WEBVTT

00:00:00.500 --> 00:00:01.920
GILBERT STRANG: OK.

00:00:01.920 --> 00:00:07.520
So this is the next step for
a first-order differential

00:00:07.520 --> 00:00:09.060
equation.

00:00:09.060 --> 00:00:13.560
We take-- instead
of an exponential,

00:00:13.560 --> 00:00:16.000
now we have an oscillating.

00:00:16.000 --> 00:00:20.370
Exponentials, the previous
lecture, grew or decayed,

00:00:20.370 --> 00:00:22.170
now we have an oscillate.

00:00:22.170 --> 00:00:28.900
We have AC, alternating
current in this problem,

00:00:28.900 --> 00:00:35.180
instead of real exponentials,
we have oscillation, vibration,

00:00:35.180 --> 00:00:40.840
all the applications that
involve circular motion,

00:00:40.840 --> 00:00:44.370
going around and around instead
of going off exponentially.

00:00:44.370 --> 00:00:45.610
OK.

00:00:45.610 --> 00:00:48.530
So here's the point.

00:00:48.530 --> 00:00:51.700
Again I'm looking for
a particular solution.

00:00:51.700 --> 00:00:54.920
The particular solution--
it would be nice

00:00:54.920 --> 00:00:57.140
if we could say the
particular solution

00:00:57.140 --> 00:01:00.100
was just some multiple
of the cosine.

00:01:00.100 --> 00:01:01.550
But that won't work.

00:01:01.550 --> 00:01:04.750
So that makes this
problem one step harder

00:01:04.750 --> 00:01:06.300
than the exponential.

00:01:06.300 --> 00:01:09.400
We need to allow
the signs in there.

00:01:09.400 --> 00:01:14.750
Because, if I look for a cosine,
if I tried only this part,

00:01:14.750 --> 00:01:16.730
I could match that.

00:01:16.730 --> 00:01:20.090
I'd have a times the cosine,
that would be cosine.

00:01:20.090 --> 00:01:25.520
But the derivative of a
cosine is a sine function.

00:01:25.520 --> 00:01:27.960
So signs are going
to get in there

00:01:27.960 --> 00:01:31.580
and we have to allow
them into the solution.

00:01:31.580 --> 00:01:32.080
OK.

00:01:32.080 --> 00:01:34.680
So that's the right
thing to assume.

00:01:34.680 --> 00:01:37.300
Actually there will
be, you'll see,

00:01:37.300 --> 00:01:42.280
three different ways to write
the answer to this problem.

00:01:42.280 --> 00:01:45.740
And this is the first sort
of most straightforward,

00:01:45.740 --> 00:01:48.730
but not the best
in the long run.

00:01:48.730 --> 00:01:49.230
OK.

00:01:49.230 --> 00:01:52.590
Straightforward one, I'm
going to substitute that

00:01:52.590 --> 00:01:57.560
into the equation and find
M and N. That's my job.

00:01:57.560 --> 00:01:59.510
Find these numbers.

00:01:59.510 --> 00:02:01.560
So put that into the equation.

00:02:01.560 --> 00:02:04.270
On the left side I
want the derivative,

00:02:04.270 --> 00:02:09.870
so that we'll be omega--
well derivative of the cosine

00:02:09.870 --> 00:02:17.218
is minus omega m sine omega t.

00:02:17.218 --> 00:02:20.390
The derivative brought
out this factor omega.

00:02:20.390 --> 00:02:23.020
The derivative of
cosine was sine.

00:02:23.020 --> 00:02:25.240
Now the derivative
of this brings out

00:02:25.240 --> 00:02:31.700
a factor of omega--
omega N cosine omega t.

00:02:31.700 --> 00:02:36.320
And that should equal
a times y-- there's y,

00:02:36.320 --> 00:02:46.350
so I just multiply by
a-- a M cosine omega t,

00:02:46.350 --> 00:02:50.950
and a N sine omega t.

00:02:50.950 --> 00:02:52.690
That's the ay part.

00:02:52.690 --> 00:02:59.110
And now I have the source
term plus cosine omega t.

00:02:59.110 --> 00:03:04.300
And that has to be
true for all time.

00:03:04.300 --> 00:03:07.970
And now I need the equation.

00:03:07.970 --> 00:03:09.570
What do I do with this?

00:03:09.570 --> 00:03:11.700
I'm looking for two
things, M and N.

00:03:11.700 --> 00:03:13.680
I'm looking for two equations.

00:03:13.680 --> 00:03:16.990
So I match the cosine terms.

00:03:16.990 --> 00:03:22.880
I match that term, that cosine
term, and the source term.

00:03:22.880 --> 00:03:27.760
So they all multiply
cosine omega t.

00:03:27.760 --> 00:03:33.090
So I want omega
n-- so I bring this

00:03:33.090 --> 00:03:38.200
over on the other side--
minus a m plus omega n

00:03:38.200 --> 00:03:43.230
equals-- here I just have
one cosine-- equals 1.

00:03:43.230 --> 00:03:46.390
Minus a M plus omega N equals 1.

00:03:46.390 --> 00:03:49.010
And now I'll match
the sine terms.

00:03:49.010 --> 00:03:56.310
So in the sine terms, I have
a minus omega M, sine omega t.

00:03:56.310 --> 00:03:58.580
And I have to bring
this on the other side,

00:03:58.580 --> 00:04:03.090
so that'll be a minus
a N sine omega t.

00:04:03.090 --> 00:04:07.060
And there's no sine
omega t in the source.

00:04:07.060 --> 00:04:09.340
There's my two equations.

00:04:09.340 --> 00:04:12.050
Those are my two
equations for M and N.

00:04:12.050 --> 00:04:14.310
So I just solve
those two equations

00:04:14.310 --> 00:04:18.600
and I've got the particular
solution that I look for.

00:04:18.600 --> 00:04:21.300
So, it's two equations,
two unknowns.

00:04:21.300 --> 00:04:24.160
It's the basic problem
of linear algebra.

00:04:24.160 --> 00:04:26.470
I'm inclined to just write
down the answer, which

00:04:26.470 --> 00:04:30.000
I prepared in advance.

00:04:30.000 --> 00:04:34.460
And it turns out
to be minus a over

00:04:34.460 --> 00:04:37.720
omega squared plus a squared.

00:04:37.720 --> 00:04:42.550
And N turns out to have
that same omega squared

00:04:42.550 --> 00:04:46.900
plus a squared, and
above it goes omega.

00:04:51.000 --> 00:04:58.870
If you check this equation,
for example, omega times the M

00:04:58.870 --> 00:05:03.700
will give me a a omega
with a minus with a minus.

00:05:03.700 --> 00:05:08.310
And then a times N will
also have an a omega.

00:05:08.310 --> 00:05:10.650
And the same omega
squared plus a

00:05:10.650 --> 00:05:15.120
squared, they cancel to give 0.

00:05:15.120 --> 00:05:17.240
And this equation
is also solved.

00:05:17.240 --> 00:05:23.060
So one more important
problem solved.

00:05:23.060 --> 00:05:27.440
Well, we found the
particular solution.

00:05:27.440 --> 00:05:32.210
I haven't added in-- I haven't
match the initial condition.

00:05:32.210 --> 00:05:35.980
Now in many, many cases,
it's this particular solution

00:05:35.980 --> 00:05:37.410
that's of interest.

00:05:37.410 --> 00:05:42.510
This here-- let me put a
box around our solution--

00:05:42.510 --> 00:05:46.600
and we substituted that in
the differential equation.

00:05:46.600 --> 00:05:49.670
We discovered M. We
discovered N. We've

00:05:49.670 --> 00:05:52.090
got this particular solution.

00:05:52.090 --> 00:05:56.310
And that's the oscillation
that keeps going.

00:05:56.310 --> 00:06:01.490
That if we're listening
to radio or if we

00:06:01.490 --> 00:06:06.230
have alternating current,
this is what we see,

00:06:06.230 --> 00:06:09.070
the null solution.

00:06:09.070 --> 00:06:13.720
The thing that's coming
with no source term.

00:06:13.720 --> 00:06:17.270
Usually a is negative
and that disappears.

00:06:17.270 --> 00:06:19.670
That's called the
transient term.

00:06:19.670 --> 00:06:26.410
So the null solution would be
have an ae to the at as always.

00:06:26.410 --> 00:06:31.570
But I'm not so interested in
that because it disappears.

00:06:31.570 --> 00:06:33.810
You don't hear it
after a minute.

00:06:33.810 --> 00:06:38.010
And this is the solution that
you're-- this is what your ear

00:06:38.010 --> 00:06:39.380
is hearing.

00:06:39.380 --> 00:06:41.170
OK.

00:06:41.170 --> 00:06:43.810
So we've got one
form of the answer.

00:06:43.810 --> 00:06:48.140
Now, that's a pretty nice
form, but it's not perfect.

00:06:48.140 --> 00:06:53.660
I can't see exactly--
this can be simplified

00:06:53.660 --> 00:06:55.640
in a really nice way.

00:06:55.640 --> 00:06:58.340
So when we work with
sines and cosines,

00:06:58.340 --> 00:07:00.900
it's this next step
that's important.

00:07:03.610 --> 00:07:07.200
I believe that that
same yp of t can

00:07:07.200 --> 00:07:13.330
be written in a different
way as what also--

00:07:13.330 --> 00:07:20.810
another form, a different--
well I should say,

00:07:20.810 --> 00:07:26.880
another form for
the same y of t.

00:07:26.880 --> 00:07:35.830
Another form will
be the same y of t.

00:07:35.830 --> 00:07:40.610
You see what I don't like is
having a cosine and a sine

00:07:40.610 --> 00:07:45.500
because those are out of
phase, and they're combining it

00:07:45.500 --> 00:07:47.390
into something, and
I want to find out

00:07:47.390 --> 00:07:48.810
what their combining into.

00:07:48.810 --> 00:07:50.210
And it's really nice.

00:07:50.210 --> 00:08:00.420
They're combining into a single
cosine, but not just omega t,

00:08:00.420 --> 00:08:02.770
there's a lag, a phase shift.

00:08:02.770 --> 00:08:06.120
The angle involved is
often called the phase.

00:08:06.120 --> 00:08:11.230
So the two, sine
and cosine, combine

00:08:11.230 --> 00:08:14.680
to give a phase shift
with some amplitude,

00:08:14.680 --> 00:08:18.450
maybe I'll call it G, the gain.

00:08:18.450 --> 00:08:20.930
Or often it would be
called capital R just

00:08:20.930 --> 00:08:25.860
for-- because it's-- sort of
what you're seeing here is

00:08:25.860 --> 00:08:27.260
polar coordinates.

00:08:27.260 --> 00:08:31.596
So I want to match
this, which has the G

00:08:31.596 --> 00:08:36.799
and the alpha-- polar
coordinates is really

00:08:36.799 --> 00:08:38.950
the right way to think of this.

00:08:38.950 --> 00:08:44.270
G and an alpha, a
magnitude and an angle.

00:08:44.270 --> 00:08:48.950
I want to match that with
the form I already had.

00:08:48.950 --> 00:08:52.950
So I'll use a little
trigonometry here

00:08:52.950 --> 00:08:57.010
to remember that
this is equal to-- I

00:08:57.010 --> 00:09:00.820
have a G. Do you
remember a formula

00:09:00.820 --> 00:09:03.220
for the cosine of a minus b?

00:09:03.220 --> 00:09:09.640
The cosine of a difference
is cosine of omega t,

00:09:09.640 --> 00:09:13.610
cosine of alpha,
plus-- it's a plus

00:09:13.610 --> 00:09:15.720
here because it's
a minus there--

00:09:15.720 --> 00:09:18.920
sine omega t sine alpha.

00:09:21.900 --> 00:09:26.680
So I have just written this
out in the two-term form,

00:09:26.680 --> 00:09:29.820
and I did that so that I could
match the two-term form I

00:09:29.820 --> 00:09:30.500
already had.

00:09:30.500 --> 00:09:33.120
So can I just do that matching?

00:09:33.120 --> 00:09:39.260
The cosine omega t, the
M must be G cosine alpha.

00:09:42.640 --> 00:09:48.565
And they N must be G sine alpha.

00:09:54.180 --> 00:09:58.220
So I now have two equations.

00:09:58.220 --> 00:10:02.550
The M and the N, I still
remember what those are.

00:10:02.550 --> 00:10:03.940
I figured those out.

00:10:03.940 --> 00:10:09.060
But now I want to convert the
M N form to the G alpha form,

00:10:09.060 --> 00:10:10.750
and this is what I have to do.

00:10:10.750 --> 00:10:14.200
And it's the usual thing
with polar coordinates.

00:10:14.200 --> 00:10:18.740
How can I get-- how do I
discover what G is there,

00:10:18.740 --> 00:10:22.260
and what alpha is?

00:10:22.260 --> 00:10:27.820
The trick is-- the one
fundamental identity

00:10:27.820 --> 00:10:31.520
when you see cosines and sines--
is to remember that cosine

00:10:31.520 --> 00:10:33.140
squared plus sine squared is 1.

00:10:33.140 --> 00:10:35.240
I'm going to use
that, have to use it.

00:10:35.240 --> 00:10:37.610
So I'll square both sides.

00:10:37.610 --> 00:10:41.260
I'll have M squared,
and I'll add.

00:10:41.260 --> 00:10:47.050
So I'll have M squared
plus N squared is

00:10:47.050 --> 00:10:52.640
G squared cosine squared alpha.

00:10:52.640 --> 00:10:57.850
G squared times
cosine squared alpha--

00:10:57.850 --> 00:11:03.010
when I square that one--
and sine squared alpha

00:11:03.010 --> 00:11:04.240
when I square that one.

00:11:04.240 --> 00:11:06.770
And again, the
point is that's one.

00:11:06.770 --> 00:11:09.840
So that's just G squared.

00:11:09.840 --> 00:11:11.430
So what do I learned?

00:11:11.430 --> 00:11:13.820
G is the square root of this.

00:11:13.820 --> 00:11:19.380
G is the square root of
M squared plus N squared.

00:11:19.380 --> 00:11:24.360
And I'm always freedom plug-in
the M and the N that I found.

00:11:24.360 --> 00:11:28.310
OK that's-- ah,
what about alpha?

00:11:28.310 --> 00:11:29.890
That's the angle.

00:11:29.890 --> 00:11:33.890
So I have to-- again,
I'm thinking trig here.

00:11:33.890 --> 00:11:37.220
How am I going to
get alpha here?

00:11:37.220 --> 00:11:39.250
I want to get G
out of this formula

00:11:39.250 --> 00:11:42.150
now, and just
focus on the alpha.

00:11:42.150 --> 00:11:45.404
Previously I got alpha
out of it and got the G.

00:11:45.404 --> 00:11:49.540
Now the way to do
is take the ratio.

00:11:49.540 --> 00:11:51.930
If I take the ratio
of that to that,

00:11:51.930 --> 00:11:55.940
divide one by the other,
the G's will cancel.

00:11:55.940 --> 00:11:58.550
So I'll take the
ratio of that to that

00:11:58.550 --> 00:12:14.220
to get G sine alpha divided by
g cosine alpha is then N over M.

00:12:14.220 --> 00:12:17.240
And the G's cancel as I wanted.

00:12:17.240 --> 00:12:19.540
So now I have an
equation for alpha.

00:12:19.540 --> 00:12:23.910
Or more exactly, I have an
equation for tangent of alpha.

00:12:23.910 --> 00:12:33.260
Sine over cosine is tangent
of alpha is N over M.

00:12:33.260 --> 00:12:36.630
So this is called
the-- you could call it

00:12:36.630 --> 00:12:38.910
the sinusoidal identity.

00:12:38.910 --> 00:12:41.210
What is that word sinusoid?

00:12:41.210 --> 00:12:47.020
Sinusoid is a word for any
mixture of sines and cosines,

00:12:47.020 --> 00:12:52.990
any mixture of sines and
cosines of the same omega t.

00:12:52.990 --> 00:12:57.560
So the sinusoidal
identity says that I

00:12:57.560 --> 00:13:03.170
can rewrite that solution
into this solution.

00:13:08.040 --> 00:13:11.480
And I really see the key
number in the whole thing

00:13:11.480 --> 00:13:14.940
is the gain, the magnitude.

00:13:14.940 --> 00:13:18.360
It's how loud the
station comes through

00:13:18.360 --> 00:13:20.300
if we're tuning a radio.

00:13:20.300 --> 00:13:23.720
So and again, this
is the response

00:13:23.720 --> 00:13:29.410
that's keeps going because
the cosine oscillates forever.

00:13:29.410 --> 00:13:32.390
There will be also
something coming

00:13:32.390 --> 00:13:36.660
from the initial condition
that we expect to die out.

00:13:36.660 --> 00:13:38.750
So I mentioned at the
very start that there

00:13:38.750 --> 00:13:46.110
were three forms of the
answer to this cosine input,

00:13:46.110 --> 00:13:48.240
and I've given you two.

00:13:48.240 --> 00:13:51.280
I've given you the M and N form.

00:13:51.280 --> 00:13:54.270
You could say rectangular
coordinates-- cosines

00:13:54.270 --> 00:13:55.380
and sines.

00:13:55.380 --> 00:13:59.500
I've given you the
polar form, which

00:13:59.500 --> 00:14:04.350
is a gain, a
magnitude, and a phase.

00:14:04.350 --> 00:14:09.280
And the third one
involves complex numbers.

00:14:09.280 --> 00:14:14.570
I have to make that a separate
lecture, maybe even two.

00:14:14.570 --> 00:14:18.910
So complex numbers,
where do they come in?

00:14:18.910 --> 00:14:22.030
It's a totally real equation.

00:14:22.030 --> 00:14:25.030
If I think about all
this that I've done,

00:14:25.030 --> 00:14:34.710
it was all totally
real, but there's

00:14:34.710 --> 00:14:38.510
a link-- the key fact about
complex numbers, Euler's

00:14:38.510 --> 00:14:43.620
great formula will give me a
connection between cosine omega

00:14:43.620 --> 00:14:50.580
t and sine omega t with
e to the I omega t.

00:14:50.580 --> 00:14:54.970
So at the price of introducing
that complex number,

00:14:54.970 --> 00:15:04.250
imaginary number i, or j
for electrical engineers,

00:15:04.250 --> 00:15:06.570
we're back to exponentials.

00:15:06.570 --> 00:15:08.220
We're back to exponentials.

00:15:08.220 --> 00:15:10.170
So that'll come in
the next lecture.

00:15:10.170 --> 00:15:17.720
This is one more example
of a nice source function.

00:15:17.720 --> 00:15:21.545
Maybe I could just say, what
are the nicest source functions?

00:15:24.270 --> 00:15:28.040
So this is the source
function here and it was nice.

00:15:28.040 --> 00:15:30.080
Exponential was even nicer.

00:15:30.080 --> 00:15:32.170
Constant was best of all.

00:15:32.170 --> 00:15:35.970
And I want to-- another
one I want to introduce

00:15:35.970 --> 00:15:38.410
is a delta function.

00:15:38.410 --> 00:15:41.310
So that's-- a delta
function is an impulse,

00:15:41.310 --> 00:15:43.390
something that
happens in an instant.

00:15:43.390 --> 00:15:46.870
And that's an interesting, very
interesting and very important

00:15:46.870 --> 00:15:47.740
possibility.

00:15:47.740 --> 00:15:48.240
OK.

00:15:48.240 --> 00:15:50.070
Thank you.