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PROFESSOR: And this
last little bit
00:00:23.070 --> 00:00:24.902
is something which is
not yet on the Web.
00:00:24.902 --> 00:00:26.860
But, anyway, when I was
walking out of the room
00:00:26.860 --> 00:00:29.740
last time, I noticed
that I'd written down
00:00:29.740 --> 00:00:32.070
the wrong formula for c_1 - c_2.
00:00:32.070 --> 00:00:35.380
There's a misprint, there's
a minus sign that's wrong.
00:00:35.380 --> 00:00:39.180
I claimed last time
that c_1 - c_2 was +1/2.
00:00:39.180 --> 00:00:40.580
But, actually, it's -1/2.
00:00:40.580 --> 00:00:42.080
If you go through
the calculation
00:00:42.080 --> 00:00:45.780
that we did with the
antiderivative of sin x cos x,
00:00:45.780 --> 00:00:48.500
we get these two
possible answers.
00:00:48.500 --> 00:00:52.900
And if they're to be equal,
then if we just subtract them
00:00:52.900 --> 00:00:56.300
we get c_1 - c_2 + 1/2 = 0.
00:00:56.300 --> 00:01:01.740
So c_1 - c_2 = 1/2.
00:01:01.740 --> 00:01:03.930
So, those are all
of the corrections.
00:01:03.930 --> 00:01:06.530
Again, everything here
will be on the Web.
00:01:06.530 --> 00:01:14.660
But just wanted to make
it all clear to you.
00:01:14.660 --> 00:01:15.410
So here we are.
00:01:15.410 --> 00:01:19.900
This is our last day of the
second unit, Applications
00:01:19.900 --> 00:01:22.010
of Differentiation.
00:01:22.010 --> 00:01:30.210
And I have one of the most fun
topics to introduce to you.
00:01:30.210 --> 00:01:32.530
Which is differential equations.
00:01:32.530 --> 00:01:35.410
Now, we have a whole course
on differential equations,
00:01:35.410 --> 00:01:38.140
which is called 18.03.
00:01:38.140 --> 00:01:43.840
And so we're only going
to do just a little bit.
00:01:43.840 --> 00:01:52.600
But I'm going to teach
you one technique.
00:01:52.600 --> 00:01:58.250
Which fits in precisely with
what we've been doing already.
00:01:58.250 --> 00:02:05.280
Which is differentials.
00:02:05.280 --> 00:02:08.480
The first and simplest kind
of differential equation
00:02:08.480 --> 00:02:12.770
is the rate of change
of x with respect to y
00:02:12.770 --> 00:02:16.770
is equal to some function f(x).
00:02:16.770 --> 00:02:19.020
Now, that's a perfectly
good differential equation.
00:02:19.020 --> 00:02:21.420
And we already
discussed last time
00:02:21.420 --> 00:02:26.204
that the solution, that
is, the function y,
00:02:26.204 --> 00:02:27.620
is going to be the
antiderivative,
00:02:27.620 --> 00:02:33.100
or the integral, of x.
00:02:33.100 --> 00:02:35.880
Now, for the purposes
of today, we're
00:02:35.880 --> 00:02:40.597
going to consider this
problem to be solved.
00:02:40.597 --> 00:02:41.930
That is, you can always do this.
00:02:41.930 --> 00:02:44.200
You can always take
antiderivatives.
00:02:44.200 --> 00:02:51.560
And for our purposes
now, that is for now,
00:02:51.560 --> 00:03:08.380
we only have one technique
to find antiderivatives.
00:03:08.380 --> 00:03:15.170
And that's called substitution.
00:03:15.170 --> 00:03:18.750
It has a very small
variant, which
00:03:18.750 --> 00:03:27.940
we called advanced guessing.
00:03:27.940 --> 00:03:29.820
And that works just as well.
00:03:29.820 --> 00:03:32.490
And that's basically all
that you'll ever need to do.
00:03:32.490 --> 00:03:36.457
As a practical matter, these are
the ones you'll face for now.
00:03:36.457 --> 00:03:38.540
Ones that you can actually
see what the answer is,
00:03:38.540 --> 00:03:42.640
or you'll have to
make a substitution.
00:03:42.640 --> 00:03:48.120
Now, the first tricky example,
or the first maybe interesting
00:03:48.120 --> 00:03:50.190
example of a
differential equation,
00:03:50.190 --> 00:04:00.750
which I'll call Example 2,
is going to be the following.
00:04:00.750 --> 00:04:07.510
d/dx + x acting on
y is equal to 0.
00:04:07.510 --> 00:04:10.580
So that's our first
differential equation that
00:04:10.580 --> 00:04:12.780
were going to try to solve.
00:04:12.780 --> 00:04:18.630
Apart from this standard
antiderivative approach.
00:04:18.630 --> 00:04:24.350
This operation here has a name.
00:04:24.350 --> 00:04:27.490
This actually has a name,
it's called the annihilation
00:04:27.490 --> 00:04:33.460
operator.
00:04:33.460 --> 00:04:43.780
And it's called that
in quantum mechanics.
00:04:43.780 --> 00:04:45.950
And there's a corresponding
creation operator
00:04:45.950 --> 00:04:50.780
where you change the
sign from plus to minus.
00:04:50.780 --> 00:04:53.660
And this is one of the simplest
differential equations.
00:04:53.660 --> 00:04:55.550
The reason why it's
studied in quantum
00:04:55.550 --> 00:04:58.360
mechanics all it that it
has very simple solutions
00:04:58.360 --> 00:05:00.500
that you can just write out.
00:05:00.500 --> 00:05:02.920
So we're going to
solve this equation.
00:05:02.920 --> 00:05:05.680
It's the one that
governs the ground state
00:05:05.680 --> 00:05:08.670
of the harmonic oscillator.
00:05:08.670 --> 00:05:10.790
So it has a lot of fancy
words associated with it,
00:05:10.790 --> 00:05:12.706
but it's a fairly simple
differential equation
00:05:12.706 --> 00:05:14.430
and it works perfectly
by the method
00:05:14.430 --> 00:05:17.270
that we're going to propose.
00:05:17.270 --> 00:05:20.820
So the first step
in this solution
00:05:20.820 --> 00:05:26.800
is just to rewrite the
equation by putting
00:05:26.800 --> 00:05:29.090
one of the terms on
the right-hand side.
00:05:29.090 --> 00:05:32.300
So this is dy/dx = -xy.
00:05:35.700 --> 00:05:37.700
Now, here is where
you see the difference
00:05:37.700 --> 00:05:41.210
between this type of equation
and the previous type.
00:05:41.210 --> 00:05:43.130
In the previous
equation, we just
00:05:43.130 --> 00:05:45.660
had a function of x on
the right-hand side.
00:05:45.660 --> 00:05:50.260
But here, the rate of change
depends on both x and y.
00:05:50.260 --> 00:05:51.830
So it's not clear
at all that we can
00:05:51.830 --> 00:05:55.110
solve this kind of equation.
00:05:55.110 --> 00:05:57.370
But there is a
remarkable trick which
00:05:57.370 --> 00:05:59.420
works very well in this case.
00:05:59.420 --> 00:06:02.710
Which is to use multiplication.
00:06:02.710 --> 00:06:06.340
To use this idea of differential
that we talked about last time.
00:06:06.340 --> 00:06:14.700
Namely, we divide by
y and multiply by dx.
00:06:14.700 --> 00:06:17.660
So now we've separated
the equation.
00:06:17.660 --> 00:06:20.750
We've separated out
the differentials.
00:06:20.750 --> 00:06:22.960
And what's going to
be important for us
00:06:22.960 --> 00:06:27.410
is that the left-hand side is
expressed solely in terms of y
00:06:27.410 --> 00:06:30.170
and the right-hand side is
expressed solely in terms of x.
00:06:30.170 --> 00:06:33.400
And we'll go through
this in careful detail.
00:06:33.400 --> 00:06:36.410
So now, the idea is if you've
set up the equation in terms
00:06:36.410 --> 00:06:39.680
of differentials as opposed
to ratios of differentials,
00:06:39.680 --> 00:06:44.680
or rates of change, now I
can use Leibniz's notation
00:06:44.680 --> 00:06:47.600
and integrate these
differentials.
00:06:47.600 --> 00:06:55.420
Take their antiderivatives.
00:06:55.420 --> 00:07:02.070
And we know what
each of these is.
00:07:02.070 --> 00:07:17.920
Namely, the left-hand side is
just-- Ah, well, that's tough.
00:07:17.920 --> 00:07:24.250
OK.
00:07:24.250 --> 00:07:29.410
I had an au pair who actually
did a lot of Tae Kwan Do.
00:07:29.410 --> 00:07:32.870
She could definitely defeat
any of you in any encounter,
00:07:32.870 --> 00:07:34.940
I promise.
00:07:34.940 --> 00:07:35.440
OK.
00:07:35.440 --> 00:07:37.870
Anyway.
00:07:37.870 --> 00:07:38.970
So, let's go back.
00:07:38.970 --> 00:07:41.970
We want to take the
antiderivative of this.
00:07:41.970 --> 00:07:48.130
So remember, this
is the function
00:07:48.130 --> 00:07:50.140
whose derivative is 1/y.
00:07:50.140 --> 00:07:52.320
And now there's a
slight novelty here.
00:07:52.320 --> 00:07:54.860
Here we're differentiating
the variable as x,
00:07:54.860 --> 00:07:58.220
and here we're differentiating
the variable as y.
00:07:58.220 --> 00:08:02.860
So the antiderivative
here is ln y.
00:08:02.860 --> 00:08:07.970
And the antiderivative on
the other side is -x^2 / 2.
00:08:07.970 --> 00:08:10.400
And they differ by a constant.
00:08:10.400 --> 00:08:17.620
So we have this
relationship here.
00:08:17.620 --> 00:08:19.660
Now, that's almost
the end of the story.
00:08:19.660 --> 00:08:23.100
We have to exponentiate to
express y in terms of x.
00:08:23.100 --> 00:08:26.970
So, e^(ln y) = e^(-x^2 / 2) + c.
00:08:29.880 --> 00:08:36.010
And now I can rewrite that as
y is equal to-- I'll write as A
00:08:36.010 --> 00:08:40.120
e^(-x^2 / 2), where A = e^c.
00:08:43.870 --> 00:08:47.940
And incidentally, we're just
taking the case y positive
00:08:47.940 --> 00:08:48.440
here.
00:08:48.440 --> 00:08:50.920
We'll talk about
what happens when
00:08:50.920 --> 00:08:55.590
y is negative in a few minutes.
00:08:55.590 --> 00:08:57.270
So here's the answer
to the question,
00:08:57.270 --> 00:09:02.040
almost, except for this fact
that I picked out y positive.
00:09:02.040 --> 00:09:09.890
Really, the solution is y
is equal to any multiple
00:09:09.890 --> 00:09:11.780
of e^(-x^2 / 2).
00:09:11.780 --> 00:09:19.310
Any constant a; a
positive, negative, or 0.
00:09:19.310 --> 00:09:22.240
Any constant will do.
00:09:22.240 --> 00:09:24.640
And we should double-check
that to make sure.
00:09:24.640 --> 00:09:32.930
If you take d/dx of y right,
that's going to be a d/dx
00:09:32.930 --> 00:09:33.950
e^(-x^2 / 2).
00:09:36.570 --> 00:09:38.220
And now by the
chain rule, you can
00:09:38.220 --> 00:09:42.080
see that this is a times
the factor of -x, that's
00:09:42.080 --> 00:09:45.080
the derivative of the
exponent, with respect to x,
00:09:45.080 --> 00:09:48.190
times the exponential.
00:09:48.190 --> 00:09:50.770
And now you just rearrange that.
00:09:50.770 --> 00:09:54.430
That's -xy.
00:09:54.430 --> 00:09:55.860
So it does check.
00:09:55.860 --> 00:09:57.360
These are solutions
to the equation.
00:09:57.360 --> 00:09:58.560
The a didn't matter.
00:09:58.560 --> 00:10:05.370
It didn't matter whether it
was positive or negative.
00:10:05.370 --> 00:10:08.750
This function is known as
the normal distribution,
00:10:08.750 --> 00:10:11.310
so it fits beautifully
with a lot of probability
00:10:11.310 --> 00:10:15.710
and probabilistic interpretation
of quantum mechanics.
00:10:15.710 --> 00:10:22.630
This is sort of where
the particle is.
00:10:22.630 --> 00:10:25.380
So next, what I'd
like to do is just
00:10:25.380 --> 00:10:30.920
go through the method in general
and point out when it works.
00:10:30.920 --> 00:10:32.960
And then I'll make
a few comments just
00:10:32.960 --> 00:10:37.650
to make sure that you understand
the technicalities of dealing
00:10:37.650 --> 00:10:39.590
with constants and so forth.
00:10:39.590 --> 00:10:42.385
So, first of all, the
general method of separation
00:10:42.385 --> 00:10:53.830
of variables.
00:10:53.830 --> 00:10:55.360
And here's when it works.
00:10:55.360 --> 00:10:58.470
It works when you're
faced with a differential
00:10:58.470 --> 00:11:03.410
equation of the form f(x) g(y).
00:11:03.410 --> 00:11:05.920
That's the situation
that we had.
00:11:05.920 --> 00:11:08.150
And I'll just illustrate that.
00:11:08.150 --> 00:11:09.820
Just to remind you here.
00:11:09.820 --> 00:11:11.650
Here's our equation.
00:11:11.650 --> 00:11:13.120
It's in that form.
00:11:13.120 --> 00:11:22.190
And the function f(x) is -x,
and the function g(y) is just y.
00:11:22.190 --> 00:11:26.670
And now, the way the method
works is, this separation step.
00:11:26.670 --> 00:11:30.370
From here to here,
this is the key step.
00:11:30.370 --> 00:11:35.070
This is the only
conceptually remarkable step,
00:11:35.070 --> 00:11:37.360
which all has to
do with the fact
00:11:37.360 --> 00:11:40.160
that Leibniz fixed his
notations up so that this
00:11:40.160 --> 00:11:42.190
works perfectly.
00:11:42.190 --> 00:11:48.320
And so that involves taking
the y, so dividing by g(y),
00:11:48.320 --> 00:11:53.170
and multiplying by
dx, it's comfortable
00:11:53.170 --> 00:11:55.540
because it feels like
ordinary arithmetic,
00:11:55.540 --> 00:11:59.430
even though these
are differentials.
00:11:59.430 --> 00:12:02.710
And then, we just
antidifferentiate.
00:12:02.710 --> 00:12:10.560
So we have a function, H, which
is the integral of dy / g(y),
00:12:10.560 --> 00:12:12.440
and we have another
function which
00:12:12.440 --> 00:12:15.910
is F. Note they are functions of
completely different variables
00:12:15.910 --> 00:12:16.740
here.
00:12:16.740 --> 00:12:20.850
Integral of f(x) dx.
00:12:20.850 --> 00:12:23.864
Now, in our example we did that.
00:12:23.864 --> 00:12:25.530
We carried out this
antidifferentiation,
00:12:25.530 --> 00:12:29.680
and this function
turned out to be ln y,
00:12:29.680 --> 00:12:39.630
and this function turned
out to be -x^2 / 2.
00:12:39.630 --> 00:12:42.350
And then we write
the relationship.
00:12:42.350 --> 00:12:45.790
Which is that if these
are both antiderivatives
00:12:45.790 --> 00:12:48.890
of the same thing, then they
have to differ by a constant.
00:12:48.890 --> 00:12:55.870
Or, in other words, H(y)
has to equal to F(x) + c.
00:12:55.870 --> 00:13:10.130
Where c is constant.
00:13:10.130 --> 00:13:15.150
Now, notice that
this kind of equation
00:13:15.150 --> 00:13:20.450
is what we call an
implicit equation.
00:13:20.450 --> 00:13:23.930
It's not quite a
formula for y, directly.
00:13:23.930 --> 00:13:26.330
It defines y implicitly.
00:13:26.330 --> 00:13:29.630
That's that top line up here.
00:13:29.630 --> 00:13:33.020
That's the implicit equation.
00:13:33.020 --> 00:13:35.110
In order to make it an
explicit equation, which
00:13:35.110 --> 00:13:38.780
is what is underneath, what I
have to do is take the inverse.
00:13:38.780 --> 00:13:41.260
So I write it as y
= H^(-1)(F(x) + c).
00:13:45.080 --> 00:13:48.140
Now, in real life the calculus
part is often pretty easy.
00:13:48.140 --> 00:13:52.620
And it can be quite messy
to do the inverse operation.
00:13:52.620 --> 00:13:55.850
So sometimes we just leave it
alone in the implicit form.
00:13:55.850 --> 00:13:58.250
But it's also
satisfying, sometimes,
00:13:58.250 --> 00:14:09.290
to write it in the
final form here.
00:14:09.290 --> 00:14:14.160
Now I've got to give you a few
little pieces of commentary
00:14:14.160 --> 00:14:16.640
before-- For those of you
walked in a little bit late,
00:14:16.640 --> 00:14:25.660
this will all be on the Web.
00:14:25.660 --> 00:14:31.180
So just a few pieces
of commentary.
00:14:31.180 --> 00:14:36.230
So if you like, some remarks.
00:14:36.230 --> 00:14:51.140
The first remark is that I
could have written natural log
00:14:51.140 --> 00:14:55.850
of absolute y is
equal to -x^2 / 2 + c.
00:14:58.390 --> 00:15:01.871
We learned last time that
the antiderivative works also
00:15:01.871 --> 00:15:02.870
for the negative values.
00:15:02.870 --> 00:15:08.490
So this would work
for y not equal to 0.
00:15:08.490 --> 00:15:10.900
Both for positive
and negative values.
00:15:10.900 --> 00:15:13.990
And you can see that that
would have captured most
00:15:13.990 --> 00:15:15.580
of the rest of the solution.
00:15:15.580 --> 00:15:21.870
Namely, |y| would be
equal to A e^(-x^2 / 2),
00:15:21.870 --> 00:15:24.800
by the same reasoning as before.
00:15:24.800 --> 00:15:29.052
And then that would mean that
y was plus or minus A e^(-x^2 /
00:15:29.052 --> 00:15:34.800
2), which is really
just what we got.
00:15:34.800 --> 00:15:38.120
Because, in fact, I
didn't bother with this.
00:15:38.120 --> 00:15:40.087
Because actually in
most-- and the reason why
00:15:40.087 --> 00:15:42.420
I'm going through this, by
the way, carefully this time,
00:15:42.420 --> 00:15:44.878
is that you're going to be
faced with this very frequently.
00:15:44.878 --> 00:15:47.290
The exponential function
comes up all the time.
00:15:47.290 --> 00:15:49.710
And so, therefore, you want
to be completely comfortable
00:15:49.710 --> 00:15:52.300
dealing with it.
00:15:52.300 --> 00:15:54.510
So this time I had
the positive A,
00:15:54.510 --> 00:15:56.540
while the negative A
fits in either this way,
00:15:56.540 --> 00:15:57.740
or I can throw it in.
00:15:57.740 --> 00:15:59.970
Because I know that that's
going to work that way.
00:15:59.970 --> 00:16:03.370
But of course, I
double-checked to be confident.
00:16:03.370 --> 00:16:07.380
Now, this still
leaves out one value.
00:16:07.380 --> 00:16:12.295
So, this still leaves
out-- So, if you like,
00:16:12.295 --> 00:16:14.420
what I have here now is a
is equal to plus or minus
00:16:14.420 --> 00:16:18.540
capital A. The capital A
one being the positive one.
00:16:18.540 --> 00:16:20.920
But this still
leaves out one case.
00:16:20.920 --> 00:16:23.480
Which is y = 0.
00:16:23.480 --> 00:16:27.180
Which is an extremely boring
solution, but nevertheless
00:16:27.180 --> 00:16:28.760
a solution to this problem.
00:16:28.760 --> 00:16:32.330
If you plug in 0 here
for y, you get 0.
00:16:32.330 --> 00:16:34.600
If you plug in 0
here for y, you get
00:16:34.600 --> 00:16:36.520
that these two sides are equal.
00:16:36.520 --> 00:16:38.080
0 = 0.
00:16:38.080 --> 00:16:40.630
Not a very interesting
answer to the question.
00:16:40.630 --> 00:16:42.290
But it's still an answer.
00:16:42.290 --> 00:16:43.930
And so y = 0 is left out..
00:16:43.930 --> 00:16:52.520
Well, that's not so surprising
that we missed that solution.
00:16:52.520 --> 00:16:56.360
Because in the process of
carrying out these operations,
00:16:56.360 --> 00:16:58.250
I divided by y.
00:16:58.250 --> 00:17:02.290
I did that right here.
00:17:02.290 --> 00:17:03.600
So, that's what happens.
00:17:03.600 --> 00:17:05.941
If you're going to do various
non-linear operations,
00:17:05.941 --> 00:17:08.190
in particular, if you're
going to divide by something,
00:17:08.190 --> 00:17:10.564
if it happens to be 0 you're
going to miss that solution.
00:17:10.564 --> 00:17:13.860
You might have problems
with that solution.
00:17:13.860 --> 00:17:16.870
But we have to live with that
because we want to get ahead.
00:17:16.870 --> 00:17:20.520
And we want to get the
formulas for various solutions.
00:17:20.520 --> 00:17:22.840
So that's the first remark
that I wanted to make.
00:17:22.840 --> 00:17:30.340
And now, the second
one is almost related
00:17:30.340 --> 00:17:33.380
to what I was just
discussing right here.
00:17:33.380 --> 00:17:37.240
That I'm erasing.
00:17:37.240 --> 00:17:39.420
And that's the following.
00:17:39.420 --> 00:17:52.100
I could have also written
ln y + c_1 = -x^2 / 2 + c_2.
00:17:52.100 --> 00:17:54.300
Where c_1 and c_2 are
different constants.
00:17:54.300 --> 00:17:57.210
When I'm faced with this
antidifferentiation,
00:17:57.210 --> 00:17:59.460
I just taught you last
time, that you want
00:17:59.460 --> 00:18:02.230
to have an arbitrary constant.
00:18:02.230 --> 00:18:06.760
Here and there, in both slots.
00:18:06.760 --> 00:18:09.480
So I perfectly well could
have written this down.
00:18:09.480 --> 00:18:17.990
But notice that I can rewrite
this as ln y = -x^2 / 2 + c_2 -
00:18:17.990 --> 00:18:20.190
c_1.
00:18:20.190 --> 00:18:22.250
I can subtract.
00:18:22.250 --> 00:18:25.385
And then, if I just combine
these two guys together
00:18:25.385 --> 00:18:29.060
and name them c, I have
a different constant.
00:18:29.060 --> 00:18:32.010
In other words, it's
superfluous and redundant
00:18:32.010 --> 00:18:35.060
to have two arbitrary
constants here,
00:18:35.060 --> 00:18:38.260
because they can always
be combined into one.
00:18:38.260 --> 00:18:47.010
So two constants
are superfluous.
00:18:47.010 --> 00:18:54.430
Can always be combined.
00:18:54.430 --> 00:18:56.490
So we just never do
it this first way.
00:18:56.490 --> 00:19:05.260
It's just extra writing,
it's a waste of time.
00:19:05.260 --> 00:19:08.120
There's one other subtle
remark, which you won't actually
00:19:08.120 --> 00:19:10.020
appreciate until you've
done several problems
00:19:10.020 --> 00:19:11.280
in this direction.
00:19:11.280 --> 00:19:14.490
Which is that the
constant appears
00:19:14.490 --> 00:19:18.900
additive here, in this first
solution to the problem.
00:19:18.900 --> 00:19:22.590
But when I do this nonlinear
operation of exponentiation,
00:19:22.590 --> 00:19:26.390
it now becomes
multiplicative constant.
00:19:26.390 --> 00:19:31.006
And so, in general, there's
a free constant somewhere
00:19:31.006 --> 00:19:31.630
in the problem.
00:19:31.630 --> 00:19:35.490
But it's not always
an additive constant.
00:19:35.490 --> 00:19:38.330
It's only an additive constant
right at the first step
00:19:38.330 --> 00:19:39.890
when you take the
antiderivative.
00:19:39.890 --> 00:19:42.265
And then after that, when you
do all your other nonlinear
00:19:42.265 --> 00:19:45.440
operations, it can turn
into anything at all.
00:19:45.440 --> 00:19:47.980
So you should always expect it
to be something slightly more
00:19:47.980 --> 00:19:49.563
interesting than an
additive constant.
00:19:49.563 --> 00:19:59.060
Although occasionally it
stays an additive constant.
00:19:59.060 --> 00:20:01.180
The last little
bit of commentary
00:20:01.180 --> 00:20:06.010
that I want to make just goes
back to the original problem
00:20:06.010 --> 00:20:06.810
here.
00:20:06.810 --> 00:20:09.680
Which is right here.
00:20:09.680 --> 00:20:11.190
The example 1.
00:20:11.190 --> 00:20:14.490
And I want to solve it, even
though this is simpleminded.
00:20:14.490 --> 00:20:21.490
But Example 1 via separation.
00:20:21.490 --> 00:20:25.290
So that you see our variables.
00:20:25.290 --> 00:20:28.440
So that you see what it does.
00:20:28.440 --> 00:20:34.230
The situation is this.
00:20:34.230 --> 00:20:35.890
And the separation
just means you
00:20:35.890 --> 00:20:38.570
put the dx on the other side.
00:20:38.570 --> 00:20:44.030
So this is dy = f(x) dx.
00:20:44.030 --> 00:20:54.680
And then we integrate.
00:20:54.680 --> 00:20:58.170
And the antiderivative
of dy is just y.
00:20:58.170 --> 00:21:03.490
So this is the solution
to the problem.
00:21:03.490 --> 00:21:05.170
And it's just what
we wrote before;
00:21:05.170 --> 00:21:07.480
it's just a funny notation.
00:21:07.480 --> 00:21:19.480
And it comes to the same
thing as the antiderivative.
00:21:19.480 --> 00:21:23.240
OK, so now we're going to
go on to a trickier problem.
00:21:23.240 --> 00:21:24.090
A trickier example.
00:21:24.090 --> 00:21:26.420
We need one or two more
just to get some practice
00:21:26.420 --> 00:21:29.330
with this method.
00:21:29.330 --> 00:21:31.730
Everybody happy so far?
00:21:31.730 --> 00:21:32.240
Question.
00:21:32.240 --> 00:21:53.472
STUDENT: [INAUDIBLE]
00:21:53.472 --> 00:21:55.180
PROFESSOR: So, the
question is, how do we
00:21:55.180 --> 00:21:58.150
deal with this ambiguity.
00:21:58.150 --> 00:22:03.020
I'm summarizing very,
very, briefly what I heard.
00:22:03.020 --> 00:22:06.530
Well, you know, sometimes
a > 0, sometimes a < 0,
00:22:06.530 --> 00:22:07.550
sometimes it's not.
00:22:07.550 --> 00:22:12.810
So there's a name for this guy.
00:22:12.810 --> 00:22:20.136
Which is that this is what's
called the general solution.
00:22:20.136 --> 00:22:22.010
In other words, the
whole family of solutions
00:22:22.010 --> 00:22:24.460
is the answer to the question.
00:22:24.460 --> 00:22:28.020
Now, it could be that you're
given extra information.
00:22:28.020 --> 00:22:31.760
If you're given extra
information, that might be,
00:22:31.760 --> 00:22:33.527
and this is very typical
in such problems,
00:22:33.527 --> 00:22:35.610
you have the rate of change
of the function, which
00:22:35.610 --> 00:22:36.510
is what we've given.
00:22:36.510 --> 00:22:39.780
But you might also have
the place where it starts.
00:22:39.780 --> 00:22:44.579
Which would be,
say, it starts at 3.
00:22:44.579 --> 00:22:46.620
Now, if you have that
extra piece of information,
00:22:46.620 --> 00:22:50.670
then you can nail down
exactly which function it is.
00:22:50.670 --> 00:22:52.420
If you do that,
if you plug in 3,
00:22:52.420 --> 00:22:57.860
you see that a times
e^(-0^2 / 2) is equal to 3.
00:22:57.860 --> 00:23:00.300
So a = 3.
00:23:00.300 --> 00:23:02.720
And the answer is
y = 3e^(-x^2 / 2).
00:23:06.140 --> 00:23:08.940
And similarly, if it's negative,
if it starts out negative,
00:23:08.940 --> 00:23:10.100
it'll stay negative.
00:23:10.100 --> 00:23:11.100
For instance.
00:23:11.100 --> 00:23:14.870
If it starts out 0, it'll stay
0, this particular function
00:23:14.870 --> 00:23:16.100
here.
00:23:16.100 --> 00:23:18.110
So the answer to
your question is how
00:23:18.110 --> 00:23:19.700
you deal with the ambiguity.
00:23:19.700 --> 00:23:23.620
The answer is that you simply
say what the solution is.
00:23:23.620 --> 00:23:25.200
And the solution is
not one function,
00:23:25.200 --> 00:23:26.324
it's a family of functions.
00:23:26.324 --> 00:23:30.340
It's a list and you have to have
what's known as a parameter.
00:23:30.340 --> 00:23:32.440
And that parameter
gets nailed down
00:23:32.440 --> 00:23:35.240
if you tell me more
information about the function.
00:23:35.240 --> 00:23:37.654
Not the rate of change, but
something about the values
00:23:37.654 --> 00:23:38.320
of the function.
00:23:46.620 --> 00:23:53.660
STUDENT: [INAUDIBLE]
00:23:53.660 --> 00:23:55.720
PROFESSOR: The general
solution is this solution.
00:23:55.720 --> 00:23:56.553
STUDENT: [INAUDIBLE]
00:23:56.553 --> 00:23:58.176
PROFESSOR: And I'm
showing you here
00:23:58.176 --> 00:24:00.300
that you could get to most
of the general solution.
00:24:00.300 --> 00:24:04.570
There's one thing that's left
out, namely the case a = 0.
00:24:04.570 --> 00:24:08.120
So, in other words, I would
not go through this method.
00:24:08.120 --> 00:24:10.690
I would only use this,
which is simpler.
00:24:10.690 --> 00:24:13.590
But then I have to understand
that I haven't gotten
00:24:13.590 --> 00:24:15.410
all of the solutions this way.
00:24:15.410 --> 00:24:19.325
I'm going to need to throw in
all the rest of the solutions.
00:24:19.325 --> 00:24:20.950
So in the back of
your head, you always
00:24:20.950 --> 00:24:23.779
have to have something
like this in mind.
00:24:23.779 --> 00:24:25.570
So that you can generate
all the solutions.
00:24:25.570 --> 00:24:28.510
This is very suggestive, right?
00:24:28.510 --> 00:24:31.840
The restriction, it turns
that the restriction A > 0 is
00:24:31.840 --> 00:24:40.660
superfluous, is unnecessary.
00:24:40.660 --> 00:24:46.180
But that, we only get by
further thought and by checking.
00:24:46.180 --> 00:24:46.890
Another question?
00:24:46.890 --> 00:24:47.389
Over here.
00:24:47.389 --> 00:24:52.210
STUDENT: [INAUDIBLE]
00:24:52.210 --> 00:24:54.630
PROFESSOR: The aim of
differential equations
00:24:54.630 --> 00:24:55.600
is to solve them.
00:24:55.600 --> 00:24:59.372
Just as with
algebraic equations.
00:24:59.372 --> 00:25:01.330
Usually, differential
equations are telling you
00:25:01.330 --> 00:25:04.300
something about the balance
between an acceleration
00:25:04.300 --> 00:25:05.980
and a velocity.
00:25:05.980 --> 00:25:09.840
If you have a falling object,
it might have a resistance.
00:25:09.840 --> 00:25:11.210
It's telling you something.
00:25:11.210 --> 00:25:13.700
So, actually, sometimes
in applied problems,
00:25:13.700 --> 00:25:16.450
formulating what differential
equation describe
00:25:16.450 --> 00:25:18.310
this situation is
very important.
00:25:18.310 --> 00:25:21.910
In order to see that
that's the right thing,
00:25:21.910 --> 00:25:24.330
you have to have solved
it to see that it fits
00:25:24.330 --> 00:25:25.780
the data that you're getting.
00:25:25.780 --> 00:25:28.570
STUDENT: [INAUDIBLE]
00:25:28.570 --> 00:25:31.720
PROFESSOR: The question is, can
you solve for x instead of y.
00:25:31.720 --> 00:25:36.250
The answer is, sure.
00:25:36.250 --> 00:25:38.356
That's the same
thing as-- so that
00:25:38.356 --> 00:25:40.230
would be the inverse
function of the function
00:25:40.230 --> 00:25:42.520
that we're officially
looking for.
00:25:42.520 --> 00:25:43.960
But yeah, it's legal.
00:25:43.960 --> 00:25:46.150
In other words,
oftentimes we're stuck
00:25:46.150 --> 00:25:48.835
with just the implicit,
some implicit formula
00:25:48.835 --> 00:25:51.540
and sometimes we're stuck with
a formula x is a function of y
00:25:51.540 --> 00:25:54.730
versus y is a function of x.
00:25:54.730 --> 00:25:57.850
The way in which the
function is specified
00:25:57.850 --> 00:26:00.780
is something that
can be complicated.
00:26:00.780 --> 00:26:02.810
As you'll see in
the next example,
00:26:02.810 --> 00:26:04.760
it's not necessarily
the best thing
00:26:04.760 --> 00:26:07.530
to think about a function--
y as a function of x.
00:26:07.530 --> 00:26:12.170
Well, in the fourth example.
00:26:12.170 --> 00:26:27.000
Alright, we're going to go on
and do our next example here.
00:26:27.000 --> 00:26:32.440
So the third example
is going to be taken
00:26:32.440 --> 00:26:36.090
as a kind of geometry problem.
00:26:36.090 --> 00:26:38.990
I'll draw a picture of it.
00:26:38.990 --> 00:26:44.180
Suppose you have a curve
with the following property.
00:26:44.180 --> 00:26:50.730
If you take a point on the
curve, and you take the ray,
00:26:50.730 --> 00:26:56.224
you take the ray from the origin
to the curve, well, that's not
00:26:56.224 --> 00:26:57.390
going to be one that I want.
00:26:57.390 --> 00:27:00.350
I think I'm going to want
something which is steeper.
00:27:00.350 --> 00:27:02.130
Because what I'm
going to insist is
00:27:02.130 --> 00:27:09.050
that the tangent line be
twice as steep as the ray
00:27:09.050 --> 00:27:10.490
from the origin.
00:27:10.490 --> 00:27:19.600
So, in other words,
slope of tangent line
00:27:19.600 --> 00:27:31.540
equals twice slope
of ray from origin.
00:27:31.540 --> 00:27:34.110
So the slope of this
orange line is twice
00:27:34.110 --> 00:27:39.410
the slope of the pink line.
00:27:39.410 --> 00:27:41.240
Now, these kinds of
geometric problems
00:27:41.240 --> 00:27:48.700
can be written very succinctly
with differential equations.
00:27:48.700 --> 00:27:51.530
Namely, it's just the
following. dy / dx,
00:27:51.530 --> 00:27:55.340
that's the slope of the
tangent line, is equal to,
00:27:55.340 --> 00:27:58.030
well remember what the
slope of this ray is,
00:27:58.030 --> 00:28:00.700
if this point-- I
need a notation.
00:28:00.700 --> 00:28:04.520
At this point is (x, y) which
is a point on the curve.
00:28:04.520 --> 00:28:07.860
So the slope of this
pink line is what?
00:28:07.860 --> 00:28:09.650
STUDENT: [INAUDIBLE]
00:28:09.650 --> 00:28:12.610
PROFESSOR: y/x.
00:28:12.610 --> 00:28:20.810
So if it's twice it,
there's the equation.
00:28:20.810 --> 00:28:28.040
OK, now, we only have one method
for solving these equations.
00:28:28.040 --> 00:28:29.890
So let's use it.
00:28:29.890 --> 00:28:31.620
It says to separate variables.
00:28:31.620 --> 00:28:41.000
So I write dy / y here,
is equal to 2 dx / x.
00:28:41.000 --> 00:28:42.530
That's the basic separation.
00:28:42.530 --> 00:28:47.990
That's the procedure that
we're always going to use.
00:28:47.990 --> 00:28:54.640
And now if I
integrate that, I find
00:28:54.640 --> 00:29:03.250
that on the right-hand side
I have the logarithm of y.
00:29:03.250 --> 00:29:05.380
And on the left-hand--
Sorry, on the left-hand side
00:29:05.380 --> 00:29:06.590
I have the logarithm of y.
00:29:06.590 --> 00:29:10.500
On the right-hand side, I
have twice the logarithm
00:29:10.500 --> 00:29:20.150
of x, plus a constant.
00:29:20.150 --> 00:29:27.330
So let's see what
happens to this example.
00:29:27.330 --> 00:29:29.846
This is an implicit
equation, and of course we
00:29:29.846 --> 00:29:31.970
have the problems of the
plus or minus signs, which
00:29:31.970 --> 00:29:38.070
I'm not going to worry
about until later.
00:29:38.070 --> 00:29:40.320
So let's exponentiate
and see what happens.
00:29:40.320 --> 00:29:43.600
We get e^(ln y)
= e^(2 ln x + c).
00:29:47.340 --> 00:29:51.940
So, again, this is y
on the left-hand side.
00:29:51.940 --> 00:29:54.010
And on the right-hand
side, if you think about it
00:29:54.010 --> 00:29:55.770
for a second, it's (e^(ln x))^2.
00:29:59.050 --> 00:30:00.370
Which is x^2.
00:30:00.370 --> 00:30:02.680
So this is x^2, and
then there's an e^c.
00:30:02.680 --> 00:30:06.390
So that's another one
of these A factors here.
00:30:06.390 --> 00:30:13.240
A = e^c.
00:30:13.240 --> 00:30:20.160
So the answer is, well,
I'll draw the picture.
00:30:20.160 --> 00:30:22.530
And I'm going to
cheat as I did before.
00:30:22.530 --> 00:30:24.550
We skipped the case y negative.
00:30:24.550 --> 00:30:30.236
We really only did the
case y positive, so far.
00:30:30.236 --> 00:30:31.860
But if you think
about it for a second,
00:30:31.860 --> 00:30:33.490
and we'll check it
in a second, you're
00:30:33.490 --> 00:30:36.390
going to get all of
these parabolas here.
00:30:36.390 --> 00:30:40.970
So the solution is this
family of functions.
00:30:40.970 --> 00:30:44.330
And they can be bending down.
00:30:44.330 --> 00:30:45.660
As well as up.
00:30:45.660 --> 00:30:48.140
So these are the solutions
to this equation.
00:30:48.140 --> 00:30:50.410
Every single one of these
curves has the property
00:30:50.410 --> 00:30:52.750
that if you pick a point
on it, the tangent line
00:30:52.750 --> 00:30:58.050
has twice the slope of
the ray to the origin.
00:30:58.050 --> 00:31:01.840
And the formula, if you like,
of the general solution is y =
00:31:01.840 --> 00:31:08.960
ax^2, a is any constant.
00:31:08.960 --> 00:31:09.460
Question?
00:31:09.460 --> 00:31:21.844
STUDENT: [INAUDIBLE]
00:31:21.844 --> 00:31:22.510
PROFESSOR: Yeah.
00:31:22.510 --> 00:31:28.960
So again - so first
of all, so there
00:31:28.960 --> 00:31:30.110
are two approaches to this.
00:31:30.110 --> 00:31:32.900
One is to check it, and
make sure that it's right.
00:31:32.900 --> 00:31:35.140
When a formula works for
some family of values,
00:31:35.140 --> 00:31:36.740
sometimes it works for others.
00:31:36.740 --> 00:31:39.650
But another one is to realize
that these things will usually
00:31:39.650 --> 00:31:40.970
work out this way.
00:31:40.970 --> 00:31:45.459
Because in this argument here,
I allow the absolute value.
00:31:45.459 --> 00:31:47.750
And that would have been a
perfectly legal thing for me
00:31:47.750 --> 00:31:48.250
to do.
00:31:48.250 --> 00:31:51.220
I could have put in
absolute values here.
00:31:51.220 --> 00:31:55.690
In which case, I would've gotten
that the absolute value of this
00:31:55.690 --> 00:31:56.890
was equal to that.
00:31:56.890 --> 00:32:02.370
And now you see I've covered
the plus and minus cases.
00:32:02.370 --> 00:32:03.880
So it's that same idea.
00:32:03.880 --> 00:32:11.180
This implies that y is equal
to either Ax^2 or -Ax^2,
00:32:11.180 --> 00:32:14.100
depending on which
sign you pick.
00:32:14.100 --> 00:32:21.210
So that allows me for the
curves above and curves below.
00:32:21.210 --> 00:32:25.470
Because it's really true that
the antiderivative here is this
00:32:25.470 --> 00:32:26.240
function.
00:32:26.240 --> 00:32:28.820
It's defined for y negative.
00:32:28.820 --> 00:32:33.840
So let's just double-check.
00:32:33.840 --> 00:32:39.460
In this case, what's happening,
we have y = ax^2 and we want
00:32:39.460 --> 00:32:44.410
to compute dy/dx to make sure
that it satisfies the equation
00:32:44.410 --> 00:32:46.040
that I started out with.
00:32:46.040 --> 00:32:50.890
And what I see here
is that this is 2ax.
00:32:50.890 --> 00:32:53.370
And now I'm going to write
this in a suggestive way.
00:32:53.370 --> 00:33:00.330
I'm going to write
it as 2ax^2 / x.
00:33:00.330 --> 00:33:06.610
And, sure enough,
this is 2y / x.
00:33:06.610 --> 00:33:08.810
It does not matter
whether a-- it
00:33:08.810 --> 00:33:17.370
works for a positive,
a negative, a equals 0.
00:33:17.370 --> 00:33:24.180
It's OK.
00:33:24.180 --> 00:33:29.770
Again, we didn't pick up by
this method the a = 0 case.
00:33:29.770 --> 00:33:35.350
And that's not surprising
because we divided by y.
00:33:35.350 --> 00:33:39.660
There's another thing to watch
out about, about this example.
00:33:39.660 --> 00:33:41.990
So there's another warning.
00:33:41.990 --> 00:33:44.910
Which I have to give you.
00:33:44.910 --> 00:33:47.130
And this is a subtlety
which you definitely
00:33:47.130 --> 00:33:50.090
won't get to in any
detail until you
00:33:50.090 --> 00:33:54.070
get to a higher level ordinary
differential equations course,
00:33:54.070 --> 00:33:56.980
but I do want to warn
you about it right now.
00:33:56.980 --> 00:34:05.310
Which is that if you
look at the equation,
00:34:05.310 --> 00:34:14.100
you need to watch out that
it's undefined at x = 0.
00:34:14.100 --> 00:34:15.700
It's undefined at x = 0.
00:34:15.700 --> 00:34:20.350
We also divided by x,
and x is also a problem.
00:34:20.350 --> 00:34:24.690
Now, that actually has
an important consequence.
00:34:24.690 --> 00:34:27.820
Which is that, strangely,
knowing the value here
00:34:27.820 --> 00:34:31.040
and knowing the rate of change
doesn't specify this function.
00:34:31.040 --> 00:34:33.180
This is bad.
00:34:33.180 --> 00:34:36.160
And it violates one of
our pieces of intuition.
00:34:36.160 --> 00:34:38.720
And what's going wrong is
that the rate of change
00:34:38.720 --> 00:34:40.600
was not specified.
00:34:40.600 --> 00:34:43.560
It's undefined at x = 0.
00:34:43.560 --> 00:34:45.260
So there's a problem
here, and in fact
00:34:45.260 --> 00:34:48.220
if you think carefully about
what this function is doing,
00:34:48.220 --> 00:34:53.510
it could come in on one branch
and leave on a completely
00:34:53.510 --> 00:34:56.050
different branch.
00:34:56.050 --> 00:35:01.541
It doesn't really have to
obey any rule across x = 0.
00:35:01.541 --> 00:35:03.540
So you should really be
thinking of these things
00:35:03.540 --> 00:35:05.860
as rays emanating
from the origin.
00:35:05.860 --> 00:35:10.140
The origin was a special point
in the whole geometric problem.
00:35:10.140 --> 00:35:15.080
Rather than as being
complete parabolas.
00:35:15.080 --> 00:35:16.460
But that's a very subtle point.
00:35:16.460 --> 00:35:23.270
I don't expect you to be able
to say anything about it.
00:35:23.270 --> 00:35:25.920
But I just want to warn
you that it really is true
00:35:25.920 --> 00:35:30.640
that when x = 0 there's a
problem for this differential
00:35:30.640 --> 00:35:33.810
equation.
00:35:33.810 --> 00:35:46.570
So now, let me say
our next problem.
00:35:46.570 --> 00:35:47.690
Next example.
00:35:47.690 --> 00:35:52.370
Just another geometry question.
00:35:52.370 --> 00:36:01.430
So here's Example 4.
00:36:01.430 --> 00:36:04.430
I'm just going to use the
example that we've already got.
00:36:04.430 --> 00:36:09.090
Because there's only
so much time left here.
00:36:09.090 --> 00:36:23.480
The fourth example is to
take the curves perpendicular
00:36:23.480 --> 00:36:31.610
to the parabolas.
00:36:31.610 --> 00:36:33.332
This is another
geometry problem.
00:36:33.332 --> 00:36:35.040
And by specifying that
the the curves are
00:36:35.040 --> 00:36:37.020
perpendicular to
these parabolas,
00:36:37.020 --> 00:36:44.500
I'm telling you
what their slope is.
00:36:44.500 --> 00:36:47.000
So let's think about that.
00:36:47.000 --> 00:36:48.900
What's the new equation?
00:36:48.900 --> 00:36:56.270
The new diff. eq.
is the following.
00:36:56.270 --> 00:37:01.550
It's that the slope is equal
to the negative reciprocal
00:37:01.550 --> 00:37:05.610
of the slope of
the tangent line.
00:37:05.610 --> 00:37:14.850
Of tangent to the parabola.
00:37:14.850 --> 00:37:16.800
So that's the equation.
00:37:16.800 --> 00:37:19.270
That's actually fairly
easy to write down,
00:37:19.270 --> 00:37:26.570
because it's -1
divided by 2 y/x.
00:37:26.570 --> 00:37:32.281
That's the slope
of the parabola.
00:37:32.281 --> 00:37:32.780
2y/x.
00:37:36.860 --> 00:37:38.300
So let's rewrite that.
00:37:38.300 --> 00:37:52.160
Now, this is-- the x goes in
the numerator, so it's -x/(2y).
00:37:52.160 --> 00:37:57.990
And now I want to
solve this one.
00:37:57.990 --> 00:38:01.780
Well, again, there's
only one technique.
00:38:01.780 --> 00:38:10.240
Which is we're going
to separate variables.
00:38:10.240 --> 00:38:12.010
And we separate the
differentials here,
00:38:12.010 --> 00:38:18.201
so we get 2y dy = -x dx.
00:38:18.201 --> 00:38:20.200
That's just looking at
the equation that I have,
00:38:20.200 --> 00:38:22.640
which is right over here.
00:38:22.640 --> 00:38:30.760
dy/dx = -x/(2y), and
cross-multiplying to get this.
00:38:30.760 --> 00:38:33.100
And now I can take
the antiderivative.
00:38:33.100 --> 00:38:35.850
This is y^2.
00:38:35.850 --> 00:38:40.670
And the antiderivative
over here is -x^2 / 2 + c.
00:38:44.410 --> 00:38:57.410
And so, the solutions are x^2
/ 2 + y^2 is equal to some c,
00:38:57.410 --> 00:39:02.600
some constant c.
00:39:02.600 --> 00:39:06.460
Now, this time, things
don't work the same.
00:39:06.460 --> 00:39:09.600
And you can't expect them
always to work the same.
00:39:09.600 --> 00:39:11.550
I claimed that
this must be true.
00:39:11.550 --> 00:39:16.910
But unfortunately I cannot
insist that every c will work.
00:39:16.910 --> 00:39:21.040
As you can see here, only
the positive numbers c
00:39:21.040 --> 00:39:24.980
can work here.
00:39:24.980 --> 00:39:28.720
So the picture is that
something slightly different
00:39:28.720 --> 00:39:29.530
happened here.
00:39:29.530 --> 00:39:31.310
And you have to live with this.
00:39:31.310 --> 00:39:33.570
Is that sometimes not all
the constants will work.
00:39:33.570 --> 00:39:36.270
Because there's more to
the problem than just
00:39:36.270 --> 00:39:37.880
the antidifferentiation.
00:39:37.880 --> 00:39:40.870
And here there are fewer answers
rather than more answers.
00:39:40.870 --> 00:39:43.110
In the other case we had
to add in some answers,
00:39:43.110 --> 00:39:45.030
here we have to take them away.
00:39:45.030 --> 00:39:46.890
Some of them don't
make any sense.
00:39:46.890 --> 00:39:48.644
And the only ones we
can get are the ones
00:39:48.644 --> 00:39:50.060
which are of this
form, where this
00:39:50.060 --> 00:39:53.804
is, say, some radius squared.
00:39:53.804 --> 00:39:55.470
Well maybe I shouldn't
call it a radius.
00:39:55.470 --> 00:39:58.800
I'll just call it
a parameter, a^2.
00:39:58.800 --> 00:40:05.940
And these are of
course ellipses.
00:40:05.940 --> 00:40:09.080
And you can see
that the ellipses,
00:40:09.080 --> 00:40:13.990
the length here is
the square root of 2a
00:40:13.990 --> 00:40:18.160
and the semi-axis,
vertical semi-axis, is a.
00:40:18.160 --> 00:40:20.110
So this is the kind of
ellipse that we've got.
00:40:20.110 --> 00:40:24.140
And I draw it on the
previous diagram,
00:40:24.140 --> 00:40:28.100
I think it's somewhat
suggestive here.
00:40:28.100 --> 00:40:30.280
There, ellipses
are kind of eggs.
00:40:30.280 --> 00:40:32.820
They're a little bit
longer than they are high.
00:40:32.820 --> 00:40:37.160
And they go like this.
00:40:37.160 --> 00:40:40.960
And if I drew them
pretty much right,
00:40:40.960 --> 00:40:43.230
they should be
making right angles.
00:40:43.230 --> 00:40:49.710
At all of these places.
00:40:49.710 --> 00:40:53.530
OK, last little bit here.
00:40:53.530 --> 00:40:57.930
Again, you've got to be very
careful with these solutions.
00:40:57.930 --> 00:41:06.760
And so there's a
warning here too.
00:41:06.760 --> 00:41:08.500
So let's take a
look at the-- This
00:41:08.500 --> 00:41:10.924
is the implicit solution
to the equation.
00:41:10.924 --> 00:41:13.090
And this is the one that
tells us what the shape is.
00:41:13.090 --> 00:41:16.350
But we can also have
the explicit solution.
00:41:16.350 --> 00:41:18.170
And if I solve for
the explicit solution,
00:41:18.170 --> 00:41:25.400
it's y is equal to either plus
square root of (a^2 - x^2 / 2),
00:41:25.400 --> 00:41:32.080
or y is equal to minus the
square root of (a^2 - x^2 / 2).
00:41:32.080 --> 00:41:39.770
These are the
explicit solutions.
00:41:39.770 --> 00:41:41.350
And now, we notice
something that we
00:41:41.350 --> 00:41:43.850
should have noticed before.
00:41:43.850 --> 00:41:50.490
Which is that an ellipse
is not a function.
00:41:50.490 --> 00:41:54.070
It's only the top
half, if you like,
00:41:54.070 --> 00:41:56.610
that's giving you a
solution to this equation.
00:41:56.610 --> 00:41:58.740
Or maybe the bottom
half that's giving it
00:41:58.740 --> 00:42:00.450
the solution to the equation.
00:42:00.450 --> 00:42:07.480
So the one over here, this
one is the top halves.
00:42:07.480 --> 00:42:15.270
And this one over here
is the bottom halves.
00:42:15.270 --> 00:42:18.570
And there's something
else that's interesting.
00:42:18.570 --> 00:42:28.560
Which is that we have a
problem at y = 0. y = 0
00:42:28.560 --> 00:42:33.550
is where x = square root of 2a.
00:42:33.550 --> 00:42:35.960
That's when we get
to this end here.
00:42:35.960 --> 00:42:38.800
And what happens is the solution
comes around and it stops.
00:42:38.800 --> 00:42:44.820
It has a vertical slope.
00:42:44.820 --> 00:42:48.910
Vertical slope.
00:42:48.910 --> 00:42:56.420
And the solution stops.
00:42:56.420 --> 00:43:00.020
But really, that's
not so unreasonable.
00:43:00.020 --> 00:43:01.890
After all, look at the formula.
00:43:01.890 --> 00:43:03.660
There was a y in the
denominator here.
00:43:03.660 --> 00:43:08.500
When y = 0, the slope
should be infinite.
00:43:08.500 --> 00:43:12.530
And so this equation
is just giving us
00:43:12.530 --> 00:43:14.490
what it geometrically
and intuitively
00:43:14.490 --> 00:43:15.800
should be giving us.
00:43:15.800 --> 00:43:22.790
At that stage.
00:43:22.790 --> 00:43:26.221
So that is the introduction
to ordinary differential
00:43:26.221 --> 00:43:26.720
equations.
00:43:26.720 --> 00:43:28.740
Again, there's
only one technique
00:43:28.740 --> 00:43:32.580
which is-- We're not done yet,
we have a whole four minutes
00:43:32.580 --> 00:43:34.030
left and we're going to review.
00:43:34.030 --> 00:43:39.220
Now, so fortunately, this
review is very short.
00:43:39.220 --> 00:43:42.170
Fortunately for you,
I handed out to you
00:43:42.170 --> 00:43:44.920
exactly what you're going
to be covering on the test.
00:43:44.920 --> 00:43:48.440
And it's what's printed
here but there's a whole two
00:43:48.440 --> 00:43:51.000
pages of discussion.
00:43:51.000 --> 00:43:59.500
And I want to give you very,
very clear-cut instructions
00:43:59.500 --> 00:44:00.000
here.
00:44:00.000 --> 00:44:04.780
This is usually the hardest
test of this course.
00:44:04.780 --> 00:44:07.170
People usually do
terribly on it.
00:44:07.170 --> 00:44:11.840
And I'm going to
try to stop that
00:44:11.840 --> 00:44:14.850
by making it a
little bit easier.
00:44:14.850 --> 00:44:17.620
And now here's what
we're going to do.
00:44:17.620 --> 00:44:21.150
I'm telling you exactly
what type of problems
00:44:21.150 --> 00:44:22.447
are going to be on the test.
00:44:22.447 --> 00:44:23.280
These are these six.
00:44:23.280 --> 00:44:25.940
It's also written on
your sheet, your handout.
00:44:25.940 --> 00:44:29.012
It's also just what was
asked on last year's test.
00:44:29.012 --> 00:44:31.220
You should go and you should
look at last year's test
00:44:31.220 --> 00:44:33.420
and see what types
of problems they are.
00:44:33.420 --> 00:44:36.210
I really, really, am going
to ask the same questions,
00:44:36.210 --> 00:44:38.900
or the same type of questions.
00:44:38.900 --> 00:44:41.860
Not the same questions.
00:44:41.860 --> 00:44:44.960
So that's what's going
to happen on the test.
00:44:44.960 --> 00:44:48.680
And let me just tell
you, say one thing, which
00:44:48.680 --> 00:44:51.210
is the main theme of the class.
00:44:51.210 --> 00:44:52.290
And I will open up.
00:44:52.290 --> 00:44:54.330
We'll have time for one
question after that.
00:44:54.330 --> 00:44:58.540
The main theme of this unit
is simply the following.
00:44:58.540 --> 00:45:05.460
That information about
derivative and sometimes
00:45:05.460 --> 00:45:11.010
maybe the second
derivative, tells us
00:45:11.010 --> 00:45:17.315
information about f itself.
00:45:17.315 --> 00:45:19.940
And that's just what were doing
here with ordinary differential
00:45:19.940 --> 00:45:20.490
equations.
00:45:20.490 --> 00:45:22.865
And that was what we were
doing way at the beginning when
00:45:22.865 --> 00:45:23.825
we did approximations.