1
00:00:08,000 --> 00:00:14,000
For the rest of the term,
we are going to be studying not

2
00:00:11,000 --> 00:00:17,000
just one differential equation
at a time, but rather what are

3
00:00:15,000 --> 00:00:21,000
called systems of differential
equations.

4
00:00:18,000 --> 00:00:24,000
Those are like systems of
linear equations.

5
00:00:21,000 --> 00:00:27,000
They have to be solved
simultaneously,

6
00:00:23,000 --> 00:00:29,000
in other words,
not just one at a time.

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00:00:25,000 --> 00:00:31,000
So, how does a system look when
you write it down?

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00:00:30,000 --> 00:00:36,000
Well, since we are going to be
talking about systems of

9
00:00:34,000 --> 00:00:40,000
ordinary differential equations,
there still will be only one

10
00:00:38,000 --> 00:00:44,000
independent variable,
but there will be several

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00:00:42,000 --> 00:00:48,000
dependent variables.
I am going to call,

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00:00:45,000 --> 00:00:51,000
let's say two.
The dependent variables are

13
00:00:48,000 --> 00:00:54,000
going to be, I will call them x
and y, and then the first order

14
00:00:53,000 --> 00:00:59,000
system, something involving just
first derivatives,

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00:00:57,000 --> 00:01:03,000
will look like this.
On the left-hand side

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00:01:02,000 --> 00:01:08,000
will be x prime,

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00:01:04,000 --> 00:01:10,000
in other words.
On the right-hand side will be

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00:01:07,000 --> 00:01:13,000
the dependent variables and then
also the independent variables.

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00:01:12,000 --> 00:01:18,000
I will indicate that,
I will separate it all from the

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00:01:16,000 --> 00:01:22,000
others by putting a semicolon
there.

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00:01:18,000 --> 00:01:24,000
And the same way y prime,
the derivative of y with

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00:01:22,000 --> 00:01:28,000
respect to t,
will be some other function of

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00:01:25,000 --> 00:01:31,000
(x, y) and t.
Let's write down explicitly

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00:01:30,000 --> 00:01:36,000
that x and y are dependent
variables.

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00:01:37,000 --> 00:01:43,000
And what they depend upon is
the independent variable t,

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00:01:41,000 --> 00:01:47,000
time.
A system like this is going to

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00:01:43,000 --> 00:01:49,000
be called first order.
And we are going to consider

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00:01:47,000 --> 00:01:53,000
basically only first-order
systems for a secret reason that

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00:01:52,000 --> 00:01:58,000
I will explain at the end of the
period.

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00:01:56,000 --> 00:02:02,000
This is a first-order system,
meaning that the only kind of

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00:02:00,000 --> 00:02:06,000
derivatives that are up here are
first derivatives.

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00:02:04,000 --> 00:02:10,000
So x prime is dx over dt
and so on.

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00:02:08,000 --> 00:02:14,000
Now, there is still more
terminology.

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00:02:10,000 --> 00:02:16,000
Of course, practically all the
equations after the term

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00:02:15,000 --> 00:02:21,000
started, virtually all the
equations we have been

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00:02:18,000 --> 00:02:24,000
considering are linear
equations, so it must be true

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00:02:22,000 --> 00:02:28,000
that linear systems are the best
kind.

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00:02:25,000 --> 00:02:31,000
And, boy, they certainly are.
When are we going to call a

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00:02:31,000 --> 00:02:37,000
system linear?
I think in the beginning you

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00:02:34,000 --> 00:02:40,000
should learn a little
terminology before we launch in

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00:02:39,000 --> 00:02:45,000
and actually try to start to
solve these things.

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00:02:43,000 --> 00:02:49,000
Well, the x and y,
the dependent variables must

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00:02:46,000 --> 00:02:52,000
occur linearly.
In other words,

44
00:02:49,000 --> 00:02:55,000
it must look like this,
ax plus by.

45
00:02:53,000 --> 00:02:59,000
Now, the t can be a mess.
And so I will throw in an extra

46
00:02:58,000 --> 00:03:04,000
function of t there.
And y prime will be some

47
00:03:03,000 --> 00:03:09,000
other linear combination of x
and y, plus some other messy

48
00:03:08,000 --> 00:03:14,000
function of t.
But even the a,

49
00:03:10,000 --> 00:03:16,000
b, c, and d are allowed to be
functions of t.

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00:03:14,000 --> 00:03:20,000
They could be one over t cubed
or sine t

51
00:03:19,000 --> 00:03:25,000
or something like that.
So I have to distinguish those

52
00:03:23,000 --> 00:03:29,000
cases.
The case where a,

53
00:03:25,000 --> 00:03:31,000
b, c, and d are constants,
that I will call --

54
00:03:30,000 --> 00:03:36,000
Well, there are different
things you can call it.

55
00:03:34,000 --> 00:03:40,000
We will simply call it a
constant coefficient system.

56
00:03:40,000 --> 00:03:46,000
A system with coefficients
would probably be better

57
00:03:45,000 --> 00:03:51,000
English.
On the other hand,

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00:03:47,000 --> 00:03:53,000
a, b, c, and d,
this system will still be

59
00:03:51,000 --> 00:03:57,000
called linear if these are
functions of t.

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00:03:56,000 --> 00:04:02,000
Can also be functions of t.

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00:04:05,000 --> 00:04:11,000
So it would be a perfectly good
linear system to have x prime

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00:04:08,000 --> 00:04:14,000
equals tx plus sine t times y
plus e to the minus t squared.

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00:04:15,000 --> 00:04:21,000
You would never see something
like that but it is okay.

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00:04:18,000 --> 00:04:24,000
What else do you need to know?
Well, what would a homogenous

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00:04:22,000 --> 00:04:28,000
system be?
A homogenous system is one

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00:04:24,000 --> 00:04:30,000
without these extra guys.
That doesn't mean there is no t

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00:04:28,000 --> 00:04:34,000
in it.
There could be t in the a,

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00:04:32,000 --> 00:04:38,000
b, c and d, but these terms
with no x and y in them must not

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00:04:38,000 --> 00:04:44,000
occur.
So, a linear homogenous.

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00:04:47,000 --> 00:04:53,000
And that is the kind we are
going to start studying first in

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00:04:50,000 --> 00:04:56,000
the same way when we studied
higher order equations.

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00:04:53,000 --> 00:04:59,000
We studied first homogenous.
You had to know how to solve

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00:04:57,000 --> 00:05:03,000
those first, and then you could
learn how to solve the more

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00:05:00,000 --> 00:05:06,000
general kind.
So linear homogenous means that

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00:05:04,000 --> 00:05:10,000
r1 is zero and r2 is zero for
all time.

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00:05:07,000 --> 00:05:13,000
They are identically zero.
They are not there.

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00:05:10,000 --> 00:05:16,000
You don't see them.
Have I left anything out?

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00:05:13,000 --> 00:05:19,000
Yes, the initial conditions.
Since that is quite general,

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00:05:18,000 --> 00:05:24,000
let's talk about what would
initial conditions look like?

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00:05:28,000 --> 00:05:34,000
Well, in a general way,
the reason you have to have

81
00:05:31,000 --> 00:05:37,000
initial conditions is to get
values for the arbitrary

82
00:05:34,000 --> 00:05:40,000
constants that appear in the
solution.

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00:05:37,000 --> 00:05:43,000
The question is,
how many arbitrary constants

84
00:05:40,000 --> 00:05:46,000
are going to appear in the
solutions of these equations?

85
00:05:43,000 --> 00:05:49,000
Well, I will just give you the
answer.

86
00:05:46,000 --> 00:05:52,000
Two.
The number of arbitrary

87
00:05:48,000 --> 00:05:54,000
constants that appear is the
total order of the system.

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00:05:51,000 --> 00:05:57,000
For example,
if this were a second

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00:05:53,000 --> 00:05:59,000
derivative and this were a first
derivative, I would expect three

90
00:05:58,000 --> 00:06:04,000
arbitrary constants in the
system --

91
00:06:02,000 --> 00:06:08,000
-- because the total,
the sum of two and one makes

92
00:06:05,000 --> 00:06:11,000
three.
So you must have as many

93
00:06:07,000 --> 00:06:13,000
initial conditions as you have
arbitrary constants in the

94
00:06:11,000 --> 00:06:17,000
solution.
And that, of course,

95
00:06:13,000 --> 00:06:19,000
explains when we studied
second-order equations,

96
00:06:17,000 --> 00:06:23,000
we had to have two initial
conditions.

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00:06:19,000 --> 00:06:25,000
I had to specify the initial
starting point and the initial

98
00:06:24,000 --> 00:06:30,000
velocity.
And the reason we had to have

99
00:06:26,000 --> 00:06:32,000
two conditions was because the
general solution had two

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00:06:30,000 --> 00:06:36,000
arbitrary constants in it.
The same thing happens here but

101
00:06:36,000 --> 00:06:42,000
the answer is it is more
natural, the conditions here are

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00:06:40,000 --> 00:06:46,000
more natural.
I don't have to specify the

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00:06:43,000 --> 00:06:49,000
velocity.
Why not?

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00:06:44,000 --> 00:06:50,000
Well, because an initial
condition, of course,

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00:06:48,000 --> 00:06:54,000
would want me to say what the
starting value of x is,

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00:06:52,000 --> 00:06:58,000
some number,
and it will also want to know

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00:06:55,000 --> 00:07:01,000
what the starting value of y is
at that same point.

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00:07:00,000 --> 00:07:06,000
Well, there are my two
conditions.

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00:07:02,000 --> 00:07:08,000
And since this is going to have
two arbitrary constants in it,

110
00:07:07,000 --> 00:07:13,000
it is these initial conditions
that will satisfy,

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00:07:10,000 --> 00:07:16,000
the arbitrary constants will
have to be picked so as to

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00:07:14,000 --> 00:07:20,000
satisfy those initial
conditions.

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00:07:17,000 --> 00:07:23,000
In some sense,
the giving of initial

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00:07:19,000 --> 00:07:25,000
conditions for a system is a
more natural activity than

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00:07:23,000 --> 00:07:29,000
giving the initial conditions of
a second order system.

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00:07:29,000 --> 00:07:35,000
You don't have to be the least
bit cleaver about it.

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00:07:32,000 --> 00:07:38,000
Anybody would give these two
numbers.

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00:07:35,000 --> 00:07:41,000
Whereas, somebody faced with a
second order system might

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00:07:38,000 --> 00:07:44,000
scratch his head.
And, in fact,

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00:07:40,000 --> 00:07:46,000
there are other kinds of
conditions.

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00:07:43,000 --> 00:07:49,000
There are boundary conditions
you learned a little bit about

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00:07:47,000 --> 00:07:53,000
instead of initial conditions
for a second order equation.

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00:07:51,000 --> 00:07:57,000
I cannot think of any more
general terminology,

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00:07:54,000 --> 00:08:00,000
so it sounds like we are going
to actually have to get to work.

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00:08:00,000 --> 00:08:06,000
Okay, let's get to work.
I want to set up a system and

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00:08:04,000 --> 00:08:10,000
solve it.
And since one of the things in

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00:08:07,000 --> 00:08:13,000
this course is supposed to be
simple modeling,

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00:08:10,000 --> 00:08:16,000
it should be a system that
models something.

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00:08:13,000 --> 00:08:19,000
In general, the kinds of models
we are going to use when we

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00:08:18,000 --> 00:08:24,000
study systems are the same ones
we used in studying just

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00:08:22,000 --> 00:08:28,000
first-order equations.
Mixing, radioactive decay,

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00:08:25,000 --> 00:08:31,000
temperature,
the motion of temperature.

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00:08:30,000 --> 00:08:36,000
Heat, heat conduction,
in other words.

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00:08:32,000 --> 00:08:38,000
Diffusion.
I have given you a diffusion

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00:08:35,000 --> 00:08:41,000
problem for your first homework
on this subject.

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00:08:39,000 --> 00:08:45,000
What else did we do?
That's all I can think of for

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00:08:43,000 --> 00:08:49,000
the moment, but I am sure they
will occur to me.

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00:08:46,000 --> 00:08:52,000
When, out of those physical
ideas, are we going to get a

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00:08:50,000 --> 00:08:56,000
system?
The answer is,

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00:08:52,000 --> 00:08:58,000
whenever there are two of
something that there was only

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00:08:56,000 --> 00:09:02,000
one of before.
For example,

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00:08:59,000 --> 00:09:05,000
if I have mixing with two tanks
where the fluid goes like that.

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00:09:03,000 --> 00:09:09,000
Say you want to have a big tank
and a little tank here and you

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00:09:07,000 --> 00:09:13,000
want to put some stuff into the
little tank so that it will get

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00:09:10,000 --> 00:09:16,000
mixed in the big tank without
having to climb a big ladder and

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00:09:14,000 --> 00:09:20,000
stop and drop the stuff in.
That will require two tanks,

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00:09:17,000 --> 00:09:23,000
the concentration of the
substance in each tank,

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00:09:20,000 --> 00:09:26,000
therefore, that will require a
system of equations rather than

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00:09:24,000 --> 00:09:30,000
just one.
Or, to give something closer to

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00:09:28,000 --> 00:09:34,000
home, closer to this backboard,
anyway, suppose you have dah,

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00:09:33,000 --> 00:09:39,000
dah, dah, don't groan,
at least not audibly,

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00:09:36,000 --> 00:09:42,000
something that looks like that.
And next to it put an EMF

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00:09:40,000 --> 00:09:46,000
there.
That is just a first order.

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00:09:43,000 --> 00:09:49,000
That just leads to a single
first order equation.

155
00:09:47,000 --> 00:09:53,000
But suppose it is a two loop
circuit.

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00:09:58,000 --> 00:10:04,000
Now I need a pair of equations.
Each of these loops gives a

157
00:10:02,000 --> 00:10:08,000
first order differential
equation, but they have to be

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00:10:06,000 --> 00:10:12,000
solved simultaneously to find
the current or the charges on

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00:10:10,000 --> 00:10:16,000
the condensers.
And if I want a system of three

160
00:10:14,000 --> 00:10:20,000
equations, throw in another
loop.

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00:10:16,000 --> 00:10:22,000
Now, suppose I put in a coil
instead.

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00:10:19,000 --> 00:10:25,000
What is this going to lead to?
This is going to give me a

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00:10:23,000 --> 00:10:29,000
system of three equations of
which this will be first order,

164
00:10:27,000 --> 00:10:33,000
first order.
And this will be second order

165
00:10:32,000 --> 00:10:38,000
because it has a coil.
You are up to that,

166
00:10:35,000 --> 00:10:41,000
right?
You've had coils,

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00:10:37,000 --> 00:10:43,000
inductance?
Good.

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00:10:39,000 --> 00:10:45,000
So the whole thing is going to
count as first-order,

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00:10:43,000 --> 00:10:49,000
first-order,
second-order.

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00:10:45,000 --> 00:10:51,000
To find out how complicated it
is, you have to add up the

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00:10:50,000 --> 00:10:56,000
orders.
That is one and one,

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00:10:52,000 --> 00:10:58,000
and two.
This is really fourth-order

173
00:10:55,000 --> 00:11:01,000
stuff that we are talking about
here.

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00:11:00,000 --> 00:11:06,000
We can expect it to be a little
complicated.

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00:11:03,000 --> 00:11:09,000
Well, now let's take a modest
little problem.

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00:11:06,000 --> 00:11:12,000
I am going to return to a
problem we considered earlier in

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00:11:10,000 --> 00:11:16,000
the problem of heat conduction.
I had forgotten whether it was

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00:11:14,000 --> 00:11:20,000
on the problem set or I did it
in class, but I am choosing it

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00:11:19,000 --> 00:11:25,000
because it leads to something we
will be able to solve.

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00:11:23,000 --> 00:11:29,000
And because it illustrates how
to add a little sophistication

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00:11:27,000 --> 00:11:33,000
to something that was
unsophisticated before.

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00:11:32,000 --> 00:11:38,000
A pot of water.
External temperature Te of t.

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00:11:35,000 --> 00:11:41,000
I am talking about the

184
00:11:38,000 --> 00:11:44,000
temperature of something.
And what I am talking about the

185
00:11:43,000 --> 00:11:49,000
temperature of will be an egg
that is cooking inside,

186
00:11:47,000 --> 00:11:53,000
but with a difference.
This egg is not homogenous

187
00:11:51,000 --> 00:11:57,000
inside.
Instead it has a white and it

188
00:11:54,000 --> 00:12:00,000
has a yolk in the middle.
In other words,

189
00:11:59,000 --> 00:12:05,000
it is a real egg and not a
phony egg.

190
00:12:01,000 --> 00:12:07,000
That is a small pot,
or it is an ostrich egg.

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00:12:04,000 --> 00:12:10,000
[LAUGHTER] That is the yoke.
The yolk is contained in a

192
00:12:08,000 --> 00:12:14,000
little membrane inside.
And there are little yucky

193
00:12:11,000 --> 00:12:17,000
things that hold it in position.
And we are going to let the

194
00:12:15,000 --> 00:12:21,000
temperature of the yolk,
if you can see in the back of

195
00:12:19,000 --> 00:12:25,000
the room, be T1.
That is the temperature of the

196
00:12:22,000 --> 00:12:28,000
yolk.
The temperature of the white,

197
00:12:25,000 --> 00:12:31,000
which we will assume is
uniform, is going to be T2.

198
00:12:30,000 --> 00:12:36,000
Oh, that's the water bath.
The temperature of the white is

199
00:12:34,000 --> 00:12:40,000
T2, and then the temperature of
the external water bath.

200
00:12:39,000 --> 00:12:45,000
In other words,
the reason for introducing two

201
00:12:42,000 --> 00:12:48,000
variables instead of just the
one variable for the overall

202
00:12:47,000 --> 00:12:53,000
temperature of the egg we had is
because egg white is liquid pure

203
00:12:52,000 --> 00:12:58,000
protein, more or less,
and the T1, the yolk has a lot

204
00:12:57,000 --> 00:13:03,000
of fat and cholesterol and other
stuff like that which is

205
00:13:01,000 --> 00:13:07,000
supposed to be bad for you.
It certainly has different

206
00:13:06,000 --> 00:13:12,000
conducting.
It is liquid,

207
00:13:07,000 --> 00:13:13,000
at the beginning at any rate,
but it certainly has different

208
00:13:11,000 --> 00:13:17,000
constants of conductivity than
the egg white would.

209
00:13:14,000 --> 00:13:20,000
And the condition of heat
through the shell of the egg

210
00:13:17,000 --> 00:13:23,000
would be different from the
conduction of heat through the

211
00:13:20,000 --> 00:13:26,000
membrane that keeps the yoke
together.

212
00:13:22,000 --> 00:13:28,000
So it is quite reasonable to
consider that the white and the

213
00:13:26,000 --> 00:13:32,000
yolk will be at different
temperatures and will have

214
00:13:29,000 --> 00:13:35,000
different conductivity
properties.

215
00:13:32,000 --> 00:13:38,000
I am going to use Newton's laws
but with this further

216
00:13:36,000 --> 00:13:42,000
refinement.
In other words,

217
00:13:38,000 --> 00:13:44,000
introducing two temperatures.
Whereas, before we only had one

218
00:13:43,000 --> 00:13:49,000
temperature.
But let's use Newton's law.

219
00:13:46,000 --> 00:13:52,000
Let's see.
The question is how does T1,

220
00:13:49,000 --> 00:13:55,000
the temperature of the yolk,
vary with time?

221
00:13:52,000 --> 00:13:58,000
Well, the yolk is getting all
its heat from the white.

222
00:13:57,000 --> 00:14:03,000
Therefore, Newton's law of
conduction will be some constant

223
00:14:01,000 --> 00:14:07,000
of conductivity for the yolk
times T2 minus T1.

224
00:14:08,000 --> 00:14:14,000
The yolk does not know anything
about the external temperature

225
00:14:12,000 --> 00:14:18,000
of the water bath.
It is completely surrounded,

226
00:14:15,000 --> 00:14:21,000
snug and secure within itself.
But how about the temperature

227
00:14:20,000 --> 00:14:26,000
of the egg white?
That gets heat and gives heat

228
00:14:23,000 --> 00:14:29,000
to two sources,
from the external water and

229
00:14:26,000 --> 00:14:32,000
also from the internal yolk
inside.

230
00:14:30,000 --> 00:14:36,000
So you have to take into
account both of those.

231
00:14:33,000 --> 00:14:39,000
Its conduction of the heat
through that membrane,

232
00:14:36,000 --> 00:14:42,000
we will use the same a,
which is going to be a times T1

233
00:14:40,000 --> 00:14:46,000
minus T2.
Remember the order in which you

234
00:14:44,000 --> 00:14:50,000
have to write these is governed
by the yolk outside to the

235
00:14:48,000 --> 00:14:54,000
white.
Therefore, that has to come

236
00:14:51,000 --> 00:14:57,000
first when I write it in order
that a be a positive constant.

237
00:14:55,000 --> 00:15:01,000
But it is also getting heat
from the water bath.

238
00:15:00,000 --> 00:15:06,000
And, presumably,
the conductivity through the

239
00:15:03,000 --> 00:15:09,000
shell is different from what it
is through this membrane around

240
00:15:08,000 --> 00:15:14,000
the yolk.
So I am going to call that by a

241
00:15:11,000 --> 00:15:17,000
different constant.
This is the conductivity

242
00:15:14,000 --> 00:15:20,000
through the shell into the
white.

243
00:15:17,000 --> 00:15:23,000
And that is going to be T,
the external temperature minus

244
00:15:21,000 --> 00:15:27,000
the temperature of the egg
white.

245
00:15:24,000 --> 00:15:30,000
Here I have a system of
equations because I want to make

246
00:15:28,000 --> 00:15:34,000
two dependent variables by
refining the original problem.

247
00:15:34,000 --> 00:15:40,000
Now, you always have to write a
system in standard form to solve

248
00:15:39,000 --> 00:15:45,000
it.
You will see that the left-hand

249
00:15:42,000 --> 00:15:48,000
side will give the dependent
variables in a certain order.

250
00:15:47,000 --> 00:15:53,000
In this case,
the temperature of the yolk and

251
00:15:51,000 --> 00:15:57,000
then the temperature of the
white.

252
00:15:54,000 --> 00:16:00,000
The law is that in order not to
make mistakes --

253
00:16:00,000 --> 00:16:06,000
And it's a very frequent source
of error so learn from the

254
00:16:03,000 --> 00:16:09,000
beginning not to do this.
You must write the variables on

255
00:16:07,000 --> 00:16:13,000
the right-hand side in the same
order left to right in which

256
00:16:11,000 --> 00:16:17,000
they occur top to bottom here.
In other words,

257
00:16:14,000 --> 00:16:20,000
this is not a good way to leave
that.

258
00:16:16,000 --> 00:16:22,000
This is the first attempt in
writing this system,

259
00:16:20,000 --> 00:16:26,000
but the final version should
like this.

260
00:16:22,000 --> 00:16:28,000
T1 prime,
I won't bother writing dT / dt,

261
00:16:25,000 --> 00:16:31,000
is equal to --
T1 must come first,

262
00:16:29,000 --> 00:16:35,000
so minus a times T1 plus a
times T2.

263
00:16:34,000 --> 00:16:40,000
And the same law for the second
one.

264
00:16:38,000 --> 00:16:44,000
It must come in the same order.
Now, the coefficient of T1,

265
00:16:44,000 --> 00:16:50,000
that is easy.
That's a times T1.

266
00:16:47,000 --> 00:16:53,000
The coefficient of T2 is minus
a minus b,

267
00:16:52,000 --> 00:16:58,000
so minus (a plus b) times T2.

268
00:16:56,000 --> 00:17:02,000
But I am not done yet.
There is still this external

269
00:17:02,000 --> 00:17:08,000
temperature I must put into the
equation.

270
00:17:06,000 --> 00:17:12,000
Now, that is not a variable.
This is some given function of

271
00:17:11,000 --> 00:17:17,000
t.
And what the function of t is,

272
00:17:14,000 --> 00:17:20,000
of course, depends upon what
the problem is.

273
00:17:18,000 --> 00:17:24,000
So that, for example,
what might be some

274
00:17:22,000 --> 00:17:28,000
possibilities,
well, suppose the problem was I

275
00:17:26,000 --> 00:17:32,000
wanted to coddle the egg.
I think there is a generation

276
00:17:32,000 --> 00:17:38,000
gap here.
How many of you know what a

277
00:17:35,000 --> 00:17:41,000
coddled egg is?
How many of you don't know?

278
00:17:38,000 --> 00:17:44,000
Well, I'm just saying my
daughter didn't know.

279
00:17:42,000 --> 00:17:48,000
I mentioned it to her.
I said I think I'm going to do

280
00:17:46,000 --> 00:17:52,000
a coddled egg tomorrow in class.
And she said what is that?

281
00:17:51,000 --> 00:17:57,000
And so I said a cuddled egg?
She said why would someone

282
00:17:55,000 --> 00:18:01,000
cuddle an egg?
I said coddle.

283
00:17:59,000 --> 00:18:05,000
And she said,
oh, you mean like a person,

284
00:18:02,000 --> 00:18:08,000
like what you do to somebody
you like or don't like or I

285
00:18:07,000 --> 00:18:13,000
don't know.
Whatever.

286
00:18:09,000 --> 00:18:15,000
I thought a while and said,
yeah, more like that.

287
00:18:13,000 --> 00:18:19,000
[LAUGHTER] Anyway,
for the enrichment of your

288
00:18:17,000 --> 00:18:23,000
cooking skills,
to coddle an egg,

289
00:18:20,000 --> 00:18:26,000
it is considered to produce a
better quality product than

290
00:18:25,000 --> 00:18:31,000
boiling an egg.
That is why people do it.

291
00:18:30,000 --> 00:18:36,000
You heat up the water to
boiling, the egg should be at

292
00:18:34,000 --> 00:18:40,000
room temperature,
and then you carefully lower

293
00:18:37,000 --> 00:18:43,000
the egg into the water.
And you turn off the heat so

294
00:18:41,000 --> 00:18:47,000
the water bath cools
exponentially while the egg

295
00:18:45,000 --> 00:18:51,000
inside is rising in temperature.
And then you wait four minutes

296
00:18:50,000 --> 00:18:56,000
or six minutes or whatever and
take it out.

297
00:18:53,000 --> 00:18:59,000
You have a perfect egg.
So for coddling,

298
00:18:56,000 --> 00:19:02,000
spelled so, what will the
external temperature be?

299
00:19:02,000 --> 00:19:08,000
Well, it starts out at time
zero at 100 degrees centigrade

300
00:19:06,000 --> 00:19:12,000
because the water is supposed to
be boiling.

301
00:19:09,000 --> 00:19:15,000
The reason you have it boiling
is for calibration so that you

302
00:19:13,000 --> 00:19:19,000
can know what temperature it is
without having to use a

303
00:19:17,000 --> 00:19:23,000
thermometer, unless you're on
Pike's Peak or some place.

304
00:19:20,000 --> 00:19:26,000
It starts out at 100 degrees.
And after that,

305
00:19:24,000 --> 00:19:30,000
since the light is off,
it cools exponential because

306
00:19:27,000 --> 00:19:33,000
that is another law.
You only have to know what K is

307
00:19:32,000 --> 00:19:38,000
for your particular pot and you
will be able to solve the

308
00:19:37,000 --> 00:19:43,000
coddled egg problem.
In other words,

309
00:19:40,000 --> 00:19:46,000
you will then be able to solve
these equations and know how the

310
00:19:45,000 --> 00:19:51,000
temperature rises.
I am going to solve a different

311
00:19:49,000 --> 00:19:55,000
problem because I don't want to
have to deal with this

312
00:19:54,000 --> 00:20:00,000
inhomogeneous term.
Let's use, as a different

313
00:19:58,000 --> 00:20:04,000
problem, a person cooks an egg.
Coddles the egg by the first

314
00:20:04,000 --> 00:20:10,000
process, decides the egg is
done, let's say hardboiled,

315
00:20:09,000 --> 00:20:15,000
and then you are supposed to
drop a hardboiled egg into cold

316
00:20:14,000 --> 00:20:20,000
water.
Not just to cool it but also

317
00:20:17,000 --> 00:20:23,000
because I think it prevents that
dark thing from forming that

318
00:20:23,000 --> 00:20:29,000
looks sort of unattractive.
Let's ice bath.

319
00:20:28,000 --> 00:20:34,000
The only reason for dropping
the egg into an ice bath is so

320
00:20:32,000 --> 00:20:38,000
that you could have a homogenous
equation to solve.

321
00:20:36,000 --> 00:20:42,000
And since this a first system
we are going to solve,

322
00:20:40,000 --> 00:20:46,000
let's make life easy for
ourselves.

323
00:20:43,000 --> 00:20:49,000
Now, all my work in preparing
this example,

324
00:20:47,000 --> 00:20:53,000
and it took considerably longer
time than actually solving the

325
00:20:52,000 --> 00:20:58,000
problem, was in picking values
for a and b which would make

326
00:20:56,000 --> 00:21:02,000
everything come out nice.
It's harder than it looks.

327
00:21:02,000 --> 00:21:08,000
The values that we are going to
use, which make no physical

328
00:21:07,000 --> 00:21:13,000
sense whatsoever,
but a equals 2 and b

329
00:21:11,000 --> 00:21:17,000
equals 3.
These are called nice numbers.

330
00:21:15,000 --> 00:21:21,000
What is the equation?
What is the system?

331
00:21:18,000 --> 00:21:24,000
Can somebody read it off for
me?

332
00:21:21,000 --> 00:21:27,000
It is T1 prime equals, what
is it, minus 2T1 plus 2T2.

333
00:21:26,000 --> 00:21:32,000
That's good.

334
00:21:30,000 --> 00:21:36,000
Minus 2T1 plus 2T2.

335
00:21:40,000 --> 00:21:46,000
T2 prime is,
what is it?

336
00:21:42,000 --> 00:21:48,000
I think this is 2T1.
And the other one is minus a

337
00:21:48,000 --> 00:21:54,000
plus b, so minus 5.

338
00:21:51,000 --> 00:21:57,000
This is a system.
Now, on Wednesday I will teach

339
00:21:57,000 --> 00:22:03,000
you a fancy way of solving this.
But, to be honest,

340
00:22:03,000 --> 00:22:09,000
the fancy way will take roughly
about as long as the way I am

341
00:22:07,000 --> 00:22:13,000
going to do it now.
The main reason for doing it is

342
00:22:10,000 --> 00:22:16,000
that it introduces new
vocabulary which everyone wants

343
00:22:14,000 --> 00:22:20,000
you to have.
And also, more important

344
00:22:16,000 --> 00:22:22,000
reasons, it gives more insight
into the solution than this

345
00:22:20,000 --> 00:22:26,000
method.
This method just produces the

346
00:22:22,000 --> 00:22:28,000
answer, but you want insight,
also.

347
00:22:24,000 --> 00:22:30,000
And that is just as important.
But for now,

348
00:22:28,000 --> 00:22:34,000
let's use a method which always
works and which in 40 years,

349
00:22:33,000 --> 00:22:39,000
after you have forgotten all
other fancy methods,

350
00:22:36,000 --> 00:22:42,000
will still be available to you
because it is method you can

351
00:22:40,000 --> 00:22:46,000
figure out yourself.
You don't have to remember

352
00:22:43,000 --> 00:22:49,000
anything.
The method is to eliminate one

353
00:22:46,000 --> 00:22:52,000
of the dependent variables.
It is just the way you solve

354
00:22:50,000 --> 00:22:56,000
systems of linear equations in
general if you aren't doing

355
00:22:54,000 --> 00:23:00,000
something fancy with
determinants and matrices.

356
00:22:59,000 --> 00:23:05,000
If you just eliminate
variables.

357
00:23:01,000 --> 00:23:07,000
We are going to eliminate one
of these variables.

358
00:23:05,000 --> 00:23:11,000
Let's eliminate T2.
You could also eliminate T1.

359
00:23:08,000 --> 00:23:14,000
The main thing is eliminate one
of them so you will have just

360
00:23:13,000 --> 00:23:19,000
one left to work with.
How do I eliminate T2?

361
00:23:16,000 --> 00:23:22,000
Beg your pardon?
Is something wrong?

362
00:23:19,000 --> 00:23:25,000
If somebody thinks something is
wrong raise his hand.

363
00:23:23,000 --> 00:23:29,000
No?

364
00:23:30,000 --> 00:23:36,000
Why do I want to get rid of T1?
Well, I can add them.

365
00:23:33,000 --> 00:23:39,000
But, on the left-hand side,
I will have T1 prime plus T2

366
00:23:36,000 --> 00:23:42,000
prime. What good is that?

367
00:23:39,000 --> 00:23:45,000
[LAUGHTER]

368
00:23:48,000 --> 00:23:54,000
I think you will want to do it
my way.

369
00:23:49,000 --> 00:23:55,000
[APPLAUSE]

370
00:24:03,000 --> 00:24:09,000
Solve for T2 in terms of T1.
That is going to be T1 prime

371
00:24:08,000 --> 00:24:14,000
plus 2T1 divided by 2.

372
00:24:12,000 --> 00:24:18,000
Now, take that and substitute
it into the second equation.

373
00:24:18,000 --> 00:24:24,000
Wherever you see a T2,
put that in,

374
00:24:21,000 --> 00:24:27,000
and what you will be left with
is something just in T1.

375
00:24:28,000 --> 00:24:34,000
To be honest,
I don't know any other good way

376
00:24:31,000 --> 00:24:37,000
of doing this.
There is a fancy method that I

377
00:24:34,000 --> 00:24:40,000
think is talked about in your
book, which leads to extraneous

378
00:24:39,000 --> 00:24:45,000
solutions and so on,
but you don't want to know

379
00:24:43,000 --> 00:24:49,000
about that.
This will work for a simple

380
00:24:46,000 --> 00:24:52,000
linear equation with constant
coefficients,

381
00:24:49,000 --> 00:24:55,000
always.
Substitute in.

382
00:24:51,000 --> 00:24:57,000
What do I do?
Now, here I do not advise doing

383
00:24:54,000 --> 00:25:00,000
this mentally.
It is just too easy to make a

384
00:24:57,000 --> 00:25:03,000
mistake.
Here, I will do it carefully,

385
00:25:04,000 --> 00:25:10,000
writing everything out just as
you would.

386
00:25:10,000 --> 00:25:16,000
T1 prime plus 2T1 over 2,
prime, equals 2T1 minus 5 time

387
00:25:18,000 --> 00:25:24,000
T1 prime plus 2T1 over two.

388
00:25:27,000 --> 00:25:33,000
I took that and substituted

389
00:25:32,000 --> 00:25:38,000
into this equation.
Now, I don't like those two's.

390
00:25:38,000 --> 00:25:44,000
Let's get rid of them by
multiplying.

391
00:25:42,000 --> 00:25:48,000
This will become 4.

392
00:25:52,000 --> 00:25:58,000
And now write this out.
What is this when you look at

393
00:25:57,000 --> 00:26:03,000
it?
This is an equation just in T1.

394
00:26:00,000 --> 00:26:06,000
It has constant coefficients.
And what is its order?

395
00:26:05,000 --> 00:26:11,000
Its order is two because T1
prime primed.

396
00:26:10,000 --> 00:26:16,000
In other words,
I can eliminate T2 okay,

397
00:26:13,000 --> 00:26:19,000
but the equation I am going to
get is no longer a first-order.

398
00:26:19,000 --> 00:26:25,000
It becomes a second-order
differential equation.

399
00:26:24,000 --> 00:26:30,000
And that's a basic law.
Even if you have a system of

400
00:26:30,000 --> 00:26:36,000
more equations,
three or four or whatever,

401
00:26:33,000 --> 00:26:39,000
the law is that after you do
the elimination successfully and

402
00:26:37,000 --> 00:26:43,000
end up with a single equation,
normally the order of that

403
00:26:42,000 --> 00:26:48,000
equation will be the sum of the
orders of the things you started

404
00:26:46,000 --> 00:26:52,000
with.
So two first-order equations

405
00:26:49,000 --> 00:26:55,000
will always produce a
second-order equation in just

406
00:26:53,000 --> 00:26:59,000
one dependent variable,
three will produce a third

407
00:26:56,000 --> 00:27:02,000
order equation and so on.
So you trade one complexity for

408
00:27:02,000 --> 00:27:08,000
another.
You trade the complexity of

409
00:27:04,000 --> 00:27:10,000
having to deal with two
equations simultaneously instead

410
00:27:09,000 --> 00:27:15,000
of just one for the complexity
of having to deal with a single

411
00:27:13,000 --> 00:27:19,000
higher order equation which is
more trouble to solve.

412
00:27:17,000 --> 00:27:23,000
It is like all mathematical
problems.

413
00:27:20,000 --> 00:27:26,000
Unless you are very lucky,
if you push them down one way,

414
00:27:24,000 --> 00:27:30,000
they are really simple now,
they just pop up some place

415
00:27:28,000 --> 00:27:34,000
else.
You say, oh,

416
00:27:30,000 --> 00:27:36,000
I didn't save anything after
all.

417
00:27:32,000 --> 00:27:38,000
That is the law of conservation
of mathematical difficulty.

418
00:27:36,000 --> 00:27:42,000
[LAUGHTER] You saw that even
with the Laplace transform.

419
00:27:40,000 --> 00:27:46,000
In the beginning it looks
great, you've got these tables,

420
00:27:44,000 --> 00:27:50,000
take the equation,
horrible to solve.

421
00:27:46,000 --> 00:27:52,000
Take some transform,
trivial to solve for capital Y.

422
00:27:50,000 --> 00:27:56,000
Now I have to find the inverse
Laplace transform.

423
00:27:53,000 --> 00:27:59,000
And suddenly all the work is
there, partial fractions,

424
00:27:57,000 --> 00:28:03,000
funny formulas and so on.
It is very hard in mathematics

425
00:28:02,000 --> 00:28:08,000
to get away with something.
It happens now and then and

426
00:28:06,000 --> 00:28:12,000
everybody cheers.
Let's write this out now in the

427
00:28:09,000 --> 00:28:15,000
form in which it looks like an
equation we can actually solve.

428
00:28:13,000 --> 00:28:19,000
Just be careful.
Now it is all right to use the

429
00:28:17,000 --> 00:28:23,000
method by which you collect
terms.

430
00:28:19,000 --> 00:28:25,000
There is only one term
involving T1 double prime.

431
00:28:23,000 --> 00:28:29,000
It's the one that comes from
here.

432
00:28:25,000 --> 00:28:31,000
How about the terms in T1
prime?

433
00:28:27,000 --> 00:28:33,000
There is a 2.
Here, there is minus 5 T1

434
00:28:33,000 --> 00:28:39,000
prime.
If I put it on the other side

435
00:28:37,000 --> 00:28:43,000
it makes plus 5 T1 prime plus
this two makes 7 T1 prime.

436
00:28:44,000 --> 00:28:50,000
And how many T1's are there?
Well, none on the left-hand

437
00:28:51,000 --> 00:28:57,000
side.
On the right-hand side I have 4

438
00:28:55,000 --> 00:29:01,000
here minus 10.
4 minus 10 is negative 6.

439
00:29:02,000 --> 00:29:08,000
Negative 6 T1 put on this
left-hand side the way we want

440
00:29:06,000 --> 00:29:12,000
to do makes plus 6 T1.

441
00:29:15,000 --> 00:29:21,000
There are no inhomogeneous
terms, so that is equal to zero.

442
00:29:18,000 --> 00:29:24,000
If I had gotten a negative
number for one of these

443
00:29:22,000 --> 00:29:28,000
coefficients,
I would instantly know if I had

444
00:29:25,000 --> 00:29:31,000
made a mistake.
Why?

445
00:29:26,000 --> 00:29:32,000
Why must those numbers come out
to be positive?

446
00:29:30,000 --> 00:29:36,000
It is because the system must
be, the system must be,

447
00:29:33,000 --> 00:29:39,000
fill in with one word,
stable.

448
00:29:36,000 --> 00:29:42,000
And why must this system be
stable?

449
00:29:38,000 --> 00:29:44,000
In other words,
the long-term solutions must be

450
00:29:42,000 --> 00:29:48,000
zero, must all go to zero,
whatever they are.

451
00:29:45,000 --> 00:29:51,000
Why is that?
Well, because you are putting

452
00:29:48,000 --> 00:29:54,000
the egg into an ice bath.
Or, because we know it was

453
00:29:52,000 --> 00:29:58,000
living but after being
hardboiled it is dead and,

454
00:29:56,000 --> 00:30:02,000
therefore, dead systems are
stable.

455
00:30:00,000 --> 00:30:06,000
That's not a good reason but it
is, so to speak,

456
00:30:03,000 --> 00:30:09,000
the real one.
It's clear anyway that all

457
00:30:05,000 --> 00:30:11,000
solutions must tend to zero
physically.

458
00:30:08,000 --> 00:30:14,000
That's obvious.
And, therefore,

459
00:30:10,000 --> 00:30:16,000
the differential equation must
have the same property,

460
00:30:14,000 --> 00:30:20,000
and that means that its
coefficients must be positive.

461
00:30:17,000 --> 00:30:23,000
All its coefficients must be
positive.

462
00:30:20,000 --> 00:30:26,000
If this weren't there,
I would get oscillating

463
00:30:23,000 --> 00:30:29,000
solutions, which wouldn't go to
zero.

464
00:30:25,000 --> 00:30:31,000
That is physical impossible for
this egg.

465
00:30:30,000 --> 00:30:36,000
Now the rest is just solving.
The characteristic equation,

466
00:30:34,000 --> 00:30:40,000
if you can remember way,
way back in prehistoric times

467
00:30:39,000 --> 00:30:45,000
when we were solving these
equations, is this.

468
00:30:43,000 --> 00:30:49,000
And what you want to do is
factor it.

469
00:30:46,000 --> 00:30:52,000
This is where all the work was,
getting those numbers so that

470
00:30:51,000 --> 00:30:57,000
this would factor. So it's
r plus 1 times r plus 6

471
00:31:04,000 --> 00:31:10,000
And so the solutions are,
the roots are r equals

472
00:31:07,000 --> 00:31:13,000
negative 1.
I am just making marks on the

473
00:31:10,000 --> 00:31:16,000
board, but you have done this
often enough,

474
00:31:13,000 --> 00:31:19,000
you know what I am talking
about.

475
00:31:15,000 --> 00:31:21,000
So the characteristic roots are
those two numbers.

476
00:31:18,000 --> 00:31:24,000
And, therefore,
the solution is,

477
00:31:20,000 --> 00:31:26,000
I could write down immediately
with its arbitrary constant as

478
00:31:24,000 --> 00:31:30,000
c1 times e to the negative t
plus c2 times e to the negative

479
00:31:28,000 --> 00:31:34,000
6t. Now, I have got to get T2.

480
00:31:34,000 --> 00:31:40,000
Here the first worry is T2 is
going to give me two more

481
00:31:39,000 --> 00:31:45,000
arbitrary constants.
It better not.

482
00:31:42,000 --> 00:31:48,000
The system is only allowed to
have two arbitrary constants in

483
00:31:47,000 --> 00:31:53,000
its solution because that is the
initial conditions we are giving

484
00:31:52,000 --> 00:31:58,000
it.
By the way, I forgot to give

485
00:31:55,000 --> 00:32:01,000
initial conditions.
Let's give initial conditions.

486
00:32:01,000 --> 00:32:07,000
Let's say the initial
temperature of the yolk,

487
00:32:05,000 --> 00:32:11,000
when it is put in the ice bath,
is 40 degrees centigrade,

488
00:32:10,000 --> 00:32:16,000
Celsius.
And T2, let's say the white

489
00:32:13,000 --> 00:32:19,000
ought to be a little hotter than
the yolk is always cooler than

490
00:32:18,000 --> 00:32:24,000
the white for a soft boiled egg,
I don't know,

491
00:32:22,000 --> 00:32:28,000
or a hardboiled egg if it
hasn't been chilled too long.

492
00:32:27,000 --> 00:32:33,000
Let's make this 45.
Realistic numbers.

493
00:32:32,000 --> 00:32:38,000
Now, the thing not to do is to
say, hey, I found T1.

494
00:32:35,000 --> 00:32:41,000
Okay, I will find T2 by the
same procedure.

495
00:32:39,000 --> 00:32:45,000
I will go through the whole
thing.

496
00:32:41,000 --> 00:32:47,000
I will eliminate T1 instead.
Then I will end up with an

497
00:32:45,000 --> 00:32:51,000
equation T2 and I will solve
that and get T2 equals blah,

498
00:32:50,000 --> 00:32:56,000
blah, blah.
That is no good,

499
00:32:52,000 --> 00:32:58,000
A, because you are working too
hard and, B, because you are

500
00:32:56,000 --> 00:33:02,000
going to get two more arbitrary
constants unrelated to these

501
00:33:01,000 --> 00:33:07,000
two.
And that is no good.

502
00:33:04,000 --> 00:33:10,000
Because the correct solution
only has two constants in it.

503
00:33:09,000 --> 00:33:15,000
Not four.
So that procedure is wrong.

504
00:33:12,000 --> 00:33:18,000
You must calculate T2 from the
T1 that you found,

505
00:33:15,000 --> 00:33:21,000
and that is the equation which
does it.

506
00:33:18,000 --> 00:33:24,000
That's the one we have to have.
Where is the chalk?

507
00:33:22,000 --> 00:33:28,000
Yes.
Maybe I can have a little thing

508
00:33:25,000 --> 00:33:31,000
so I can just carry this around
with me.

509
00:33:37,000 --> 00:33:43,000
That is the relation between T2
and T1.

510
00:33:40,000 --> 00:33:46,000
Or, if you don't like it,
either one of these equations

511
00:33:44,000 --> 00:33:50,000
will express T2 in terms of T1
for you.

512
00:33:47,000 --> 00:33:53,000
It doesn't matter.
Whichever one you use,

513
00:33:50,000 --> 00:33:56,000
however you do it,
that's the way you must

514
00:33:53,000 --> 00:33:59,000
calculate T2.
So what is it?

515
00:33:56,000 --> 00:34:02,000
T2 is calculated from that pink
box.

516
00:34:00,000 --> 00:34:06,000
It is one-half of T1 prime plus
T1.

517
00:34:05,000 --> 00:34:11,000
Now, if I take the derivative
of this, I get minus c1 times

518
00:34:11,000 --> 00:34:17,000
the exponential.
The coefficient is minus c1,

519
00:34:16,000 --> 00:34:22,000
take half of that,
that is minus a half c1

520
00:34:21,000 --> 00:34:27,000
and add it to T1.
Minus one-half c1 plus c1 gives

521
00:34:26,000 --> 00:34:32,000
me one-half c1.

522
00:34:32,000 --> 00:34:38,000
And here I take the derivative,
it is minus 6 c2.

523
00:34:38,000 --> 00:34:44,000
Take half of that,
minus 3 c2 and add this c2 to

524
00:34:44,000 --> 00:34:50,000
it, minus 3 plus 1 makes minus
2.

525
00:34:48,000 --> 00:34:54,000
That is T2.
And notice it uses the same

526
00:34:53,000 --> 00:34:59,000
arbitrary constants that T1
uses.

527
00:34:59,000 --> 00:35:05,000
So we end up with just two
because we calculated T2 from

528
00:35:02,000 --> 00:35:08,000
that formula or from the
equation which is equivalent to

529
00:35:06,000 --> 00:35:12,000
it, not from scratch.
We haven't put in the initial

530
00:35:09,000 --> 00:35:15,000
conditions yet,
but that is easy to do.

531
00:35:11,000 --> 00:35:17,000
Everybody, when working with
exponentials,

532
00:35:14,000 --> 00:35:20,000
of course, you always want the
initial conditions to be when T

533
00:35:18,000 --> 00:35:24,000
is equal to zero
because that makes all the

534
00:35:21,000 --> 00:35:27,000
exponentials one and you don't
have to worry about them.

535
00:35:25,000 --> 00:35:31,000
But this you know.
If I put in the initial

536
00:35:27,000 --> 00:35:33,000
conditions, at time zero,
T1 has the value 40.

537
00:35:32,000 --> 00:35:38,000
So 40 should be equal to c1 +
c2.

538
00:35:38,000 --> 00:35:44,000
And the other equation will say
that 45 is equal to one-half c1

539
00:35:45,000 --> 00:35:51,000
minus 2 c2.
Now we are supposed to

540
00:35:52,000 --> 00:35:58,000
solve these.
Well, this is called solving

541
00:35:57,000 --> 00:36:03,000
simultaneous linear equations.
We could use Kramer's rule,

542
00:36:05,000 --> 00:36:11,000
inverse matrices,
but why don't we just

543
00:36:09,000 --> 00:36:15,000
eliminate.
Let me see.

544
00:36:12,000 --> 00:36:18,000
If I multiply by,
45, so multiply by two,

545
00:36:17,000 --> 00:36:23,000
you get 90 equals c1
minus 4 c2.

546
00:36:23,000 --> 00:36:29,000
Then subtract this guy from
that guy.

547
00:36:27,000 --> 00:36:33,000
So, 40 taken from 90 makes 50.
And c1 taken from c1,

548
00:36:35,000 --> 00:36:41,000
because I multiplied by two,
makes zero.

549
00:36:40,000 --> 00:36:46,000
And c2 taken from minus 4 c2,
that makes minus 5 c2,

550
00:36:47,000 --> 00:36:53,000
I guess.

551
00:36:49,000 --> 00:36:55,000
I seem to get c2 is equal to
negative 10.

552
00:36:56,000 --> 00:37:02,000
And if c2 is negative 10,
then c1 must be 50.

553
00:37:04,000 --> 00:37:10,000
There are two ways of checking
the answer.

554
00:37:07,000 --> 00:37:13,000
One is to plug it into the
equations, and the other is to

555
00:37:13,000 --> 00:37:19,000
peak.
Yes, that's right.

556
00:37:15,000 --> 00:37:21,000
[LAUGHTER]

557
00:37:25,000 --> 00:37:31,000
The final answer is,
in other words,

558
00:37:27,000 --> 00:37:33,000
you put a 50 here,
25 there, negative 10 here,

559
00:37:30,000 --> 00:37:36,000
and positive 20 there.
That gives the answer to the

560
00:37:34,000 --> 00:37:40,000
problem.
It tells you,

561
00:37:35,000 --> 00:37:41,000
in other words,
how the temperature of the yolk

562
00:37:39,000 --> 00:37:45,000
varies with time and how the
temperature of the white varies

563
00:37:43,000 --> 00:37:49,000
with time.
As I said, we are going to

564
00:37:46,000 --> 00:37:52,000
learn a slick way of doing this
problem, or at least a very

565
00:37:51,000 --> 00:37:57,000
different way of doing the same
problem next time,

566
00:37:54,000 --> 00:38:00,000
but let's put that on ice for
the moment.

567
00:37:57,000 --> 00:38:03,000
And instead I would like to
spend the rest of the period

568
00:38:01,000 --> 00:38:07,000
doing for first order systems
the same thing that I did for

569
00:38:05,000 --> 00:38:11,000
you the very first day of the
term.

570
00:38:09,000 --> 00:38:15,000
Remember, I walked in assuming
that you knew how to separate

571
00:38:13,000 --> 00:38:19,000
variables the first day of the
term, and I did not talk to you

572
00:38:17,000 --> 00:38:23,000
about how to solve fancier
equations by fancier methods.

573
00:38:21,000 --> 00:38:27,000
I instead talked to you about
the geometric significance,

574
00:38:25,000 --> 00:38:31,000
what the geometric meaning of a
single first order equation was

575
00:38:29,000 --> 00:38:35,000
and how that geometric meaning
enabled you to solve it

576
00:38:33,000 --> 00:38:39,000
numerically.
And we spent a little while

577
00:38:36,000 --> 00:38:42,000
working on such problems because
nowadays with computers it is

578
00:38:40,000 --> 00:38:46,000
really important that you get a
feeling for what these things

579
00:38:44,000 --> 00:38:50,000
mean as opposed to just
algorithms for solving them.

580
00:38:47,000 --> 00:38:53,000
As I say, most differential
equations, especially systems,

581
00:38:50,000 --> 00:38:56,000
are likely to be solved by a
computer anyway.

582
00:38:54,000 --> 00:39:00,000
You have to be the guiding
genius that interprets the

583
00:38:57,000 --> 00:39:03,000
answers and can see when
mistakes are being made,

584
00:39:01,000 --> 00:39:07,000
stuff like that.
The problem is,

585
00:39:04,000 --> 00:39:10,000
therefore, what is the meaning
of this system?

586
00:39:15,000 --> 00:39:21,000
Well, you are not going to get
anywhere interpreting it

587
00:39:18,000 --> 00:39:24,000
geometrically,
unless you get rid of that t on

588
00:39:21,000 --> 00:39:27,000
the right-hand side.
And the only way of getting rid

589
00:39:25,000 --> 00:39:31,000
of the t is to declare it is not
there.

590
00:39:28,000 --> 00:39:34,000
So I hereby declare that I will
only consider,

591
00:39:31,000 --> 00:39:37,000
for the rest of the period,
that is only ten minutes,

592
00:39:34,000 --> 00:39:40,000
systems in which no t appears
explicitly on the right-hand

593
00:39:38,000 --> 00:39:44,000
side.
Because I don't know what to do

594
00:39:42,000 --> 00:39:48,000
if it does up here.
We have a word for these.

595
00:39:45,000 --> 00:39:51,000
Remember what the first order
word was?

596
00:39:48,000 --> 00:39:54,000
A first order equation where
there was no t explicitly on the

597
00:39:53,000 --> 00:39:59,000
right-hand side,
we called it,

598
00:39:55,000 --> 00:40:01,000
anybody remember?
Just curious.

599
00:39:57,000 --> 00:40:03,000
Autonomous, right.

600
00:40:05,000 --> 00:40:11,000
This is an autonomous system.
It is not a linear system

601
00:40:08,000 --> 00:40:14,000
because these are messy
functions.

602
00:40:10,000 --> 00:40:16,000
This could be x times y
or x squared minus 3y squared

603
00:40:14,000 --> 00:40:20,000
divided by sine of x plus y.

604
00:40:18,000 --> 00:40:24,000
It could be a mess.
Definitely not linear.

605
00:40:21,000 --> 00:40:27,000
But autonomous means no t.
t means the independent

606
00:40:24,000 --> 00:40:30,000
variable appears on the
right-hand side.

607
00:40:27,000 --> 00:40:33,000
Of course, it is there.
It is buried in the dx/dt and

608
00:40:30,000 --> 00:40:36,000
dy/dt.
But it is not on the right-hand

609
00:40:33,000 --> 00:40:39,000
side.
No t appears on the right-hand

610
00:40:35,000 --> 00:40:41,000
side.

611
00:40:41,000 --> 00:40:47,000
Because no t appears on the
right-hand side,

612
00:40:44,000 --> 00:40:50,000
I can now draw a picture of
this.

613
00:40:47,000 --> 00:40:53,000
But, let's see,
what does a solution look like?

614
00:40:52,000 --> 00:40:58,000
I never even talked about what
a solution was,

615
00:40:56,000 --> 00:41:02,000
did I?
Well, pretend that immediately

616
00:40:59,000 --> 00:41:05,000
after I talked about that,
I talked about this.

617
00:41:05,000 --> 00:41:11,000
What is the solution?
Well, the solution,

618
00:41:07,000 --> 00:41:13,000
maybe you took it for granted,
is a pair of functions,

619
00:41:10,000 --> 00:41:16,000
x of t, y of t if when
you plug it in

620
00:41:13,000 --> 00:41:19,000
it satisfies the equation.
And so what else is new?

621
00:41:16,000 --> 00:41:22,000
The solution is x
equals x of t,

622
00:41:19,000 --> 00:41:25,000
y equals y of t.

623
00:41:27,000 --> 00:41:33,000
If I draw a picture of that
what would it look like?

624
00:41:30,000 --> 00:41:36,000
This is where your previous
knowledge of physics above all

625
00:41:35,000 --> 00:41:41,000
18.02, maybe 18.01 if you
learned this in high school,

626
00:41:39,000 --> 00:41:45,000
what is x equals x of t and
y equals y of t?

627
00:41:44,000 --> 00:41:50,000
How do you draw a picture of

628
00:41:47,000 --> 00:41:53,000
that?
What does it represent?

629
00:41:49,000 --> 00:41:55,000
A curve.
And what will be the title of

630
00:41:52,000 --> 00:41:58,000
the chapter of the calculus book
in which that is discussed?

631
00:41:56,000 --> 00:42:02,000
Parametric equations.
This is a parameterized curve.

632
00:42:12,000 --> 00:42:18,000
So we know what the solution
looks like.

633
00:42:15,000 --> 00:42:21,000
Our solution is a parameterized
curve.

634
00:42:18,000 --> 00:42:24,000
And what does a parameterized
curve look like?

635
00:42:21,000 --> 00:42:27,000
Well, it travels,
and in a certain direction.

636
00:42:34,000 --> 00:42:40,000
Okay.
That's enough.

637
00:42:35,000 --> 00:42:41,000
Why do I have several of those
curves?

638
00:42:38,000 --> 00:42:44,000
Well, because I have several
solutions.

639
00:42:40,000 --> 00:42:46,000
In fact, given any initial
starting point,

640
00:42:43,000 --> 00:42:49,000
there is a solution that goes
through it.

641
00:42:46,000 --> 00:42:52,000
I will put in possible starting
points.

642
00:42:49,000 --> 00:42:55,000
And you can do this on the
computer screen with a little

643
00:42:53,000 --> 00:42:59,000
program you will have,
one of the visuals you'll have.

644
00:42:56,000 --> 00:43:02,000
It's being made right now.
You put down starter point,

645
00:43:01,000 --> 00:43:07,000
put down a click,
and then it just draws the

646
00:43:04,000 --> 00:43:10,000
curve passing through that
point.

647
00:43:06,000 --> 00:43:12,000
Didn't we do this early in the
term?

648
00:43:09,000 --> 00:43:15,000
Yes.
But there is a difference now

649
00:43:11,000 --> 00:43:17,000
which I will explain.
These are various possible

650
00:43:14,000 --> 00:43:20,000
starting points at time zero for
this solution,

651
00:43:17,000 --> 00:43:23,000
and then you see what happens
to it afterwards.

652
00:43:20,000 --> 00:43:26,000
In fact, through every point in
the plane will pass a solution

653
00:43:25,000 --> 00:43:31,000
curve, parameterized curve.
Now, what is then the

654
00:43:29,000 --> 00:43:35,000
representation of this?
Well, what is the meaning of x

655
00:43:32,000 --> 00:43:38,000
prime of t and y prime of t?

656
00:43:40,000 --> 00:43:46,000
I am not going to worry for the
moment about the right-hand

657
00:43:44,000 --> 00:43:50,000
side.
What does this mean by itself?

658
00:43:47,000 --> 00:43:53,000
If this is the curve,
the parameterized motion,

659
00:43:50,000 --> 00:43:56,000
then this represents its
velocity vector.

660
00:43:53,000 --> 00:43:59,000
It is the velocity of the
solution at time t.

661
00:43:58,000 --> 00:44:04,000
If I think of the solution as
being a parameterized motion.

662
00:44:03,000 --> 00:44:09,000
All I have drawn here is the
trace, the path of the motion.

663
00:44:08,000 --> 00:44:14,000
This hasn't indicated how fast
it was going.

664
00:44:11,000 --> 00:44:17,000
One solution might go whoosh
and another one might go rah.

665
00:44:16,000 --> 00:44:22,000
That is a velocity,
and that velocity changes from

666
00:44:20,000 --> 00:44:26,000
point to point.
It changes direction.

667
00:44:23,000 --> 00:44:29,000
Well, we know its direction at
each point.

668
00:44:27,000 --> 00:44:33,000
That's tangent.
What I cannot tell is the

669
00:44:31,000 --> 00:44:37,000
speed.
From this picture,

670
00:44:33,000 --> 00:44:39,000
I cannot tell what the speed
was.

671
00:44:36,000 --> 00:44:42,000
Too bad.
Now, what is then the meaning

672
00:44:39,000 --> 00:44:45,000
of the system?
What the system does,

673
00:44:41,000 --> 00:44:47,000
it prescribes at each point the
velocity vector.

674
00:44:45,000 --> 00:44:51,000
If you tell me what the point
(x, y) is in the plane then

675
00:44:50,000 --> 00:44:56,000
these equations give you the
velocity vector at that point.

676
00:44:54,000 --> 00:45:00,000
And, therefore,
what I end up with,

677
00:44:57,000 --> 00:45:03,000
the system is what you call in
physics and what you call in

678
00:45:01,000 --> 00:45:07,000
18.02 a velocity field.
So at each point there is a

679
00:45:06,000 --> 00:45:12,000
certain vector.
The vector is always tangent to

680
00:45:09,000 --> 00:45:15,000
the solution curve through
there, but I cannot predict from

681
00:45:13,000 --> 00:45:19,000
just this picture what its
length will be because at some

682
00:45:17,000 --> 00:45:23,000
points, it might be going slow.
The solution might be going

683
00:45:21,000 --> 00:45:27,000
slowly.
In other words,

684
00:45:22,000 --> 00:45:28,000
the plane is filled up with
these guys.

685
00:45:33,000 --> 00:45:39,000
Stop me.
Not enough here.

686
00:45:37,000 --> 00:45:43,000
So on and so on.
We can say a system of first

687
00:45:44,000 --> 00:45:50,000
order equations,
ODEs of first order equations,

688
00:45:52,000 --> 00:45:58,000
autonomous because there must
be no t on the right-hand side,

689
00:46:03,000 --> 00:46:09,000
is equal to a velocity field.
A field of velocity.

690
00:46:12,000 --> 00:46:18,000
The plane covered with velocity
vectors.

691
00:46:18,000 --> 00:46:24,000
And a solution is a
parameterized curve with the

692
00:46:25,000 --> 00:46:31,000
right velocity everywhere.

693
00:46:38,000 --> 00:46:44,000
Now, there obviously must be a
connection between that and the

694
00:47:39,000 --> 00:47:45,000
direction fields we studied at
the beginning of the term.

695
00:48:36,000 --> 00:48:42,000
And there is.
It is a very important

696
00:49:11,000 --> 00:49:17,000
connection.
It is too important to talk

697
00:49:49,000 --> 00:49:55,000
about in minus one minute.
When we need it,

698
00:50:32,000 --> 00:50:38,000
I will have to spend some time
talking about it then.