to have to
flip this upside down is
00:29:15.440 --> 00:29:18.870
what will actually happen.
00:29:18.870 --> 00:29:22.680
Let me make it easy to flip.
00:29:22.680 --> 00:29:26.100
Let me, I'll change c,
which is just some constant
00:29:26.100 --> 00:29:29.070
to some constant d over a.
00:29:29.070 --> 00:29:33.400
So then it's a is
everywhere down below.
00:29:33.400 --> 00:29:36.250
And I just write it
here in the middle.
00:29:36.250 --> 00:29:37.760
That makes it easier to flip.
00:29:37.760 --> 00:29:42.120
So finally I get their solution.
00:29:42.120 --> 00:29:47.500
Solution to the
population equation.
00:29:47.500 --> 00:29:53.330
But that's the famous word
for it, the Logistic equation.
00:29:53.330 --> 00:29:59.000
This is section 1.7 of the
text on the differential
00:29:59.000 --> 00:30:02.130
equations in linear algebra.
00:30:02.130 --> 00:30:05.710
It's a very, very
much studied example.
00:30:05.710 --> 00:30:07.930
It's a great example.
00:30:07.930 --> 00:30:17.680
It fits the growth of
human population with some,
00:30:17.680 --> 00:30:21.710
it's our first
level approximation
00:30:21.710 --> 00:30:30.270
to growth of or other
populations or other things.
00:30:30.270 --> 00:30:36.000
It's a linear term giving
us exponential growth,
00:30:36.000 --> 00:30:40.270
and a quadratic term of
competition slowing it down.
00:30:40.270 --> 00:30:42.450
And let's see that slow down.
00:30:42.450 --> 00:30:48.120
So now that was
a bit of algebra.
00:30:48.120 --> 00:30:50.700
Much nicer than
partial fractions.
00:30:50.700 --> 00:30:56.030
The bit of algebra just came
from this idea of going to z.
00:30:56.030 --> 00:30:57.750
And now I want to go back to y.
00:30:57.750 --> 00:31:01.750
So y is 1 over z.
00:31:01.750 --> 00:31:09.630
So it's a over d e to
the minus at plus b.
00:31:12.760 --> 00:31:14.930
That's our solution.
00:31:14.930 --> 00:31:19.930
A and b came out
of the equation.
00:31:19.930 --> 00:31:23.110
And d is going to
be the number that
00:31:23.110 --> 00:31:26.110
makes the initial value correct.
00:31:26.110 --> 00:31:33.210
So at t equals 0, I would
have y of 0, whatever
00:31:33.210 --> 00:31:38.395
the initial population
is, is a over d.
00:31:38.395 --> 00:31:42.440
T Is 0, so that's just 1 plus b.
00:31:42.440 --> 00:31:44.856
So that tells me what d is.
00:31:44.856 --> 00:31:46.520
D equals something.
00:31:49.640 --> 00:31:52.440
It comes from y of 0.
00:31:52.440 --> 00:31:54.735
So the answer, let me
circle that answer.
00:31:58.460 --> 00:32:02.370
That answer has three
numbers in it, a, b, and d.
00:32:02.370 --> 00:32:04.240
a and b come from the equation.
00:32:04.240 --> 00:32:08.680
D also involves the
initial starting thing,
00:32:08.680 --> 00:32:10.548
which is exactly what it showed.
00:32:13.620 --> 00:32:15.550
So you could say
we've solved it.
00:32:15.550 --> 00:32:19.660
But if you ever solve
an equation like this,
00:32:19.660 --> 00:32:21.220
you want to graph it.
00:32:21.220 --> 00:32:23.130
You want to graph it.
00:32:23.130 --> 00:32:25.810
So let me draw its graph.
00:32:25.810 --> 00:32:27.680
This is important picture.
00:32:30.190 --> 00:32:32.789
So here is time.
00:32:32.789 --> 00:32:33.580
Here is population.
00:32:36.120 --> 00:32:38.910
Here's, maybe it started there.
00:32:38.910 --> 00:32:40.160
This is times 0.
00:32:46.730 --> 00:32:50.220
And now I want to graph this.
00:32:50.220 --> 00:32:54.230
I want to graph that function.
00:32:56.390 --> 00:33:01.420
Really, this is where
we're going somewhere.
00:33:01.420 --> 00:33:06.790
What happens for a long time?
00:33:06.790 --> 00:33:09.890
At t equal infinity, what
happens to the population?
00:33:12.340 --> 00:33:15.500
Does it grow, like e to the t?
00:33:15.500 --> 00:33:20.450
Just remember the examples here.
00:33:20.450 --> 00:33:23.940
We had a growth like e to the t.
00:33:23.940 --> 00:33:29.540
We had a growth faster than e
to the t that actually blew up.
00:33:29.540 --> 00:33:31.750
What about this guy?
00:33:31.750 --> 00:33:38.920
What will happen as t goes to
infinity with that population?
00:33:38.920 --> 00:33:39.950
It goes to?
00:33:39.950 --> 00:33:41.320
AUDIENCE: A over b.
00:33:41.320 --> 00:33:42.320
PROFESSOR: A over b.
00:33:42.320 --> 00:33:43.840
A over b.
00:33:43.840 --> 00:33:46.080
That's the key number
in the whole thing.
00:33:49.660 --> 00:33:55.130
It keeps growing, but it
never passes a over b.
00:33:55.130 --> 00:33:59.290
This is y at infinity.
00:33:59.290 --> 00:34:01.830
That's the final population.
00:34:01.830 --> 00:34:03.500
So how does it do this?
00:34:03.500 --> 00:34:09.320
If I draw this graph-- and
what about negative time?
00:34:09.320 --> 00:34:11.199
Let's go backwards in time.
00:34:14.020 --> 00:34:18.880
What is it at t
equals minus infinity?
00:34:18.880 --> 00:34:22.690
Then you really see
the whole curve.
00:34:22.690 --> 00:34:27.495
At t equal minus infinity,
what is this doing?
00:34:27.495 --> 00:34:28.370
AUDIENCE: 0 infinity.
00:34:28.370 --> 00:34:29.148
PROFESSOR: It's 0.
00:34:29.148 --> 00:34:29.648
Good.
00:34:29.648 --> 00:34:30.148
Good.
00:34:30.148 --> 00:34:31.260
Good.
00:34:31.260 --> 00:34:34.889
T equal minus infinity,
this is enormous.
00:34:34.889 --> 00:34:36.449
This is blowing up.
00:34:36.449 --> 00:34:38.110
It's in the denominator.
00:34:38.110 --> 00:34:39.540
We're dividing by it.
00:34:39.540 --> 00:34:41.710
So the whole thing
is going to 0.
00:34:41.710 --> 00:34:45.920
So here's what the
logistic curve looks like.
00:34:45.920 --> 00:34:47.370
It creeps up.
00:34:47.370 --> 00:34:51.580
And it's beautifully, there's
a point of symmetry here.
00:34:51.580 --> 00:34:54.489
The growth is increasing here.
00:34:54.489 --> 00:34:58.140
And then, as a point of
inflection you could say,
00:34:58.140 --> 00:35:01.370
growth is bending
upwards for a while.
00:35:01.370 --> 00:35:05.150
At this point, it starts
bending downwards.
00:35:05.150 --> 00:35:08.440
From that point on, ooh,
let's see if I can draw it.
00:35:08.440 --> 00:35:11.100
It'll get closer, and
exponentially close.
00:35:11.660 --> 00:35:13.160
That wasn't a bad picture.
00:35:17.370 --> 00:35:20.090
The population here is half way.
00:35:20.090 --> 00:35:24.280
Here, the population, the
final population, is a over b.
00:35:24.280 --> 00:35:28.720
And just by beautiful symmetry,
the population here is a 1/2
00:35:28.720 --> 00:35:31.200
of a over b.
00:35:31.200 --> 00:35:32.460
At this point.
00:35:35.440 --> 00:35:38.210
If this was the actual
population of the world
00:35:38.210 --> 00:35:46.400
we live in-- I think we're
pretty close to this point.
00:35:46.400 --> 00:35:49.575
I believe, well, of course,
nobody knows the numbers,
00:35:49.575 --> 00:35:54.730
unfortunately, because
the model isn't perfect.
00:35:54.730 --> 00:36:02.160
If the model was perfect, then
we could just takes the census
00:36:02.160 --> 00:36:04.320
and we would know a and b.
00:36:04.320 --> 00:36:11.470
But the model isn't that great.
00:36:11.470 --> 00:36:17.330
But it's sort of, we're at
a very interesting time,
00:36:17.330 --> 00:36:20.390
close to a very
interesting time.
00:36:20.390 --> 00:36:24.780
I believe that with reasonable
numbers, this a over b
00:36:24.780 --> 00:36:27.310
might be maybe 12 billion.
00:36:31.590 --> 00:36:36.300
And we might be, I think we're
a little above six billion.
00:36:36.300 --> 00:36:37.410
I think so.
00:36:37.410 --> 00:36:39.870
So we're a little bit past it.
00:36:39.870 --> 00:36:42.730
This is now.
00:36:42.730 --> 00:36:45.880
This is halfway.
00:36:45.880 --> 00:36:47.510
That's the halfway point.
00:36:47.510 --> 00:36:49.490
It's perfectly symmetric.
00:36:49.490 --> 00:36:51.035
It's called an S curve.
00:36:55.350 --> 00:37:05.890
And many, many equations in
math biology involve S curves.
00:37:05.890 --> 00:37:12.324
So math biology often gives
rise, with simple models,
00:37:12.324 --> 00:37:15.470
to a kind of problem
we've had here
00:37:15.470 --> 00:37:19.380
with a quadratic term
slowing things down.
00:37:19.380 --> 00:37:24.040
Enzymes, all kinds of.
00:37:24.040 --> 00:37:26.730
Ordinary differential
equations are core ideas
00:37:26.730 --> 00:37:33.230
in a lot of topics, lot
of areas of science.
00:37:33.230 --> 00:37:35.490
OK.
00:37:35.490 --> 00:37:39.550
Do I want to say more about
the logistic equation?
00:37:39.550 --> 00:37:44.450
I guess I do want to
distinguish one thing.
00:37:44.450 --> 00:37:46.000
Yeah.
00:37:46.000 --> 00:37:49.730
One thing about logistic
equations and will of course
00:37:49.730 --> 00:37:51.970
come back to this.
00:37:51.970 --> 00:37:52.470
OK.
00:37:52.470 --> 00:37:54.222
Let me look at that
logistic equation.
00:37:56.900 --> 00:37:58.110
Here's my equation.
00:38:00.680 --> 00:38:02.600
So I've managed to solve it.
00:38:02.600 --> 00:38:03.130
Fine.
00:38:03.130 --> 00:38:04.100
Great.
00:38:04.100 --> 00:38:05.460
Even graph it.
00:38:05.460 --> 00:38:09.000
But let me come back
to the question,
00:38:09.000 --> 00:38:12.520
suppose I just look at.
00:38:12.520 --> 00:38:19.800
I can see two constant
solutions, two steady states,
00:38:19.800 --> 00:38:24.210
two solutions where
the derivative is 0.
00:38:24.210 --> 00:38:26.260
So nothing will happen.
00:38:26.260 --> 00:38:33.910
So in other words, I want to set
this thing set to 0 equal to 0
00:38:33.910 --> 00:38:37.390
to find steady solutions.
00:38:37.390 --> 00:38:39.850
Steady means the
derivative is 0.
00:38:39.850 --> 00:38:41.930
So this side has to be 0.
00:38:41.930 --> 00:38:50.820
So what are the two possible
steady states where, if y of 0
00:38:50.820 --> 00:38:53.990
is there, it'll stay there?
00:38:53.990 --> 00:38:54.830
AUDIENCE: 0.
00:38:54.830 --> 00:38:55.830
PROFESSOR: 0.
00:38:55.830 --> 00:38:58.510
Y equals 0 is one.
00:38:58.510 --> 00:38:59.730
And the other?
00:38:59.730 --> 00:39:00.960
AUDIENCE: A over b.
00:39:00.960 --> 00:39:02.880
PROFESSOR: Is a over b.
00:39:02.880 --> 00:39:04.105
So steady equal to 0.
00:39:04.105 --> 00:39:05.830
And I get two steady states.
00:39:05.830 --> 00:39:08.890
Let me call them
capital Y equals 0
00:39:08.890 --> 00:39:12.330
because that's certainly 0
of, if we have 0 population,
00:39:12.330 --> 00:39:14.750
we'll never move.
00:39:14.750 --> 00:39:20.090
Or setting this to
0, ay is by squared
00:39:20.090 --> 00:39:25.480
cancel y's divide by b a over b.
00:39:25.480 --> 00:39:29.350
So the two steady
states are here.
00:39:29.350 --> 00:39:33.220
That's a steady state and
that's a steady state.
00:39:35.960 --> 00:39:39.020
Those are the only
two in this problem.
00:39:39.020 --> 00:39:41.760
You see how easy that was
to find the steady states?
00:39:41.760 --> 00:39:44.510
That's an important thing to do.
00:39:44.510 --> 00:39:48.640
And then the other important
thing to do is to decide,
00:39:48.640 --> 00:39:53.690
are those steady states stable?
00:39:53.690 --> 00:39:56.470
When the population's
near a steady state,
00:39:56.470 --> 00:40:00.010
does it approach that, does
it go toward that steady state
00:40:00.010 --> 00:40:02.480
or away?
00:40:02.480 --> 00:40:03.860
So what's the answer?
00:40:03.860 --> 00:40:10.690
For this steady state, that
steady state, y is a over b.
00:40:10.690 --> 00:40:13.060
Is that stable or unstable?
00:40:13.060 --> 00:40:15.420
So I'll write the word stable.
00:40:15.420 --> 00:40:20.267
And I'm prepared to put in "un,"
unstable, if you want me to.
00:40:24.470 --> 00:40:26.840
This is a key, key idea.
00:40:26.840 --> 00:40:28.700
And with nonlinear
equations, you
00:40:28.700 --> 00:40:34.140
can answer this stability
stuff without formulas.
00:40:34.140 --> 00:40:35.030
Without formulas.
00:40:35.030 --> 00:40:36.000
That's the nice thing.
00:40:36.000 --> 00:40:39.090
And then that comes
in a later class.
00:40:39.090 --> 00:40:41.490
But here's a perfect example.
00:40:41.490 --> 00:40:47.630
So do we approach this
answer or do we leave it?
00:40:47.630 --> 00:40:52.090
We approach it, the solutions.
00:40:52.090 --> 00:40:54.930
This is stable, yes.
00:40:54.930 --> 00:40:59.220
And here's the other
stationary point,
00:40:59.220 --> 00:41:07.590
capital Y. The other steady
state is that nothing happens.
00:41:07.590 --> 00:41:17.790
So now if I'm close to that, if
y is a little number, like 2,
00:41:17.790 --> 00:41:23.100
will that 2 drop to 0, will
it approach this steady state,
00:41:23.100 --> 00:41:24.914
or will it leave it?
00:41:24.914 --> 00:41:25.840
AUDIENCE: Leave it.
00:41:25.840 --> 00:41:26.740
PROFESSOR: Leave it.
00:41:26.740 --> 00:41:28.245
So this steady state is.
00:41:28.245 --> 00:41:29.110
AUDIENCE: Unstable.
00:41:29.110 --> 00:41:31.075
PROFESSOR: Unstable.
00:41:31.075 --> 00:41:31.575
Unstable.
00:41:36.610 --> 00:41:37.580
Right.
00:41:37.580 --> 00:41:39.470
Right.
00:41:39.470 --> 00:41:41.610
With linear equations,
we really only
00:41:41.610 --> 00:41:44.590
had one steady state, like 0.
00:41:44.590 --> 00:41:47.480
Once it started, it
took off forever.
00:41:47.480 --> 00:41:50.400
Here, it doesn't
go infinitely high.
00:41:53.640 --> 00:41:58.700
It bends down again
to that limit,
00:41:58.700 --> 00:42:02.071
that carrying capacity is
what it's called, a over b.
00:42:04.777 --> 00:42:08.560
I guess I hope you
think a nonlinear
00:42:08.560 --> 00:42:12.720
equation like got a
little more to it.
00:42:12.720 --> 00:42:15.760
Little more interesting, but
a little more complicated,
00:42:15.760 --> 00:42:17.533
than linear equations.
00:42:17.533 --> 00:42:18.032
Yep.
00:42:18.032 --> 00:42:18.886
Yep.
00:42:18.886 --> 00:42:19.385
Yep.
00:42:20.170 --> 00:42:24.900
And similarly, the
rocket equation, we
00:42:24.900 --> 00:42:29.200
could at the right time soon in
the course, ask the same thing,
00:42:29.200 --> 00:42:32.550
a rocket equation was
something like that.
00:42:32.550 --> 00:42:34.140
What are the steady states?
00:42:34.140 --> 00:42:35.060
Are they stable?
00:42:35.060 --> 00:42:36.720
Are they unstable?
00:42:36.720 --> 00:42:38.780
Can you find a formula?
00:42:38.780 --> 00:42:39.280
Here.
00:42:39.280 --> 00:42:40.800
This.
00:42:40.800 --> 00:42:41.530
We got a formula.
00:42:44.330 --> 00:42:48.440
And there are other nonlinear
equations, which we'll see.
00:42:51.650 --> 00:42:52.150
OK.
00:42:52.150 --> 00:42:54.480
I could create more
separable equations,
00:42:54.480 --> 00:42:59.950
but I guess I hope that you
see with separable equations,
00:42:59.950 --> 00:43:03.500
you just separate them
and integrate a y integral
00:43:03.500 --> 00:43:04.600
and a t integral.
00:43:05.380 --> 00:43:11.300
Is that OK any question on
this nonlinear separable stuff?
00:43:16.490 --> 00:43:19.920
Differential equations
courses and the subject
00:43:19.920 --> 00:43:25.630
tends to be types of
equations as can solve.
00:43:25.630 --> 00:43:30.190
And then there are a hell of a
lot of equations that are not
00:43:30.190 --> 00:43:33.880
on anybody's list, where
you could maybe solve them
00:43:33.880 --> 00:43:39.320
by an infinite series, but
not by functions that we know.
00:43:39.320 --> 00:43:40.720
OK.
00:43:40.720 --> 00:43:46.380
I'm ready to do the
other topic for today.
00:43:46.380 --> 00:43:52.620
It's the topic that I
left incomplete on Monday.
00:43:52.620 --> 00:43:54.890
So I'm staying with
first order equations,
00:43:54.890 --> 00:43:59.060
but actually this topic is
essential for second order
00:43:59.060 --> 00:43:59.630
equations.
00:43:59.630 --> 00:44:05.020
So I'm going to
topic two for today.
00:44:05.020 --> 00:44:10.070
So topic two will
involve complex numbers.
00:44:10.070 --> 00:44:14.740
So we have to deal
with complex numbers.
00:44:14.740 --> 00:44:22.370
And the purpose of introducing
these complex numbers
00:44:22.370 --> 00:44:28.370
is to deal with what we met
last time when the right hand
00:44:28.370 --> 00:44:32.350
side, the forcing
term, was a cosine.
00:44:32.350 --> 00:44:37.640
Typical alternating current,
oscillating, rotating,
00:44:37.640 --> 00:44:38.330
rotation.
00:44:38.330 --> 00:44:42.110
All these things
produce trig functions.
00:44:42.110 --> 00:44:46.960
Maybe rotation is more of
a mechanical engineering
00:44:46.960 --> 00:44:49.220
phenomenon.
00:44:49.220 --> 00:44:52.100
Alternating current more
of an EE phenomenon.
00:44:52.100 --> 00:44:53.350
But they're always there.
00:44:54.231 --> 00:44:58.000
And what was the point?
00:44:58.000 --> 00:45:04.530
The point was we had
some linear equation,
00:45:04.530 --> 00:45:13.860
and we had some forcing by
something like cos omega t.
00:45:13.860 --> 00:45:19.510
Or it could be A cos omega
t and B sine omega t.
00:45:25.530 --> 00:45:31.970
Either just cosine alone, or
maybe these come together.
00:45:31.970 --> 00:45:39.290
And then the
solution was y equals
00:45:39.290 --> 00:45:41.960
some combination
of those same guys.
00:45:49.690 --> 00:45:53.980
In other words,
what I'm saying is
00:45:53.980 --> 00:46:00.080
cosines are nice right
hand forcing functions.
00:46:00.080 --> 00:46:03.530
Fortunately, because we
see them all the time.
00:46:03.530 --> 00:46:07.710
But they do lead to
cosines and sines.
00:46:07.710 --> 00:46:10.020
I emphasized that last time.
00:46:10.020 --> 00:46:13.050
If we just have cosines
in the forcing function,
00:46:13.050 --> 00:46:16.190
we can't expect that
there's any damping,
00:46:16.190 --> 00:46:18.210
we can't expect only cosines.
00:46:18.210 --> 00:46:20.170
We have to expect some sines.
00:46:20.170 --> 00:46:26.540
In other words, we have to
deal with combinations of them.
00:46:26.540 --> 00:46:32.720
And the question is, how do you
understand cos omega t plus 3.
00:46:32.720 --> 00:46:35.540
Or let me take a first example.
00:46:35.540 --> 00:46:38.598
Example-- cos t plus sine t.
00:46:43.410 --> 00:46:46.100
That's a perfect example.
00:46:46.100 --> 00:46:50.900
So what is omega here in this
example that I'm starting with?
00:46:50.900 --> 00:46:51.400
AUDIENCE: 1.
00:46:51.400 --> 00:46:52.840
PROFESSOR: 1.
00:46:52.840 --> 00:46:55.360
So I just read off
the coefficient of t
00:46:55.360 --> 00:46:59.550
is 1, 1 hertz here.
00:46:59.550 --> 00:47:01.800
But we have got
this combination.
00:47:01.800 --> 00:47:06.780
And the question is, how do
we understand that cosine
00:47:06.780 --> 00:47:07.590
plus sine?
00:47:07.590 --> 00:47:10.410
Two very simple functions, but
they're added, unfortunately.
00:47:14.240 --> 00:47:16.930
And there's a much
better way to write this
00:47:16.930 --> 00:47:19.540
so you really see it.
00:47:19.540 --> 00:47:21.630
You really see this.
00:47:21.630 --> 00:47:24.510
That's called a sinusoid.
00:47:24.510 --> 00:47:28.380
And the rule that
want to focus on now
00:47:28.380 --> 00:47:34.250
is that everything of that kind,
of this kind, of this kind,
00:47:34.250 --> 00:47:40.240
of a cosine plus a sine, can
be compressed into one term.
00:47:40.240 --> 00:47:42.120
One term.
00:47:42.120 --> 00:47:46.640
Of course, it's got
to have two constants
00:47:46.640 --> 00:47:50.090
to choose because
that had an a and a b.
00:47:50.090 --> 00:47:51.540
This had an m and an n.
00:47:51.540 --> 00:47:53.700
This had a 1 and a 1.
00:47:53.700 --> 00:47:58.750
But the term I'm looking
for is some number R
00:47:58.750 --> 00:48:09.060
times a pure cosine of omega
t, but with a phase shift.
00:48:09.060 --> 00:48:11.510
So you see there are two
numbers here to choose.
00:48:11.510 --> 00:48:15.595
It's really like going
from rectangular to polar.
00:48:19.570 --> 00:48:21.590
Say in complex
numbers, let's just
00:48:21.590 --> 00:48:25.820
remember the first fact
about a complex number.
00:48:25.820 --> 00:48:32.460
If the real part is 3, and the
imaginary part is, let's say 2,
00:48:32.460 --> 00:48:36.760
then here's a complex
number, 3 plus 2 i.
00:48:39.320 --> 00:48:41.940
So this was the real axis.
00:48:41.940 --> 00:48:44.450
This was the imaginary axis.
00:48:44.450 --> 00:48:48.530
I went along 3, I went up
2, I got to that number.
00:48:48.530 --> 00:48:50.450
There it is.
00:48:50.450 --> 00:48:54.282
I plotted the number 3 plus
2 i in the complex plane.
00:48:57.060 --> 00:49:00.820
And for me, that number
3 plus and so on,
00:49:00.820 --> 00:49:02.360
really saying
something important.
00:49:02.360 --> 00:49:06.150
And maybe it's not entirely new.
00:49:06.150 --> 00:49:09.360
I'm saying something important
about complex numbers,
00:49:09.360 --> 00:49:11.570
this is their rectangular form.
00:49:11.570 --> 00:49:12.750
Something plus something.
00:49:16.410 --> 00:49:20.700
That form is nice to add
to another complex number.
00:49:20.700 --> 00:49:27.115
If I added 3 plus 2 i to 1
plus i, what would I get?
00:49:27.115 --> 00:49:28.480
AUDIENCE: 4 plus 3 i.
00:49:28.480 --> 00:49:31.020
PROFESSOR: 4 plus 3 i.
00:49:31.020 --> 00:49:36.560
But if I multiply,
multiply, 3 plus 2 i times,
00:49:36.560 --> 00:49:38.450
let's say I square it.
00:49:38.450 --> 00:49:43.260
I multiply 3 plus
2 i by 3 plus 2 i.
00:49:43.260 --> 00:49:45.940
What do I get?
00:49:45.940 --> 00:49:50.280
If I do it with this
rectangular form, I get a mess.
00:49:50.280 --> 00:49:53.600
I can't see what's happening.
00:49:53.600 --> 00:49:55.970
It's the same over here.
00:49:55.970 --> 00:49:59.880
This is like having a 1
and a 1, with an addition.
00:49:59.880 --> 00:50:05.320
This is like a polar
form where it's one term.
00:50:05.320 --> 00:50:06.110
OK.
00:50:06.110 --> 00:50:08.440
So let me answer the
question here and then let
00:50:08.440 --> 00:50:09.930
me answer the question there.
00:50:09.930 --> 00:50:15.740
And then you've got a good shot
at what complex numbers can do,
00:50:15.740 --> 00:50:21.024
and why we like the polar form
for squaring, for multiplying,
00:50:21.024 --> 00:50:21.565
for dividing.
00:50:24.510 --> 00:50:26.820
What's the polar form?
00:50:26.820 --> 00:50:28.830
Well, I'm using
that word "polar"
00:50:28.830 --> 00:50:31.850
in the same way we
use polar coordinates.
00:50:31.850 --> 00:50:35.170
What are the polar
coordinates of this point?
00:50:35.170 --> 00:50:39.202
They're the radial
distance, which is what?
00:50:39.202 --> 00:50:40.670
So what's that distance?
00:50:40.670 --> 00:50:43.590
That's the R you could say.
00:50:43.590 --> 00:50:46.790
It corresponds to this R here.
00:50:46.790 --> 00:50:49.790
So I'm just using Pythagoras.
00:50:49.790 --> 00:50:54.132
That hypotenuse is what?
00:50:54.132 --> 00:50:55.380
AUDIENCE: Square root of 13.
00:50:55.380 --> 00:50:56.350
PROFESSOR: Square root of 13.
00:50:56.350 --> 00:50:56.960
Thanks.
00:50:56.960 --> 00:51:00.340
9 plus 4, square root of 13.
00:51:00.340 --> 00:51:03.350
And what's the
other number that's
00:51:03.350 --> 00:51:06.830
locating this in
polar coordinates?
00:51:06.830 --> 00:51:08.290
The angle.
00:51:08.290 --> 00:51:09.230
And the angle.
00:51:09.230 --> 00:51:11.500
What can we say
about that angle?
00:51:11.500 --> 00:51:19.250
Let's call it phi
is-- what's the angle?
00:51:21.920 --> 00:51:26.310
Well, it's some number.
00:51:26.310 --> 00:51:31.220
It's between 0 and pi
over 2, I'm sure of that.
00:51:31.220 --> 00:51:33.130
What do I know about that angle?
00:51:41.780 --> 00:51:45.270
I know that this
is 2 and this is 3.
00:51:45.270 --> 00:51:47.270
So that's telling me the angle.
00:51:47.270 --> 00:51:50.170
Well, what is that really
telling me immediately?
00:51:50.170 --> 00:51:51.590
It's telling me the.
00:51:51.590 --> 00:51:52.340
AUDIENCE: Tangent.
00:51:52.340 --> 00:51:54.250
PROFESSOR: Tangent of the angle.
00:51:54.250 --> 00:52:02.730
So the tangent of the
angle is 2 over 3.
00:52:02.730 --> 00:52:05.745
And the magnitude is
square root of 13.
00:52:08.320 --> 00:52:08.820
OK.
00:52:11.710 --> 00:52:16.130
So those beautiful
numbers, 2 and 3,
00:52:16.130 --> 00:52:19.500
have become a little weirder.
00:52:19.500 --> 00:52:23.120
Square root of 13,
inverse tangent of 2/3.
00:52:23.120 --> 00:52:24.880
You could say, well,
that's not so nice.
00:52:29.364 --> 00:52:30.600
What was I going to do?
00:52:30.600 --> 00:52:34.030
I was going to try
squaring that number.
00:52:34.030 --> 00:52:40.090
So if I square 3 plus 2 i, or
if I take the 10th power of 3
00:52:40.090 --> 00:52:47.380
plus 2 i, or the exponential,
all these things,
00:52:47.380 --> 00:52:49.800
then I'm happy with
polar coordinates.
00:52:49.800 --> 00:52:55.450
Like, what would be the
magnitude of the square?
00:52:55.450 --> 00:52:59.860
And where will the square
of that number, so I want
00:52:59.860 --> 00:53:04.820
to put in 3 plus 2 i squared,
which I can figure out
00:53:04.820 --> 00:53:10.730
in rectangular, of
course-- a 9, and 6 i,
00:53:10.730 --> 00:53:14.020
or 12 i, or 4 i squared,
stuff like that.
00:53:16.910 --> 00:53:18.710
It's not pleasant.
00:53:18.710 --> 00:53:24.100
What's the magnitude,
what's the R for this guy?
00:53:24.100 --> 00:53:27.761
What's the size of
that number squared?
00:53:27.761 --> 00:53:28.260
Yes?
00:53:28.260 --> 00:53:29.269
Say that again.
00:53:29.269 --> 00:53:29.810
AUDIENCE: 13.
00:53:29.810 --> 00:53:31.210
PROFESSOR: 13.
00:53:31.210 --> 00:53:33.000
Right.
00:53:33.000 --> 00:53:37.160
I just have to square this
square root so I get 13.
00:53:37.160 --> 00:53:43.935
And the angle will be, what's
the angle for the square there?
00:53:43.935 --> 00:53:45.080
I don't want a number.
00:53:50.510 --> 00:53:52.720
I guess I'm just doing this.
00:53:52.720 --> 00:53:58.880
R e to the i phi
squared is R squared.
00:53:58.880 --> 00:54:00.540
And what's the angle here?
00:54:05.680 --> 00:54:11.370
E to the i phi squared
is e to the 2 i phi.
00:54:11.370 --> 00:54:13.740
It's the angle doubled.
00:54:13.740 --> 00:54:16.870
E to the 2 i phi.
00:54:16.870 --> 00:54:20.680
The angle just went
from phi to 2 phi.
00:54:20.680 --> 00:54:23.790
The lengths went from
square root of 13 to 13.
00:54:26.320 --> 00:54:30.580
Squaring, multiplying is
nice with complex numbers.
00:54:30.580 --> 00:54:35.440
Maybe can I before I go on
and on about complex numbers,
00:54:35.440 --> 00:54:40.430
I should ask you, how many
know all this already?
00:54:40.430 --> 00:54:42.761
Complex numbers are familiar?
00:54:42.761 --> 00:54:43.260
Mostly.
00:54:46.450 --> 00:54:48.131
Correctly, with a wiggle.
00:54:48.131 --> 00:54:48.630
OK.
00:54:51.800 --> 00:54:56.470
I won't go more about
complex numbers.
00:54:56.470 --> 00:54:59.070
Let me come back to
my question here.
00:55:02.110 --> 00:55:05.520
Let me come back
to the application.
00:55:05.520 --> 00:55:08.390
So here it is with
complex numbers.
00:55:08.390 --> 00:55:10.270
Here it is with sinusoids.
00:55:10.270 --> 00:55:13.850
And the little
beautiful bit of math
00:55:13.850 --> 00:55:18.060
is that the sinusoid
question goes completely
00:55:18.060 --> 00:55:22.020
parallel to the complex
number question.
00:55:22.020 --> 00:55:25.260
So you have an idea on
those complex numbers.
00:55:25.260 --> 00:55:27.050
We'll see them again.
00:55:27.050 --> 00:55:29.160
Let me go to this.
00:55:29.160 --> 00:55:34.290
So I want this to be
the same as this, OK.
00:55:34.290 --> 00:55:37.792
Maybe I'm going to have to
use a new board for this.
00:55:37.792 --> 00:55:40.820
Can I start a new board?
00:55:40.820 --> 00:55:46.660
So I want cos t
plus sine t to be
00:55:46.660 --> 00:55:53.910
some number R times
cosine of t 1.
00:55:53.910 --> 00:55:59.990
I can see omega's 1, so I just
put to 1 minus some angle.
00:55:59.990 --> 00:56:02.100
OK.
00:56:02.100 --> 00:56:05.830
And I want to choose R and
phi to make that right.
00:56:05.830 --> 00:56:08.100
You see what I like about it?
00:56:08.100 --> 00:56:13.360
This tells me the magnitude
of the oscillation.
00:56:13.360 --> 00:56:19.080
It tells me how
loud the station is.
00:56:19.080 --> 00:56:22.140
When I see cos t and,
separately, sine t,
00:56:22.140 --> 00:56:24.990
or I might see 3
cos t and 2 sine t.
00:56:27.990 --> 00:56:31.790
3 cos t is a cosine curve.
00:56:31.790 --> 00:56:35.810
2 sine t is a sine
curve shifted by 90.
00:56:35.810 --> 00:56:38.395
I put them together, it
bumps, it bumps, bumps.
00:56:41.100 --> 00:56:43.570
Not completely clear.
00:56:43.570 --> 00:56:48.260
It seems to me just beautiful
that if I put together
00:56:48.260 --> 00:56:54.870
a cosine curve that we
know, that starts at 0,
00:56:54.870 --> 00:57:00.380
with a sine curve
that starts at 0,
00:57:00.380 --> 00:57:04.340
the combination
is a cosine curve.
00:57:04.340 --> 00:57:06.230
Isn't that nice?
00:57:06.230 --> 00:57:07.940
I mean, you know,
that sometimes math
00:57:07.940 --> 00:57:10.210
gets worse and worse
whatever you do.
00:57:10.210 --> 00:57:14.750
But this is really nice that
we can put the two into one.
00:57:14.750 --> 00:57:20.470
But you see, it's going
to-- well, let's do it.
00:57:20.470 --> 00:57:24.440
What would R and phi be here?
00:57:24.440 --> 00:57:29.280
So I'll use a trig fact here.
00:57:29.280 --> 00:57:32.160
A cosine of a
difference of angles,
00:57:32.160 --> 00:57:37.536
so this is R times cosine
t, do you member this?
00:57:37.536 --> 00:57:41.870
This was the whole point
of going to high school.
00:57:41.870 --> 00:57:45.650
Plus sine t sine phi.
00:57:52.810 --> 00:57:59.290
So now, how do I get R and phi?
00:57:59.290 --> 00:58:03.840
I use the same idea
that worked last time.
00:58:03.840 --> 00:58:08.220
I match the cosine terms
and I match the sine terms.
00:58:08.220 --> 00:58:10.680
So the cosine t has a 1.
00:58:10.680 --> 00:58:14.530
1 cosine t is R cos phi.
00:58:18.120 --> 00:58:21.360
That's what's
multiplying cosine t.
00:58:21.360 --> 00:58:24.540
And the sine t has a 1.
00:58:24.540 --> 00:58:28.250
And that has to agree
with R sine phi.
00:58:34.770 --> 00:58:41.690
So I'm in business if I
solve those two equations.
00:58:44.770 --> 00:58:48.390
And well, they're
not linear equations.
00:58:48.390 --> 00:58:49.790
But I can solve them.
00:58:49.790 --> 00:58:53.740
Of course, the one fact
that you never forget
00:58:53.740 --> 00:58:56.590
is that sine squared
plus cosine squared is 1.
00:58:56.590 --> 00:58:57.840
Right?
00:58:57.840 --> 00:59:01.510
So if I square that one, and
square that one, and add,
00:59:01.510 --> 00:59:02.670
what will I get?
00:59:02.670 --> 00:59:07.110
1 squared and 1 squared will
be 2, on the left hand side.
00:59:07.110 --> 00:59:11.400
On the right hand side, I'll
have R squared cos squared,
00:59:11.400 --> 00:59:17.870
R squared cos squared, and
plus R squared sine squared.
00:59:17.870 --> 00:59:18.964
And what's that?
00:59:21.646 --> 00:59:25.436
What's R squared cosine squared
plus r squared sine squared?
00:59:25.436 --> 00:59:26.390
AUDIENCE: R squared.
00:59:26.390 --> 00:59:28.610
PROFESSOR: It's just R squared.
00:59:28.610 --> 00:59:33.470
So all that added
up to R squared.
00:59:33.470 --> 00:59:37.370
In other words, it's just
like polar coordinates.
00:59:37.370 --> 00:59:39.010
R is the square root of 2.
00:59:43.900 --> 00:59:50.870
That's telling us the
magnitude of the response.
00:59:50.870 --> 00:59:53.060
Square root of 2.
00:59:53.060 --> 00:59:57.390
You see, it's just
like complex numbers.
00:59:57.390 --> 01:00:00.170
It's like the cosine
gave us a real part
01:00:00.170 --> 01:00:02.800
and the sine gave us
an imaginary part.
01:00:02.800 --> 01:00:08.200
And R was the hypotenuse.
01:00:08.200 --> 01:00:12.400
And that's really nice.
01:00:12.400 --> 01:00:13.930
So R is the square root of 2.
01:00:13.930 --> 01:00:14.640
OK.
01:00:14.640 --> 01:00:18.650
Now, the angle is
never quite as nice.
01:00:18.650 --> 01:00:22.450
But how can we get something
about an angle out of there?
01:00:22.450 --> 01:00:25.510
All we could get
in this case here
01:00:25.510 --> 01:00:27.710
was the tangent of the angle.
01:00:27.710 --> 01:00:30.350
And I'll be happy
with that again here
01:00:30.350 --> 01:00:35.750
because it's a totally
parallel question.
01:00:35.750 --> 01:00:39.430
How am I going to get
the tangent of the angle?
01:00:45.220 --> 01:00:48.120
What do I have?
01:00:48.120 --> 01:00:51.110
From these two equations,
I want to eliminate R.
01:00:51.110 --> 01:00:52.390
So how do I eliminate R?
01:00:57.540 --> 01:00:58.960
What do I do?
01:00:58.960 --> 01:00:59.940
Divide.
01:00:59.940 --> 01:01:00.970
Divide.
01:01:00.970 --> 01:01:04.180
I guess if I want
tangent sine over cosine,
01:01:04.180 --> 01:01:08.100
I'll divide this one in the
top by this one in the bottom.
01:01:08.100 --> 01:01:09.780
So I take the ratio.
01:01:09.780 --> 01:01:11.900
That'll cancel the Rs perfectly.
01:01:11.900 --> 01:01:13.650
It'll leave me with 10 phi.
01:01:13.650 --> 01:01:16.570
And here it happens to be 1.
01:01:16.570 --> 01:01:17.070
OK.
01:01:19.890 --> 01:01:21.840
So what have I learned?
01:01:21.840 --> 01:01:29.870
I've learned that when
these two add up together,
01:01:29.870 --> 01:01:33.250
they equal what?
01:01:33.250 --> 01:01:36.220
R square root of 2.
01:01:36.220 --> 01:01:37.350
You see how easy it is.
01:01:37.350 --> 01:01:39.600
Square root of 2 came from
the square root of 1 plus.
01:01:39.600 --> 01:01:42.350
It's like Pythagoras.
01:01:42.350 --> 01:01:45.290
Pythagoras going
in circles, really.
01:01:45.290 --> 01:01:52.190
Times the cosine of t minus.
01:01:52.190 --> 01:01:53.210
And what is phi?
01:01:55.750 --> 01:01:59.307
Its tangent is 1, so
what's the angle phi?
01:01:59.307 --> 01:02:00.140
AUDIENCE: Pi over 4.
01:02:00.140 --> 01:02:01.911
PROFESSOR: Pi over 4.
01:02:01.911 --> 01:02:02.411
Right.
01:02:06.310 --> 01:02:09.490
So that's the
sinusoidal identity
01:02:09.490 --> 01:02:13.380
when the numbers are 1 and 1.
01:02:13.380 --> 01:02:15.135
But you saw the general rule.
01:02:19.870 --> 01:02:23.640
Let me just take it.
01:02:23.640 --> 01:02:31.200
Suppose this is the output, and
cos omega t plus n sine omega
01:02:31.200 --> 01:02:32.440
t.
01:02:32.440 --> 01:02:33.940
What is the gain?
01:02:33.940 --> 01:02:36.510
What's the magnitude,
the amplitude,
01:02:36.510 --> 01:02:41.780
the loudness of the volume in
this when I'm tuning the radio?
01:02:41.780 --> 01:02:44.790
What's the R for this guy?
01:02:44.790 --> 01:02:45.545
What's this R?
01:02:49.310 --> 01:02:53.360
If we just follow the same idea.
01:02:53.360 --> 01:02:58.010
So if we have m times a
cosine and n times a sine,
01:02:58.010 --> 01:02:59.200
what's your guess?
01:02:59.200 --> 01:03:05.190
What's your guess
for R, the magnitude?
01:03:05.190 --> 01:03:08.320
I'm guessing a
square root of what?
01:03:12.560 --> 01:03:13.180
Yeah?
01:03:13.180 --> 01:03:14.030
You got.
01:03:14.030 --> 01:03:15.600
What is it?
01:03:15.600 --> 01:03:16.440
n squared--
01:03:16.440 --> 01:03:17.387
[INTERPOSING VOICES]
01:03:17.387 --> 01:03:18.470
PROFESSOR: Plus n squared.
01:03:18.470 --> 01:03:19.450
Way to go.
01:03:19.450 --> 01:03:20.810
M squared plus N squared.
01:03:23.370 --> 01:03:27.590
And the angle is
like the phase shift.
01:03:27.590 --> 01:03:30.040
I'm not great at
graphing, but let
01:03:30.040 --> 01:03:34.280
me try to go back to
my simple example.
01:03:34.280 --> 01:03:40.250
If I tried to add up on
the same graph cosine
01:03:40.250 --> 01:03:46.450
t, which would start from 1 and
drop to 0, go like that, right?
01:03:46.450 --> 01:03:48.920
Something like that
would be cosine.
01:03:48.920 --> 01:03:51.630
And now I want to
add sine t to that.
01:03:51.630 --> 01:03:56.830
So that climbs up to
1 back to 0, down.
01:03:56.830 --> 01:04:00.440
And now if I add those
two, this formula
01:04:00.440 --> 01:04:04.730
is telling me that
it comes out neat.
01:04:04.730 --> 01:04:06.490
Neatly.
01:04:06.490 --> 01:04:11.620
That one plus that one is
another sinusoid with height
01:04:11.620 --> 01:04:12.740
square root of 2.
01:04:12.740 --> 01:04:17.020
If I had different
chalk, I've got
01:04:17.020 --> 01:04:20.690
at least a little bit different.
01:04:20.690 --> 01:04:21.800
But does it start here?
01:04:21.800 --> 01:04:22.600
Of course not.
01:04:22.600 --> 01:04:24.570
It starts here, I guess.
01:04:24.570 --> 01:04:26.120
But it goes up, right?
01:04:26.120 --> 01:04:28.040
Because this comes down,
but this is going up.
01:04:28.040 --> 01:04:32.430
All together, it's up
to, where is the peak?
01:04:32.430 --> 01:04:34.360
Where is the peak on the sum?
01:04:34.360 --> 01:04:37.500
So I'm adding, everybody
sees what I'm doing?
01:04:37.500 --> 01:04:40.780
I'm adding a cosine
curve and a sine curve.
01:04:40.780 --> 01:04:42.210
And it goes up.
01:04:42.210 --> 01:04:44.760
And where does it peak?
01:04:44.760 --> 01:04:48.565
What angle is it
going to peek at?
01:04:48.565 --> 01:04:52.056
What's the biggest
value this gets to?
01:04:52.056 --> 01:04:54.450
AUDIENCE: [INAUDIBLE].
01:04:54.450 --> 01:04:56.880
PROFESSOR: At pi
over 4, it'll peak.
01:04:56.880 --> 01:05:00.430
At pi over 4, it'll be the
cosine of 0, which is 1.
01:05:00.430 --> 01:05:04.720
It's height'll be the magnitude,
the gain, square root of 2.
01:05:04.720 --> 01:05:07.570
So it'll peak at
pi over 4, which
01:05:07.570 --> 01:05:09.540
is probably about there, right?
01:05:09.540 --> 01:05:17.220
Peak at pi over 4 and, I don't
know if I got it right frankly.
01:05:17.220 --> 01:05:20.240
I did my best.
01:05:20.240 --> 01:05:23.470
That's the sum.
01:05:23.470 --> 01:05:24.680
That right there.
01:05:27.750 --> 01:05:33.380
The first key point is
it's a perfect cosine.
01:05:33.380 --> 01:05:38.990
The second key point is
it's a shifted cosine.
01:05:38.990 --> 01:05:41.680
The third key point
is its magnitude
01:05:41.680 --> 01:05:45.990
is the square root of 1 squared
plus 1 squared, or n squared
01:05:45.990 --> 01:05:49.940
plus n squared, or a
squared plus b squared.
01:05:49.940 --> 01:05:53.812
So that's the
sinusoidal identity.
01:05:53.812 --> 01:06:02.890
A key identity and being able to
deal with forcing terms, source
01:06:02.890 --> 01:06:05.744
terms, that are sinusoids.
01:06:05.744 --> 01:06:06.244
OK.
01:06:09.690 --> 01:06:13.070
Now, I'm going to take
one more step since we
01:06:13.070 --> 01:06:18.440
have just like 10
minutes left, and let
01:06:18.440 --> 01:06:21.950
the number i get
in here properly.
01:06:21.950 --> 01:06:25.170
Get a complex number
to show up here.
01:06:25.170 --> 01:06:25.670
OK.
01:06:49.860 --> 01:06:51.570
Before I start on
this, let me recap.
01:06:58.660 --> 01:07:00.485
Let me recap today's lecture.
01:07:03.540 --> 01:07:08.480
It started with nonlinear
separable equations.
01:07:08.480 --> 01:07:13.710
And a great example was the
logistic equation up there,
01:07:13.710 --> 01:07:16.550
with the S curve.
01:07:16.550 --> 01:07:19.880
That took half the lecture.
01:07:19.880 --> 01:07:22.170
The second half
of the lecture has
01:07:22.170 --> 01:07:27.000
started with things
real with sinusoids
01:07:27.000 --> 01:07:29.585
that are combinations
of cosine and sine
01:07:29.585 --> 01:07:35.560
and has written them
in a one term way.
01:07:35.560 --> 01:07:40.340
And now I want to
get the same one term
01:07:40.340 --> 01:07:46.080
picture from using
complex numbers.
01:07:46.080 --> 01:07:48.520
OK.
01:07:48.520 --> 01:07:50.030
OK.
01:07:50.030 --> 01:07:52.440
And everything I
do would be based
01:07:52.440 --> 01:08:02.870
on this great fact from Euler
that e to the i omega t.
01:08:02.870 --> 01:08:06.110
The real part is cosine omega t.
01:08:06.110 --> 01:08:09.246
And the imaginary
part is sine omega t.
01:08:23.029 --> 01:08:26.479
That's a central formula.
01:08:26.479 --> 01:08:29.319
Let me draw it rather
than proving it.
01:08:29.319 --> 01:08:30.650
Let me draw what that means.
01:08:37.760 --> 01:08:39.340
I'm in the complex plane again.
01:08:39.340 --> 01:08:45.310
Real part is the cosine.
01:08:45.310 --> 01:08:48.765
The imaginary part is the sine.
01:08:53.319 --> 01:08:56.920
That number there
is e to the i omega
01:08:56.920 --> 01:09:02.939
t because it's got that real
part and that imaginary part.
01:09:02.939 --> 01:09:04.074
And what's its magnitude?
01:09:06.649 --> 01:09:13.140
What's the R, the polar
distance for cos omega t plus,
01:09:13.140 --> 01:09:17.990
for this number, which
is for this number?
01:09:17.990 --> 01:09:19.899
What's the hypotenuse here?
01:09:19.899 --> 01:09:22.147
Everybody knows.
01:09:22.147 --> 01:09:22.979
AUDIENCE: 1.
01:09:22.979 --> 01:09:23.920
PROFESSOR: 1.
01:09:23.920 --> 01:09:25.470
Hypotenuse is 1.
01:09:25.470 --> 01:09:28.380
Cos squared plus
sine squared is 1.
01:09:28.380 --> 01:09:34.150
So e to the i omega t is
on a circle of radius 1.
01:09:36.979 --> 01:09:41.850
That's the most important
circle in the complex world,
01:09:41.850 --> 01:09:43.810
the circle of radius 1.
01:09:43.810 --> 01:09:46.540
And all these points are on it.
01:09:46.540 --> 01:09:50.765
And their angles are omega t.
01:09:50.765 --> 01:09:53.770
And as t increases,
the angle increases,
01:09:53.770 --> 01:09:55.360
and you go around the circle.
01:09:55.360 --> 01:09:56.030
You've seen it.
01:09:58.780 --> 01:10:05.877
Physics couldn't live
without this model.
01:10:05.877 --> 01:10:06.377
OK.
01:10:10.130 --> 01:10:14.220
So that's basically
what we have to know.
01:10:14.220 --> 01:10:15.650
And now, how do we use it?
01:10:18.300 --> 01:10:23.030
Well, the idea is to
deal with the equation.
01:10:23.030 --> 01:10:26.160
Like, the equation I
had last time was dy,
01:10:26.160 --> 01:10:33.430
dt equals y plus cos t.
01:10:33.430 --> 01:10:39.200
That gave us some trouble
because the solution didn't
01:10:39.200 --> 01:10:42.990
just involve cosines,
it also involved sines.
01:10:42.990 --> 01:10:43.490
Yeah.
01:10:46.700 --> 01:10:54.240
So I want to write that equation
differently, in complex form.
01:10:54.240 --> 01:10:58.130
And this is the key point here.
01:10:58.130 --> 01:11:02.290
So I'm going to look
at the equation dz,
01:11:02.290 --> 01:11:09.410
dt equals z plus e to the i t.
01:11:09.410 --> 01:11:13.380
Well, I'll make that
cos omega t just
01:11:13.380 --> 01:11:24.220
to have a little more, the
units are better, everything's
01:11:24.220 --> 01:11:26.910
better if I have
a frequency there.
01:11:26.910 --> 01:11:30.600
Units of this are seconds
and the units of this
01:11:30.600 --> 01:11:31.639
are 1 over seconds.
01:11:36.630 --> 01:11:38.870
Now, question.
01:11:38.870 --> 01:11:41.270
What's the relation
between the solution
01:11:41.270 --> 01:11:44.820
z to that complex equation
and the solution y
01:11:44.820 --> 01:11:46.026
to that equation?
01:11:50.283 --> 01:11:52.000
Of course, they
have to be related,
01:11:52.000 --> 01:11:56.740
otherwise it was stupid to
move to this complex one.
01:11:56.740 --> 01:12:01.120
My claim is that complex
equation is easy to solve.
01:12:01.120 --> 01:12:05.270
And it gives us the answer
to the real equation.
01:12:05.270 --> 01:12:09.698
And what's the connection
between y and z?
01:12:09.698 --> 01:12:11.180
AUDIENCE: So y's the real part.
01:12:11.180 --> 01:12:13.269
PROFESSOR: Y is,
exactly, say it again.
01:12:13.269 --> 01:12:14.310
AUDIENCE: The real part--
01:12:14.310 --> 01:12:15.710
PROFESSOR: Of z.
01:12:15.710 --> 01:12:17.210
Y is the real part of z.
01:12:20.750 --> 01:12:23.290
So that gives us an idea.
01:12:23.290 --> 01:12:26.856
Solve this equation and
take it's real part.
01:12:30.730 --> 01:12:34.910
If I can solve this equation
without getting into cosine
01:12:34.910 --> 01:12:40.060
and sine separately and
matching, I can stay real.
01:12:40.060 --> 01:12:44.210
I solve the equation by
totally real methods up to now.
01:12:44.210 --> 01:12:48.220
Now I'm going to say,
here's another approach.
01:12:48.220 --> 01:12:51.820
Look at the complex
equation, solve it,
01:12:51.820 --> 01:12:52.940
and take the real part.
01:12:56.180 --> 01:12:57.550
You may prefer one method.
01:12:57.550 --> 01:12:59.020
You may like to stay real.
01:13:01.780 --> 01:13:04.500
In a way, it's a little
more straightforward.
01:13:04.500 --> 01:13:08.690
But the complex one is
the one that will show us,
01:13:08.690 --> 01:13:12.160
it brings out this R,
it brings out the gain,
01:13:12.160 --> 01:13:15.030
it brings out the
important-- engineering
01:13:15.030 --> 01:13:19.310
quantities are important,
if I do it this way.
01:13:19.310 --> 01:13:24.150
Now, I believe that the
solution to that is easy.
01:13:24.150 --> 01:13:27.170
Actually, it is included
in what I did last time.
01:13:27.170 --> 01:13:30.170
It's a linear equation
with a forcing term
01:13:30.170 --> 01:13:33.260
that's a pure exponential.
01:13:33.260 --> 01:13:36.740
And what kind of
solution do I look for?
01:13:36.740 --> 01:13:38.635
I'm looking for a
particular solution.
01:13:41.460 --> 01:13:47.350
If I see an exponential
forcing term, I say, great.
01:13:47.350 --> 01:13:50.300
The solution will
be an exponential.
01:13:50.300 --> 01:13:52.960
So the solution will be sum.
01:13:52.960 --> 01:13:56.780
Z is sum capital Z
e to the i omega t.
01:14:00.960 --> 01:14:01.650
Plug it in.
01:14:05.170 --> 01:14:09.650
What happens if I plug
that in to find capital Z,
01:14:09.650 --> 01:14:12.861
which is just a number?
01:14:12.861 --> 01:14:13.360
Right.
01:14:13.360 --> 01:14:14.900
This is my method.
01:14:14.900 --> 01:14:18.140
This is a linear equation, with
one of those cool right hand
01:14:18.140 --> 01:14:22.285
sides, where the solution has
the same form with a constant,
01:14:22.285 --> 01:14:24.670
and I just have to
find that constant.
01:14:24.670 --> 01:14:26.200
So I plug it in.
01:14:26.200 --> 01:14:27.360
Dz, dt.
01:14:27.360 --> 01:14:32.225
Take the derivative of this,
z i omega will come down.
01:14:32.225 --> 01:14:35.790
E to the i omega t.
01:14:35.790 --> 01:14:39.810
Z is just this, z
to the i omega t.
01:14:39.810 --> 01:14:44.500
And this is just 1
e to the i omega t.
01:14:44.500 --> 01:14:48.450
So I plugged it in, hoping
things would be good.
01:14:48.450 --> 01:14:51.000
And they are because
I can cancel e
01:14:51.000 --> 01:14:55.060
to the i omega t, that's the
beauty of exponentials, leaving
01:14:55.060 --> 01:14:56.420
just a 1 there.
01:14:56.420 --> 01:14:59.090
So what's capital Z?
01:14:59.090 --> 01:15:00.475
What's capital Z then?
01:15:03.630 --> 01:15:05.200
I've got a z here.
01:15:05.200 --> 01:15:07.010
I better bring it over here.
01:15:07.010 --> 01:15:08.790
And I've got the 1 there.
01:15:08.790 --> 01:15:12.050
I think the z is 1 over.
01:15:12.050 --> 01:15:17.190
When I bring that z over here,
do you see what I'm getting?
01:15:17.190 --> 01:15:20.460
It's all multiplying
this e the i omega t.
01:15:20.460 --> 01:15:21.560
It's a number there.
01:15:24.950 --> 01:15:29.370
Z times i omega and
comes over as a minus z.
01:15:29.370 --> 01:15:31.300
What do I have
multiplying z here?
01:15:35.350 --> 01:15:38.480
I see the i omega.
01:15:38.480 --> 01:15:41.490
And what else have I
got multiplying the z?
01:15:41.490 --> 01:15:42.750
AUDIENCE: Minus.
01:15:42.750 --> 01:15:46.018
PROFESSOR: A negative 1 because
it came over with a minus sign.
01:15:49.340 --> 01:15:50.744
Done.
01:15:50.744 --> 01:15:51.410
Equation solved.
01:15:55.040 --> 01:15:56.980
Equation solved.
01:15:56.980 --> 01:15:59.430
Complex equation solved.
01:15:59.430 --> 01:16:03.350
So the point is, the complex
equation was a cinch.
01:16:03.350 --> 01:16:06.190
We just assumed the right
form, plugged it in,
01:16:06.190 --> 01:16:07.712
found the number, we're done.
01:16:10.660 --> 01:16:12.830
But there's one more
step, which is what?
01:16:15.350 --> 01:16:17.590
Take the real part.
01:16:17.590 --> 01:16:19.550
So I have to take the
real part of this.
01:16:19.550 --> 01:16:25.900
So the correct answer is y is
the real part of that number, 1
01:16:25.900 --> 01:16:30.166
over i omega minus 1
times e to the i omega t.
01:16:40.290 --> 01:16:47.850
I'm tempted to stop there, but
just with a little comment.
01:16:47.850 --> 01:16:49.700
How am I going to
find that real part?
01:16:54.660 --> 01:16:57.410
And what form will it have?
01:16:57.410 --> 01:16:59.160
What form will that
real part have?
01:16:59.160 --> 01:17:02.990
Yeah, maybe just to say
what form will it have?
01:17:02.990 --> 01:17:09.389
The real part, it's
going to be a sinusoid.
01:17:13.520 --> 01:17:17.110
But I have a complex number
multiplying this guy.
01:17:17.110 --> 01:17:20.537
The real part is going to
be exactly of the form we--
01:17:20.537 --> 01:17:22.120
well, of course, it
had to be the form
01:17:22.120 --> 01:17:24.470
because that was another
way to solve the equation.
01:17:24.470 --> 01:17:26.480
It's going to be some number.
01:17:26.480 --> 01:17:33.780
And I'll call it g, for
gain, times the real part.
01:17:33.780 --> 01:17:36.650
And so the real part
will be a cosine.
01:17:36.650 --> 01:17:38.990
Yeah, it's just perfect.
01:17:38.990 --> 01:17:41.650
A cosine of omega t.
01:17:41.650 --> 01:17:45.345
And there'll be a phase.
01:17:45.345 --> 01:17:45.845
Yeah.
01:17:49.585 --> 01:17:54.460
i haven't taken that step fully.
01:17:54.460 --> 01:17:57.310
I got to that fully.
01:17:57.310 --> 01:18:02.980
And then I said that that, if
I use some complex arithmetic,
01:18:02.980 --> 01:18:05.190
will come out to be this.
01:18:05.190 --> 01:18:08.100
And you see the beauty
of that answer, which
01:18:08.100 --> 01:18:12.450
was way better than a
sum of sines and cosines.
01:18:12.450 --> 01:18:13.620
We see the gain.
01:18:13.620 --> 01:18:15.080
We see the amplitude.
01:18:15.080 --> 01:18:17.040
And we see the phase shift.
01:18:17.040 --> 01:18:17.960
Yeah.
01:18:17.960 --> 01:18:20.640
So I don't know, that
would be a good exercise
01:18:20.640 --> 01:18:23.320
in complex numbers.
01:18:23.320 --> 01:18:30.460
Find g and find phi, in taking
the real part of this thing.
01:18:30.460 --> 01:18:31.020
Yeah.
01:18:31.020 --> 01:18:34.640
It's a pure exercise in
using complex numbers.
01:18:34.640 --> 01:18:36.430
I don't feel like
doing it today.
01:18:39.940 --> 01:18:43.440
If we do it, you just
see a lot of formulas.
01:18:43.440 --> 01:18:45.360
Here, you see the point.
01:18:45.360 --> 01:18:48.560
The point was that
the complex equation
01:18:48.560 --> 01:18:51.730
could be solved in one line.
01:18:51.730 --> 01:18:53.750
We just did it.
01:18:53.750 --> 01:18:57.380
But that left us the problem
of taking the real part.
01:18:57.380 --> 01:19:00.170
That was the e to
the i omega t there.
01:19:00.170 --> 01:19:02.130
Left us the problem of
taking the real part.
01:19:02.130 --> 01:19:05.360
And that's a practice
with complex arithmetic.
01:19:05.360 --> 01:19:07.190
So you've got the choice.
01:19:07.190 --> 01:19:12.030
Either stay real--
sign plus cosine.
01:19:12.030 --> 01:19:18.830
And then use the sinusoidal
identity, polar form.
01:19:18.830 --> 01:19:22.140
Or get the polar form from here.
01:19:22.140 --> 01:19:24.500
Same answer both ways.