WEBVTT
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For the rest of the term,
we are going to be studying not
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just one differential equation
at a time, but rather what are
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called systems of differential
equations.
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Those are like systems of
linear equations.
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They have to be solved
simultaneously,
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in other words,
not just one at a time.
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So, how does a system look when
you write it down?
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Well, since we are going to be
talking about systems of
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ordinary differential equations,
there still will be only one
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independent variable,
but there will be several
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dependent variables.
I am going to call,
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let's say two.
The dependent variables are
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going to be, I will call them x
and y, and then the first order
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system, something involving just
first derivatives,
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will look like this.
On the left-hand side
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will be x prime,
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in other words.
On the right-hand side will be
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the dependent variables and then
also the independent variables.
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I will indicate that,
I will separate it all from the
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others by putting a semicolon
there.
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And the same way y prime,
the derivative of y with
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respect to t,
will be some other function of
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(x, y) and t.
Let's write down explicitly
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that x and y are dependent
variables.
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And what they depend upon is
the independent variable t,
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time.
A system like this is going to
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be called first order.
And we are going to consider
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basically only first-order
systems for a secret reason that
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I will explain at the end of the
period.
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This is a first-order system,
meaning that the only kind of
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derivatives that are up here are
first derivatives.
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So x prime is dx over dt
and so on.
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Now, there is still more
terminology.
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Of course, practically all the
equations after the term
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started, virtually all the
equations we have been
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considering are linear
equations, so it must be true
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that linear systems are the best
kind.
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And, boy, they certainly are.
When are we going to call a
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system linear?
I think in the beginning you
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should learn a little
terminology before we launch in
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and actually try to start to
solve these things.
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Well, the x and y,
the dependent variables must
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occur linearly.
In other words,
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it must look like this,
ax plus by.
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Now, the t can be a mess.
And so I will throw in an extra
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function of t there.
And y prime will be some
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other linear combination of x
and y, plus some other messy
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function of t.
But even the a,
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b, c, and d are allowed to be
functions of t.
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They could be one over t cubed
or sine t
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or something like that.
So I have to distinguish those
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cases.
The case where a,
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b, c, and d are constants,
that I will call --
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Well, there are different
things you can call it.
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We will simply call it a
constant coefficient system.
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A system with coefficients
would probably be better
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English.
On the other hand,
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a, b, c, and d,
this system will still be
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called linear if these are
functions of t.
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Can also be functions of t.
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So it would be a perfectly good
linear system to have x prime
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equals tx plus sine t times y
plus e to the minus t squared.
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You would never see something
like that but it is okay.
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What else do you need to know?
Well, what would a homogenous
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system be?
A homogenous system is one
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without these extra guys.
That doesn't mean there is no t
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in it.
There could be t in the a,
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b, c and d, but these terms
with no x and y in them must not
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occur.
So, a linear homogenous.
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And that is the kind we are
going to start studying first in
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the same way when we studied
higher order equations.
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We studied first homogenous.
You had to know how to solve
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those first, and then you could
learn how to solve the more
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general kind.
So linear homogenous means that
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r1 is zero and r2 is zero for
all time.
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They are identically zero.
They are not there.
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You don't see them.
Have I left anything out?
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Yes, the initial conditions.
Since that is quite general,
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let's talk about what would
initial conditions look like?
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Well, in a general way,
the reason you have to have
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initial conditions is to get
values for the arbitrary
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constants that appear in the
solution.
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The question is,
how many arbitrary constants
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are going to appear in the
solutions of these equations?
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Well, I will just give you the
answer.
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Two.
The number of arbitrary
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constants that appear is the
total order of the system.
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For example,
if this were a second
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derivative and this were a first
derivative, I would expect three
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arbitrary constants in the
system --
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-- because the total,
the sum of two and one makes
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three.
So you must have as many
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initial conditions as you have
arbitrary constants in the
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solution.
And that, of course,
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explains when we studied
second-order equations,
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we had to have two initial
conditions.
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I had to specify the initial
starting point and the initial
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velocity.
And the reason we had to have
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two conditions was because the
general solution had two
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arbitrary constants in it.
The same thing happens here but
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the answer is it is more
natural, the conditions here are
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more natural.
I don't have to specify the
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velocity.
Why not?
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Well, because an initial
condition, of course,
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would want me to say what the
starting value of x is,
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some number,
and it will also want to know
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what the starting value of y is
at that same point.
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Well, there are my two
conditions.
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And since this is going to have
two arbitrary constants in it,
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it is these initial conditions
that will satisfy,
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the arbitrary constants will
have to be picked so as to
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satisfy those initial
conditions.
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In some sense,
the giving of initial
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conditions for a system is a
more natural activity than
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giving the initial conditions of
a second order system.
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You don't have to be the least
bit cleaver about it.
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Anybody would give these two
numbers.
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Whereas, somebody faced with a
second order system might
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scratch his head.
And, in fact,
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there are other kinds of
conditions.
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There are boundary conditions
you learned a little bit about
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instead of initial conditions
for a second order equation.
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I cannot think of any more
general terminology,
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so it sounds like we are going
to actually have to get to work.
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Okay, let's get to work.
I want to set up a system and
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solve it.
And since one of the things in
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this course is supposed to be
simple modeling,
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it should be a system that
models something.
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In general, the kinds of models
we are going to use when we
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study systems are the same ones
we used in studying just
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first-order equations.
Mixing, radioactive decay,
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temperature,
the motion of temperature.
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Heat, heat conduction,
in other words.
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Diffusion.
I have given you a diffusion
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problem for your first homework
on this subject.
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What else did we do?
That's all I can think of for
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the moment, but I am sure they
will occur to me.
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When, out of those physical
ideas, are we going to get a
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system?
The answer is,
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whenever there are two of
something that there was only
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one of before.
For example,
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if I have mixing with two tanks
where the fluid goes like that.
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Say you want to have a big tank
and a little tank here and you
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want to put some stuff into the
little tank so that it will get
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mixed in the big tank without
having to climb a big ladder and
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stop and drop the stuff in.
That will require two tanks,
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the concentration of the
substance in each tank,
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therefore, that will require a
system of equations rather than
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just one.
Or, to give something closer to
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home, closer to this backboard,
anyway, suppose you have dah,
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dah, dah, don't groan,
at least not audibly,
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something that looks like that.
And next to it put an EMF
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there.
That is just a first order.
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That just leads to a single
first order equation.
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But suppose it is a two loop
circuit.
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Now I need a pair of equations.
Each of these loops gives a
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first order differential
equation, but they have to be
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solved simultaneously to find
the current or the charges on
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the condensers.
And if I want a system of three
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equations, throw in another
loop.
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Now, suppose I put in a coil
instead.
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What is this going to lead to?
This is going to give me a
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system of three equations of
which this will be first order,
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first order.
And this will be second order
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because it has a coil.
You are up to that,
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right?
You've had coils,
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inductance?
Good.
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So the whole thing is going to
count as first-order,
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first-order,
second-order.
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To find out how complicated it
is, you have to add up the
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orders.
That is one and one,
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and two.
This is really fourth-order
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stuff that we are talking about
here.
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We can expect it to be a little
complicated.
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Well, now let's take a modest
little problem.
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I am going to return to a
problem we considered earlier in
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the problem of heat conduction.
I had forgotten whether it was
00:11:14.000 --> 00:11:20.000
on the problem set or I did it
in class, but I am choosing it
00:11:19.000 --> 00:11:25.000
because it leads to something we
will be able to solve.
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And because it illustrates how
to add a little sophistication
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to something that was
unsophisticated before.
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A pot of water.
External temperature Te of t.
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I am talking about the
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temperature of something.
And what I am talking about the
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temperature of will be an egg
that is cooking inside,
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but with a difference.
This egg is not homogenous
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inside.
Instead it has a white and it
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has a yolk in the middle.
In other words,
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it is a real egg and not a
phony egg.
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That is a small pot,
or it is an ostrich egg.
00:12:04.000 --> 00:12:10.000
[LAUGHTER] That is the yoke.
The yolk is contained in a
00:12:08.000 --> 00:12:14.000
little membrane inside.
And there are little yucky
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things that hold it in position.
And we are going to let the
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temperature of the yolk,
if you can see in the back of
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the room, be T1.
That is the temperature of the
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yolk.
The temperature of the white,
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which we will assume is
uniform, is going to be T2.
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Oh, that's the water bath.
The temperature of the white is
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T2, and then the temperature of
the external water bath.
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In other words,
the reason for introducing two
00:12:42.000 --> 00:12:48.000
variables instead of just the
one variable for the overall
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temperature of the egg we had is
because egg white is liquid pure
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protein, more or less,
and the T1, the yolk has a lot
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of fat and cholesterol and other
stuff like that which is
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supposed to be bad for you.
It certainly has different
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conducting.
It is liquid,
00:13:07.000 --> 00:13:13.000
at the beginning at any rate,
but it certainly has different
00:13:11.000 --> 00:13:17.000
constants of conductivity than
the egg white would.
00:13:14.000 --> 00:13:20.000
And the condition of heat
through the shell of the egg
00:13:17.000 --> 00:13:23.000
would be different from the
conduction of heat through the
00:13:20.000 --> 00:13:26.000
membrane that keeps the yoke
together.
00:13:22.000 --> 00:13:28.000
So it is quite reasonable to
consider that the white and the
00:13:26.000 --> 00:13:32.000
yolk will be at different
temperatures and will have
00:13:29.000 --> 00:13:35.000
different conductivity
properties.
00:13:32.000 --> 00:13:38.000
I am going to use Newton's laws
but with this further
00:13:36.000 --> 00:13:42.000
refinement.
In other words,
00:13:38.000 --> 00:13:44.000
introducing two temperatures.
Whereas, before we only had one
00:13:43.000 --> 00:13:49.000
temperature.
But let's use Newton's law.
00:13:46.000 --> 00:13:52.000
Let's see.
The question is how does T1,
00:13:49.000 --> 00:13:55.000
the temperature of the yolk,
vary with time?
00:13:52.000 --> 00:13:58.000
Well, the yolk is getting all
its heat from the white.
00:13:57.000 --> 00:14:03.000
Therefore, Newton's law of
conduction will be some constant
00:14:01.000 --> 00:14:07.000
of conductivity for the yolk
times T2 minus T1.
00:14:08.000 --> 00:14:14.000
The yolk does not know anything
about the external temperature
00:14:12.000 --> 00:14:18.000
of the water bath.
It is completely surrounded,
00:14:15.000 --> 00:14:21.000
snug and secure within itself.
But how about the temperature
00:14:20.000 --> 00:14:26.000
of the egg white?
That gets heat and gives heat
00:14:23.000 --> 00:14:29.000
to two sources,
from the external water and
00:14:26.000 --> 00:14:32.000
also from the internal yolk
inside.
00:14:30.000 --> 00:14:36.000
So you have to take into
account both of those.
00:14:33.000 --> 00:14:39.000
Its conduction of the heat
through that membrane,
00:14:36.000 --> 00:14:42.000
we will use the same a,
which is going to be a times T1
00:14:40.000 --> 00:14:46.000
minus T2.
Remember the order in which you
00:14:44.000 --> 00:14:50.000
have to write these is governed
by the yolk outside to the
00:14:48.000 --> 00:14:54.000
white.
Therefore, that has to come
00:14:51.000 --> 00:14:57.000
first when I write it in order
that a be a positive constant.
00:14:55.000 --> 00:15:01.000
But it is also getting heat
from the water bath.
00:15:00.000 --> 00:15:06.000
And, presumably,
the conductivity through the
00:15:03.000 --> 00:15:09.000
shell is different from what it
is through this membrane around
00:15:08.000 --> 00:15:14.000
the yolk.
So I am going to call that by a
00:15:11.000 --> 00:15:17.000
different constant.
This is the conductivity
00:15:14.000 --> 00:15:20.000
through the shell into the
white.
00:15:17.000 --> 00:15:23.000
And that is going to be T,
the external temperature minus
00:15:21.000 --> 00:15:27.000
the temperature of the egg
white.
00:15:24.000 --> 00:15:30.000
Here I have a system of
equations because I want to make
00:15:28.000 --> 00:15:34.000
two dependent variables by
refining the original problem.
00:15:34.000 --> 00:15:40.000
Now, you always have to write a
system in standard form to solve
00:15:39.000 --> 00:15:45.000
it.
You will see that the left-hand
00:15:42.000 --> 00:15:48.000
side will give the dependent
variables in a certain order.
00:15:47.000 --> 00:15:53.000
In this case,
the temperature of the yolk and
00:15:51.000 --> 00:15:57.000
then the temperature of the
white.
00:15:54.000 --> 00:16:00.000
The law is that in order not to
make mistakes --
00:16:00.000 --> 00:16:06.000
And it's a very frequent source
of error so learn from the
00:16:03.000 --> 00:16:09.000
beginning not to do this.
You must write the variables on
00:16:07.000 --> 00:16:13.000
the right-hand side in the same
order left to right in which
00:16:11.000 --> 00:16:17.000
they occur top to bottom here.
In other words,
00:16:14.000 --> 00:16:20.000
this is not a good way to leave
that.
00:16:16.000 --> 00:16:22.000
This is the first attempt in
writing this system,
00:16:20.000 --> 00:16:26.000
but the final version should
like this.
00:16:22.000 --> 00:16:28.000
T1 prime,
I won't bother writing dT / dt,
00:16:25.000 --> 00:16:31.000
is equal to --
T1 must come first,
00:16:29.000 --> 00:16:35.000
so minus a times T1 plus a
times T2.
00:16:34.000 --> 00:16:40.000
And the same law for the second
one.
00:16:38.000 --> 00:16:44.000
It must come in the same order.
Now, the coefficient of T1,
00:16:44.000 --> 00:16:50.000
that is easy.
That's a times T1.
00:16:47.000 --> 00:16:53.000
The coefficient of T2 is minus
a minus b,
00:16:52.000 --> 00:16:58.000
so minus (a plus b) times T2.
00:16:56.000 --> 00:17:02.000
But I am not done yet.
There is still this external
00:17:02.000 --> 00:17:08.000
temperature I must put into the
equation.
00:17:06.000 --> 00:17:12.000
Now, that is not a variable.
This is some given function of
00:17:11.000 --> 00:17:17.000
t.
And what the function of t is,
00:17:14.000 --> 00:17:20.000
of course, depends upon what
the problem is.
00:17:18.000 --> 00:17:24.000
So that, for example,
what might be some
00:17:22.000 --> 00:17:28.000
possibilities,
well, suppose the problem was I
00:17:26.000 --> 00:17:32.000
wanted to coddle the egg.
I think there is a generation
00:17:32.000 --> 00:17:38.000
gap here.
How many of you know what a
00:17:35.000 --> 00:17:41.000
coddled egg is?
How many of you don't know?
00:17:38.000 --> 00:17:44.000
Well, I'm just saying my
daughter didn't know.
00:17:42.000 --> 00:17:48.000
I mentioned it to her.
I said I think I'm going to do
00:17:46.000 --> 00:17:52.000
a coddled egg tomorrow in class.
And she said what is that?
00:17:51.000 --> 00:17:57.000
And so I said a cuddled egg?
She said why would someone
00:17:55.000 --> 00:18:01.000
cuddle an egg?
I said coddle.
00:17:59.000 --> 00:18:05.000
And she said,
oh, you mean like a person,
00:18:02.000 --> 00:18:08.000
like what you do to somebody
you like or don't like or I
00:18:07.000 --> 00:18:13.000
don't know.
Whatever.
00:18:09.000 --> 00:18:15.000
I thought a while and said,
yeah, more like that.
00:18:13.000 --> 00:18:19.000
[LAUGHTER] Anyway,
for the enrichment of your
00:18:17.000 --> 00:18:23.000
cooking skills,
to coddle an egg,
00:18:20.000 --> 00:18:26.000
it is considered to produce a
better quality product than
00:18:25.000 --> 00:18:31.000
boiling an egg.
That is why people do it.
00:18:30.000 --> 00:18:36.000
You heat up the water to
boiling, the egg should be at
00:18:34.000 --> 00:18:40.000
room temperature,
and then you carefully lower
00:18:37.000 --> 00:18:43.000
the egg into the water.
And you turn off the heat so
00:18:41.000 --> 00:18:47.000
the water bath cools
exponentially while the egg
00:18:45.000 --> 00:18:51.000
inside is rising in temperature.
And then you wait four minutes
00:18:50.000 --> 00:18:56.000
or six minutes or whatever and
take it out.
00:18:53.000 --> 00:18:59.000
You have a perfect egg.
So for coddling,
00:18:56.000 --> 00:19:02.000
spelled so, what will the
external temperature be?
00:19:02.000 --> 00:19:08.000
Well, it starts out at time
zero at 100 degrees centigrade
00:19:06.000 --> 00:19:12.000
because the water is supposed to
be boiling.
00:19:09.000 --> 00:19:15.000
The reason you have it boiling
is for calibration so that you
00:19:13.000 --> 00:19:19.000
can know what temperature it is
without having to use a
00:19:17.000 --> 00:19:23.000
thermometer, unless you're on
Pike's Peak or some place.
00:19:20.000 --> 00:19:26.000
It starts out at 100 degrees.
And after that,
00:19:24.000 --> 00:19:30.000
since the light is off,
it cools exponential because
00:19:27.000 --> 00:19:33.000
that is another law.
You only have to know what K is
00:19:32.000 --> 00:19:38.000
for your particular pot and you
will be able to solve the
00:19:37.000 --> 00:19:43.000
coddled egg problem.
In other words,
00:19:40.000 --> 00:19:46.000
you will then be able to solve
these equations and know how the
00:19:45.000 --> 00:19:51.000
temperature rises.
I am going to solve a different
00:19:49.000 --> 00:19:55.000
problem because I don't want to
have to deal with this
00:19:54.000 --> 00:20:00.000
inhomogeneous term.
Let's use, as a different
00:19:58.000 --> 00:20:04.000
problem, a person cooks an egg.
Coddles the egg by the first
00:20:04.000 --> 00:20:10.000
process, decides the egg is
done, let's say hardboiled,
00:20:09.000 --> 00:20:15.000
and then you are supposed to
drop a hardboiled egg into cold
00:20:14.000 --> 00:20:20.000
water.
Not just to cool it but also
00:20:17.000 --> 00:20:23.000
because I think it prevents that
dark thing from forming that
00:20:23.000 --> 00:20:29.000
looks sort of unattractive.
Let's ice bath.
00:20:28.000 --> 00:20:34.000
The only reason for dropping
the egg into an ice bath is so
00:20:32.000 --> 00:20:38.000
that you could have a homogenous
equation to solve.
00:20:36.000 --> 00:20:42.000
And since this a first system
we are going to solve,
00:20:40.000 --> 00:20:46.000
let's make life easy for
ourselves.
00:20:43.000 --> 00:20:49.000
Now, all my work in preparing
this example,
00:20:47.000 --> 00:20:53.000
and it took considerably longer
time than actually solving the
00:20:52.000 --> 00:20:58.000
problem, was in picking values
for a and b which would make
00:20:56.000 --> 00:21:02.000
everything come out nice.
It's harder than it looks.
00:21:02.000 --> 00:21:08.000
The values that we are going to
use, which make no physical
00:21:07.000 --> 00:21:13.000
sense whatsoever,
but a equals 2 and b
00:21:11.000 --> 00:21:17.000
equals 3.
These are called nice numbers.
00:21:15.000 --> 00:21:21.000
What is the equation?
What is the system?
00:21:18.000 --> 00:21:24.000
Can somebody read it off for
me?
00:21:21.000 --> 00:21:27.000
It is T1 prime equals, what
is it, minus 2T1 plus 2T2.
00:21:26.000 --> 00:21:32.000
That's good.
00:21:30.000 --> 00:21:36.000
Minus 2T1 plus 2T2.
00:21:40.000 --> 00:21:46.000
T2 prime is,
what is it?
00:21:42.000 --> 00:21:48.000
I think this is 2T1.
And the other one is minus a
00:21:48.000 --> 00:21:54.000
plus b, so minus 5.
00:21:51.000 --> 00:21:57.000
This is a system.
Now, on Wednesday I will teach
00:21:57.000 --> 00:22:03.000
you a fancy way of solving this.
But, to be honest,
00:22:03.000 --> 00:22:09.000
the fancy way will take roughly
about as long as the way I am
00:22:07.000 --> 00:22:13.000
going to do it now.
The main reason for doing it is
00:22:10.000 --> 00:22:16.000
that it introduces new
vocabulary which everyone wants
00:22:14.000 --> 00:22:20.000
you to have.
And also, more important
00:22:16.000 --> 00:22:22.000
reasons, it gives more insight
into the solution than this
00:22:20.000 --> 00:22:26.000
method.
This method just produces the
00:22:22.000 --> 00:22:28.000
answer, but you want insight,
also.
00:22:24.000 --> 00:22:30.000
And that is just as important.
But for now,
00:22:28.000 --> 00:22:34.000
let's use a method which always
works and which in 40 years,
00:22:33.000 --> 00:22:39.000
after you have forgotten all
other fancy methods,
00:22:36.000 --> 00:22:42.000
will still be available to you
because it is method you can
00:22:40.000 --> 00:22:46.000
figure out yourself.
You don't have to remember
00:22:43.000 --> 00:22:49.000
anything.
The method is to eliminate one
00:22:46.000 --> 00:22:52.000
of the dependent variables.
It is just the way you solve
00:22:50.000 --> 00:22:56.000
systems of linear equations in
general if you aren't doing
00:22:54.000 --> 00:23:00.000
something fancy with
determinants and matrices.
00:22:59.000 --> 00:23:05.000
If you just eliminate
variables.
00:23:01.000 --> 00:23:07.000
We are going to eliminate one
of these variables.
00:23:05.000 --> 00:23:11.000
Let's eliminate T2.
You could also eliminate T1.
00:23:08.000 --> 00:23:14.000
The main thing is eliminate one
of them so you will have just
00:23:13.000 --> 00:23:19.000
one left to work with.
How do I eliminate T2?
00:23:16.000 --> 00:23:22.000
Beg your pardon?
Is something wrong?
00:23:19.000 --> 00:23:25.000
If somebody thinks something is
wrong raise his hand.
00:23:23.000 --> 00:23:29.000
No?
00:23:30.000 --> 00:23:36.000
Why do I want to get rid of T1?
Well, I can add them.
00:23:33.000 --> 00:23:39.000
But, on the left-hand side,
I will have T1 prime plus T2
00:23:36.000 --> 00:23:42.000
prime. What good is that?
00:23:39.000 --> 00:23:45.000
[LAUGHTER]
00:23:48.000 --> 00:23:54.000
I think you will want to do it
my way.
00:23:49.000 --> 00:23:55.000
[APPLAUSE]
00:24:03.000 --> 00:24:09.000
Solve for T2 in terms of T1.
That is going to be T1 prime
00:24:08.000 --> 00:24:14.000
plus 2T1 divided by 2.
00:24:12.000 --> 00:24:18.000
Now, take that and substitute
it into the second equation.
00:24:18.000 --> 00:24:24.000
Wherever you see a T2,
put that in,
00:24:21.000 --> 00:24:27.000
and what you will be left with
is something just in T1.
00:24:28.000 --> 00:24:34.000
To be honest,
I don't know any other good way
00:24:31.000 --> 00:24:37.000
of doing this.
There is a fancy method that I
00:24:34.000 --> 00:24:40.000
think is talked about in your
book, which leads to extraneous
00:24:39.000 --> 00:24:45.000
solutions and so on,
but you don't want to know
00:24:43.000 --> 00:24:49.000
about that.
This will work for a simple
00:24:46.000 --> 00:24:52.000
linear equation with constant
coefficients,
00:24:49.000 --> 00:24:55.000
always.
Substitute in.
00:24:51.000 --> 00:24:57.000
What do I do?
Now, here I do not advise doing
00:24:54.000 --> 00:25:00.000
this mentally.
It is just too easy to make a
00:24:57.000 --> 00:25:03.000
mistake.
Here, I will do it carefully,
00:25:04.000 --> 00:25:10.000
writing everything out just as
you would.
00:25:10.000 --> 00:25:16.000
T1 prime plus 2T1 over 2,
prime, equals 2T1 minus 5 time
00:25:18.000 --> 00:25:24.000
T1 prime plus 2T1 over two.
00:25:27.000 --> 00:25:33.000
I took that and substituted
00:25:32.000 --> 00:25:38.000
into this equation.
Now, I don't like those two's.
00:25:38.000 --> 00:25:44.000
Let's get rid of them by
multiplying.
00:25:42.000 --> 00:25:48.000
This will become 4.
00:25:52.000 --> 00:25:58.000
And now write this out.
What is this when you look at
00:25:57.000 --> 00:26:03.000
it?
This is an equation just in T1.
00:26:00.000 --> 00:26:06.000
It has constant coefficients.
And what is its order?
00:26:05.000 --> 00:26:11.000
Its order is two because T1
prime primed.
00:26:10.000 --> 00:26:16.000
In other words,
I can eliminate T2 okay,
00:26:13.000 --> 00:26:19.000
but the equation I am going to
get is no longer a first-order.
00:26:19.000 --> 00:26:25.000
It becomes a second-order
differential equation.
00:26:24.000 --> 00:26:30.000
And that's a basic law.
Even if you have a system of
00:26:30.000 --> 00:26:36.000
more equations,
three or four or whatever,
00:26:33.000 --> 00:26:39.000
the law is that after you do
the elimination successfully and
00:26:37.000 --> 00:26:43.000
end up with a single equation,
normally the order of that
00:26:42.000 --> 00:26:48.000
equation will be the sum of the
orders of the things you started
00:26:46.000 --> 00:26:52.000
with.
So two first-order equations
00:26:49.000 --> 00:26:55.000
will always produce a
second-order equation in just
00:26:53.000 --> 00:26:59.000
one dependent variable,
three will produce a third
00:26:56.000 --> 00:27:02.000
order equation and so on.
So you trade one complexity for
00:27:02.000 --> 00:27:08.000
another.
You trade the complexity of
00:27:04.000 --> 00:27:10.000
having to deal with two
equations simultaneously instead
00:27:09.000 --> 00:27:15.000
of just one for the complexity
of having to deal with a single
00:27:13.000 --> 00:27:19.000
higher order equation which is
more trouble to solve.
00:27:17.000 --> 00:27:23.000
It is like all mathematical
problems.
00:27:20.000 --> 00:27:26.000
Unless you are very lucky,
if you push them down one way,
00:27:24.000 --> 00:27:30.000
they are really simple now,
they just pop up some place
00:27:28.000 --> 00:27:34.000
else.
You say, oh,
00:27:30.000 --> 00:27:36.000
I didn't save anything after
all.
00:27:32.000 --> 00:27:38.000
That is the law of conservation
of mathematical difficulty.
00:27:36.000 --> 00:27:42.000
[LAUGHTER] You saw that even
with the Laplace transform.
00:27:40.000 --> 00:27:46.000
In the beginning it looks
great, you've got these tables,
00:27:44.000 --> 00:27:50.000
take the equation,
horrible to solve.
00:27:46.000 --> 00:27:52.000
Take some transform,
trivial to solve for capital Y.
00:27:50.000 --> 00:27:56.000
Now I have to find the inverse
Laplace transform.
00:27:53.000 --> 00:27:59.000
And suddenly all the work is
there, partial fractions,
00:27:57.000 --> 00:28:03.000
funny formulas and so on.
It is very hard in mathematics
00:28:02.000 --> 00:28:08.000
to get away with something.
It happens now and then and
00:28:06.000 --> 00:28:12.000
everybody cheers.
Let's write this out now in the
00:28:09.000 --> 00:28:15.000
form in which it looks like an
equation we can actually solve.
00:28:13.000 --> 00:28:19.000
Just be careful.
Now it is all right to use the
00:28:17.000 --> 00:28:23.000
method by which you collect
terms.
00:28:19.000 --> 00:28:25.000
There is only one term
involving T1 double prime.
00:28:23.000 --> 00:28:29.000
It's the one that comes from
here.
00:28:25.000 --> 00:28:31.000
How about the terms in T1
prime?
00:28:27.000 --> 00:28:33.000
There is a 2.
Here, there is minus 5 T1
00:28:33.000 --> 00:28:39.000
prime.
If I put it on the other side
00:28:37.000 --> 00:28:43.000
it makes plus 5 T1 prime plus
this two makes 7 T1 prime.
00:28:44.000 --> 00:28:50.000
And how many T1's are there?
Well, none on the left-hand
00:28:51.000 --> 00:28:57.000
side.
On the right-hand side I have 4
00:28:55.000 --> 00:29:01.000
here minus 10.
4 minus 10 is negative 6.
00:29:02.000 --> 00:29:08.000
Negative 6 T1 put on this
left-hand side the way we want
00:29:06.000 --> 00:29:12.000
to do makes plus 6 T1.
00:29:15.000 --> 00:29:21.000
There are no inhomogeneous
terms, so that is equal to zero.
00:29:18.000 --> 00:29:24.000
If I had gotten a negative
number for one of these
00:29:22.000 --> 00:29:28.000
coefficients,
I would instantly know if I had
00:29:25.000 --> 00:29:31.000
made a mistake.
Why?
00:29:26.000 --> 00:29:32.000
Why must those numbers come out
to be positive?
00:29:30.000 --> 00:29:36.000
It is because the system must
be, the system must be,
00:29:33.000 --> 00:29:39.000
fill in with one word,
stable.
00:29:36.000 --> 00:29:42.000
And why must this system be
stable?
00:29:38.000 --> 00:29:44.000
In other words,
the long-term solutions must be
00:29:42.000 --> 00:29:48.000
zero, must all go to zero,
whatever they are.
00:29:45.000 --> 00:29:51.000
Why is that?
Well, because you are putting
00:29:48.000 --> 00:29:54.000
the egg into an ice bath.
Or, because we know it was
00:29:52.000 --> 00:29:58.000
living but after being
hardboiled it is dead and,
00:29:56.000 --> 00:30:02.000
therefore, dead systems are
stable.
00:30:00.000 --> 00:30:06.000
That's not a good reason but it
is, so to speak,
00:30:03.000 --> 00:30:09.000
the real one.
It's clear anyway that all
00:30:05.000 --> 00:30:11.000
solutions must tend to zero
physically.
00:30:08.000 --> 00:30:14.000
That's obvious.
And, therefore,
00:30:10.000 --> 00:30:16.000
the differential equation must
have the same property,
00:30:14.000 --> 00:30:20.000
and that means that its
coefficients must be positive.
00:30:17.000 --> 00:30:23.000
All its coefficients must be
positive.
00:30:20.000 --> 00:30:26.000
If this weren't there,
I would get oscillating
00:30:23.000 --> 00:30:29.000
solutions, which wouldn't go to
zero.
00:30:25.000 --> 00:30:31.000
That is physical impossible for
this egg.
00:30:30.000 --> 00:30:36.000
Now the rest is just solving.
The characteristic equation,
00:30:34.000 --> 00:30:40.000
if you can remember way,
way back in prehistoric times
00:30:39.000 --> 00:30:45.000
when we were solving these
equations, is this.
00:30:43.000 --> 00:30:49.000
And what you want to do is
factor it.
00:30:46.000 --> 00:30:52.000
This is where all the work was,
getting those numbers so that
00:30:51.000 --> 00:30:57.000
this would factor. So it's
r plus 1 times r plus 6
00:31:04.000 --> 00:31:10.000
And so the solutions are,
the roots are r equals
00:31:07.000 --> 00:31:13.000
negative 1.
I am just making marks on the
00:31:10.000 --> 00:31:16.000
board, but you have done this
often enough,
00:31:13.000 --> 00:31:19.000
you know what I am talking
about.
00:31:15.000 --> 00:31:21.000
So the characteristic roots are
those two numbers.
00:31:18.000 --> 00:31:24.000
And, therefore,
the solution is,
00:31:20.000 --> 00:31:26.000
I could write down immediately
with its arbitrary constant as
00:31:24.000 --> 00:31:30.000
c1 times e to the negative t
plus c2 times e to the negative
00:31:28.000 --> 00:31:34.000
6t. Now, I have got to get T2.
00:31:34.000 --> 00:31:40.000
Here the first worry is T2 is
going to give me two more
00:31:39.000 --> 00:31:45.000
arbitrary constants.
It better not.
00:31:42.000 --> 00:31:48.000
The system is only allowed to
have two arbitrary constants in
00:31:47.000 --> 00:31:53.000
its solution because that is the
initial conditions we are giving
00:31:52.000 --> 00:31:58.000
it.
By the way, I forgot to give
00:31:55.000 --> 00:32:01.000
initial conditions.
Let's give initial conditions.
00:32:01.000 --> 00:32:07.000
Let's say the initial
temperature of the yolk,
00:32:05.000 --> 00:32:11.000
when it is put in the ice bath,
is 40 degrees centigrade,
00:32:10.000 --> 00:32:16.000
Celsius.
And T2, let's say the white
00:32:13.000 --> 00:32:19.000
ought to be a little hotter than
the yolk is always cooler than
00:32:18.000 --> 00:32:24.000
the white for a soft boiled egg,
I don't know,
00:32:22.000 --> 00:32:28.000
or a hardboiled egg if it
hasn't been chilled too long.
00:32:27.000 --> 00:32:33.000
Let's make this 45.
Realistic numbers.
00:32:32.000 --> 00:32:38.000
Now, the thing not to do is to
say, hey, I found T1.
00:32:35.000 --> 00:32:41.000
Okay, I will find T2 by the
same procedure.
00:32:39.000 --> 00:32:45.000
I will go through the whole
thing.
00:32:41.000 --> 00:32:47.000
I will eliminate T1 instead.
Then I will end up with an
00:32:45.000 --> 00:32:51.000
equation T2 and I will solve
that and get T2 equals blah,
00:32:50.000 --> 00:32:56.000
blah, blah.
That is no good,
00:32:52.000 --> 00:32:58.000
A, because you are working too
hard and, B, because you are
00:32:56.000 --> 00:33:02.000
going to get two more arbitrary
constants unrelated to these
00:33:01.000 --> 00:33:07.000
two.
And that is no good.
00:33:04.000 --> 00:33:10.000
Because the correct solution
only has two constants in it.
00:33:09.000 --> 00:33:15.000
Not four.
So that procedure is wrong.
00:33:12.000 --> 00:33:18.000
You must calculate T2 from the
T1 that you found,
00:33:15.000 --> 00:33:21.000
and that is the equation which
does it.
00:33:18.000 --> 00:33:24.000
That's the one we have to have.
Where is the chalk?
00:33:22.000 --> 00:33:28.000
Yes.
Maybe I can have a little thing
00:33:25.000 --> 00:33:31.000
so I can just carry this around
with me.
00:33:37.000 --> 00:33:43.000
That is the relation between T2
and T1.
00:33:40.000 --> 00:33:46.000
Or, if you don't like it,
either one of these equations
00:33:44.000 --> 00:33:50.000
will express T2 in terms of T1
for you.
00:33:47.000 --> 00:33:53.000
It doesn't matter.
Whichever one you use,
00:33:50.000 --> 00:33:56.000
however you do it,
that's the way you must
00:33:53.000 --> 00:33:59.000
calculate T2.
So what is it?
00:33:56.000 --> 00:34:02.000
T2 is calculated from that pink
box.
00:34:00.000 --> 00:34:06.000
It is one-half of T1 prime plus
T1.
00:34:05.000 --> 00:34:11.000
Now, if I take the derivative
of this, I get minus c1 times
00:34:11.000 --> 00:34:17.000
the exponential.
The coefficient is minus c1,
00:34:16.000 --> 00:34:22.000
take half of that,
that is minus a half c1
00:34:21.000 --> 00:34:27.000
and add it to T1.
Minus one-half c1 plus c1 gives
00:34:26.000 --> 00:34:32.000
me one-half c1.
00:34:32.000 --> 00:34:38.000
And here I take the derivative,
it is minus 6 c2.
00:34:38.000 --> 00:34:44.000
Take half of that,
minus 3 c2 and add this c2 to
00:34:44.000 --> 00:34:50.000
it, minus 3 plus 1 makes minus
2.
00:34:48.000 --> 00:34:54.000
That is T2.
And notice it uses the same
00:34:53.000 --> 00:34:59.000
arbitrary constants that T1
uses.
00:34:59.000 --> 00:35:05.000
So we end up with just two
because we calculated T2 from
00:35:02.000 --> 00:35:08.000
that formula or from the
equation which is equivalent to
00:35:06.000 --> 00:35:12.000
it, not from scratch.
We haven't put in the initial
00:35:09.000 --> 00:35:15.000
conditions yet,
but that is easy to do.
00:35:11.000 --> 00:35:17.000
Everybody, when working with
exponentials,
00:35:14.000 --> 00:35:20.000
of course, you always want the
initial conditions to be when T
00:35:18.000 --> 00:35:24.000
is equal to zero
because that makes all the
00:35:21.000 --> 00:35:27.000
exponentials one and you don't
have to worry about them.
00:35:25.000 --> 00:35:31.000
But this you know.
If I put in the initial
00:35:27.000 --> 00:35:33.000
conditions, at time zero,
T1 has the value 40.
00:35:32.000 --> 00:35:38.000
So 40 should be equal to c1 +
c2.
00:35:38.000 --> 00:35:44.000
And the other equation will say
that 45 is equal to one-half c1
00:35:45.000 --> 00:35:51.000
minus 2 c2.
Now we are supposed to
00:35:52.000 --> 00:35:58.000
solve these.
Well, this is called solving
00:35:57.000 --> 00:36:03.000
simultaneous linear equations.
We could use Kramer's rule,
00:36:05.000 --> 00:36:11.000
inverse matrices,
but why don't we just
00:36:09.000 --> 00:36:15.000
eliminate.
Let me see.
00:36:12.000 --> 00:36:18.000
If I multiply by,
45, so multiply by two,
00:36:17.000 --> 00:36:23.000
you get 90 equals c1
minus 4 c2.
00:36:23.000 --> 00:36:29.000
Then subtract this guy from
that guy.
00:36:27.000 --> 00:36:33.000
So, 40 taken from 90 makes 50.
And c1 taken from c1,
00:36:35.000 --> 00:36:41.000
because I multiplied by two,
makes zero.
00:36:40.000 --> 00:36:46.000
And c2 taken from minus 4 c2,
that makes minus 5 c2,
00:36:47.000 --> 00:36:53.000
I guess.
00:36:49.000 --> 00:36:55.000
I seem to get c2 is equal to
negative 10.
00:36:56.000 --> 00:37:02.000
And if c2 is negative 10,
then c1 must be 50.
00:37:04.000 --> 00:37:10.000
There are two ways of checking
the answer.
00:37:07.000 --> 00:37:13.000
One is to plug it into the
equations, and the other is to
00:37:13.000 --> 00:37:19.000
peak.
Yes, that's right.
00:37:15.000 --> 00:37:21.000
[LAUGHTER]
00:37:25.000 --> 00:37:31.000
The final answer is,
in other words,
00:37:27.000 --> 00:37:33.000
you put a 50 here,
25 there, negative 10 here,
00:37:30.000 --> 00:37:36.000
and positive 20 there.
That gives the answer to the
00:37:34.000 --> 00:37:40.000
problem.
It tells you,
00:37:35.000 --> 00:37:41.000
in other words,
how the temperature of the yolk
00:37:39.000 --> 00:37:45.000
varies with time and how the
temperature of the white varies
00:37:43.000 --> 00:37:49.000
with time.
As I said, we are going to
00:37:46.000 --> 00:37:52.000
learn a slick way of doing this
problem, or at least a very
00:37:51.000 --> 00:37:57.000
different way of doing the same
problem next time,
00:37:54.000 --> 00:38:00.000
but let's put that on ice for
the moment.
00:37:57.000 --> 00:38:03.000
And instead I would like to
spend the rest of the period
00:38:01.000 --> 00:38:07.000
doing for first order systems
the same thing that I did for
00:38:05.000 --> 00:38:11.000
you the very first day of the
term.
00:38:09.000 --> 00:38:15.000
Remember, I walked in assuming
that you knew how to separate
00:38:13.000 --> 00:38:19.000
variables the first day of the
term, and I did not talk to you
00:38:17.000 --> 00:38:23.000
about how to solve fancier
equations by fancier methods.
00:38:21.000 --> 00:38:27.000
I instead talked to you about
the geometric significance,
00:38:25.000 --> 00:38:31.000
what the geometric meaning of a
single first order equation was
00:38:29.000 --> 00:38:35.000
and how that geometric meaning
enabled you to solve it
00:38:33.000 --> 00:38:39.000
numerically.
And we spent a little while
00:38:36.000 --> 00:38:42.000
working on such problems because
nowadays with computers it is
00:38:40.000 --> 00:38:46.000
really important that you get a
feeling for what these things
00:38:44.000 --> 00:38:50.000
mean as opposed to just
algorithms for solving them.
00:38:47.000 --> 00:38:53.000
As I say, most differential
equations, especially systems,
00:38:50.000 --> 00:38:56.000
are likely to be solved by a
computer anyway.
00:38:54.000 --> 00:39:00.000
You have to be the guiding
genius that interprets the
00:38:57.000 --> 00:39:03.000
answers and can see when
mistakes are being made,
00:39:01.000 --> 00:39:07.000
stuff like that.
The problem is,
00:39:04.000 --> 00:39:10.000
therefore, what is the meaning
of this system?
00:39:15.000 --> 00:39:21.000
Well, you are not going to get
anywhere interpreting it
00:39:18.000 --> 00:39:24.000
geometrically,
unless you get rid of that t on
00:39:21.000 --> 00:39:27.000
the right-hand side.
And the only way of getting rid
00:39:25.000 --> 00:39:31.000
of the t is to declare it is not
there.
00:39:28.000 --> 00:39:34.000
So I hereby declare that I will
only consider,
00:39:31.000 --> 00:39:37.000
for the rest of the period,
that is only ten minutes,
00:39:34.000 --> 00:39:40.000
systems in which no t appears
explicitly on the right-hand
00:39:38.000 --> 00:39:44.000
side.
Because I don't know what to do
00:39:42.000 --> 00:39:48.000
if it does up here.
We have a word for these.
00:39:45.000 --> 00:39:51.000
Remember what the first order
word was?
00:39:48.000 --> 00:39:54.000
A first order equation where
there was no t explicitly on the
00:39:53.000 --> 00:39:59.000
right-hand side,
we called it,
00:39:55.000 --> 00:40:01.000
anybody remember?
Just curious.
00:39:57.000 --> 00:40:03.000
Autonomous, right.
00:40:05.000 --> 00:40:11.000
This is an autonomous system.
It is not a linear system
00:40:08.000 --> 00:40:14.000
because these are messy
functions.
00:40:10.000 --> 00:40:16.000
This could be x times y
or x squared minus 3y squared
00:40:14.000 --> 00:40:20.000
divided by sine of x plus y.
00:40:18.000 --> 00:40:24.000
It could be a mess.
Definitely not linear.
00:40:21.000 --> 00:40:27.000
But autonomous means no t.
t means the independent
00:40:24.000 --> 00:40:30.000
variable appears on the
right-hand side.
00:40:27.000 --> 00:40:33.000
Of course, it is there.
It is buried in the dx/dt and
00:40:30.000 --> 00:40:36.000
dy/dt.
But it is not on the right-hand
00:40:33.000 --> 00:40:39.000
side.
No t appears on the right-hand
00:40:35.000 --> 00:40:41.000
side.
00:40:41.000 --> 00:40:47.000
Because no t appears on the
right-hand side,
00:40:44.000 --> 00:40:50.000
I can now draw a picture of
this.
00:40:47.000 --> 00:40:53.000
But, let's see,
what does a solution look like?
00:40:52.000 --> 00:40:58.000
I never even talked about what
a solution was,
00:40:56.000 --> 00:41:02.000
did I?
Well, pretend that immediately
00:40:59.000 --> 00:41:05.000
after I talked about that,
I talked about this.
00:41:05.000 --> 00:41:11.000
What is the solution?
Well, the solution,
00:41:07.000 --> 00:41:13.000
maybe you took it for granted,
is a pair of functions,
00:41:10.000 --> 00:41:16.000
x of t, y of t if when
you plug it in
00:41:13.000 --> 00:41:19.000
it satisfies the equation.
And so what else is new?
00:41:16.000 --> 00:41:22.000
The solution is x
equals x of t,
00:41:19.000 --> 00:41:25.000
y equals y of t.
00:41:27.000 --> 00:41:33.000
If I draw a picture of that
what would it look like?
00:41:30.000 --> 00:41:36.000
This is where your previous
knowledge of physics above all
00:41:35.000 --> 00:41:41.000
18.02, maybe 18.01 if you
learned this in high school,
00:41:39.000 --> 00:41:45.000
what is x equals x of t and
y equals y of t?
00:41:44.000 --> 00:41:50.000
How do you draw a picture of
00:41:47.000 --> 00:41:53.000
that?
What does it represent?
00:41:49.000 --> 00:41:55.000
A curve.
And what will be the title of
00:41:52.000 --> 00:41:58.000
the chapter of the calculus book
in which that is discussed?
00:41:56.000 --> 00:42:02.000
Parametric equations.
This is a parameterized curve.
00:42:12.000 --> 00:42:18.000
So we know what the solution
looks like.
00:42:15.000 --> 00:42:21.000
Our solution is a parameterized
curve.
00:42:18.000 --> 00:42:24.000
And what does a parameterized
curve look like?
00:42:21.000 --> 00:42:27.000
Well, it travels,
and in a certain direction.
00:42:34.000 --> 00:42:40.000
Okay.
That's enough.
00:42:35.000 --> 00:42:41.000
Why do I have several of those
curves?
00:42:38.000 --> 00:42:44.000
Well, because I have several
solutions.
00:42:40.000 --> 00:42:46.000
In fact, given any initial
starting point,
00:42:43.000 --> 00:42:49.000
there is a solution that goes
through it.
00:42:46.000 --> 00:42:52.000
I will put in possible starting
points.
00:42:49.000 --> 00:42:55.000
And you can do this on the
computer screen with a little
00:42:53.000 --> 00:42:59.000
program you will have,
one of the visuals you'll have.
00:42:56.000 --> 00:43:02.000
It's being made right now.
You put down starter point,
00:43:01.000 --> 00:43:07.000
put down a click,
and then it just draws the
00:43:04.000 --> 00:43:10.000
curve passing through that
point.
00:43:06.000 --> 00:43:12.000
Didn't we do this early in the
term?
00:43:09.000 --> 00:43:15.000
Yes.
But there is a difference now
00:43:11.000 --> 00:43:17.000
which I will explain.
These are various possible
00:43:14.000 --> 00:43:20.000
starting points at time zero for
this solution,
00:43:17.000 --> 00:43:23.000
and then you see what happens
to it afterwards.
00:43:20.000 --> 00:43:26.000
In fact, through every point in
the plane will pass a solution
00:43:25.000 --> 00:43:31.000
curve, parameterized curve.
Now, what is then the
00:43:29.000 --> 00:43:35.000
representation of this?
Well, what is the meaning of x
00:43:32.000 --> 00:43:38.000
prime of t and y prime of t?
00:43:40.000 --> 00:43:46.000
I am not going to worry for the
moment about the right-hand
00:43:44.000 --> 00:43:50.000
side.
What does this mean by itself?
00:43:47.000 --> 00:43:53.000
If this is the curve,
the parameterized motion,
00:43:50.000 --> 00:43:56.000
then this represents its
velocity vector.
00:43:53.000 --> 00:43:59.000
It is the velocity of the
solution at time t.
00:43:58.000 --> 00:44:04.000
If I think of the solution as
being a parameterized motion.
00:44:03.000 --> 00:44:09.000
All I have drawn here is the
trace, the path of the motion.
00:44:08.000 --> 00:44:14.000
This hasn't indicated how fast
it was going.
00:44:11.000 --> 00:44:17.000
One solution might go whoosh
and another one might go rah.
00:44:16.000 --> 00:44:22.000
That is a velocity,
and that velocity changes from
00:44:20.000 --> 00:44:26.000
point to point.
It changes direction.
00:44:23.000 --> 00:44:29.000
Well, we know its direction at
each point.
00:44:27.000 --> 00:44:33.000
That's tangent.
What I cannot tell is the
00:44:31.000 --> 00:44:37.000
speed.
From this picture,
00:44:33.000 --> 00:44:39.000
I cannot tell what the speed
was.
00:44:36.000 --> 00:44:42.000
Too bad.
Now, what is then the meaning
00:44:39.000 --> 00:44:45.000
of the system?
What the system does,
00:44:41.000 --> 00:44:47.000
it prescribes at each point the
velocity vector.
00:44:45.000 --> 00:44:51.000
If you tell me what the point
(x, y) is in the plane then
00:44:50.000 --> 00:44:56.000
these equations give you the
velocity vector at that point.
00:44:54.000 --> 00:45:00.000
And, therefore,
what I end up with,
00:44:57.000 --> 00:45:03.000
the system is what you call in
physics and what you call in
00:45:01.000 --> 00:45:07.000
18.02 a velocity field.
So at each point there is a
00:45:06.000 --> 00:45:12.000
certain vector.
The vector is always tangent to
00:45:09.000 --> 00:45:15.000
the solution curve through
there, but I cannot predict from
00:45:13.000 --> 00:45:19.000
just this picture what its
length will be because at some
00:45:17.000 --> 00:45:23.000
points, it might be going slow.
The solution might be going
00:45:21.000 --> 00:45:27.000
slowly.
In other words,
00:45:22.000 --> 00:45:28.000
the plane is filled up with
these guys.
00:45:33.000 --> 00:45:39.000
Stop me.
Not enough here.
00:45:37.000 --> 00:45:43.000
So on and so on.
We can say a system of first
00:45:44.000 --> 00:45:50.000
order equations,
ODEs of first order equations,
00:45:52.000 --> 00:45:58.000
autonomous because there must
be no t on the right-hand side,
00:46:03.000 --> 00:46:09.000
is equal to a velocity field.
A field of velocity.
00:46:12.000 --> 00:46:18.000
The plane covered with velocity
vectors.
00:46:18.000 --> 00:46:24.000
And a solution is a
parameterized curve with the
00:46:25.000 --> 00:46:31.000
right velocity everywhere.
00:46:38.000 --> 00:46:44.000
Now, there obviously must be a
connection between that and the
00:47:39.000 --> 00:47:45.000
direction fields we studied at
the beginning of the term.
00:48:36.000 --> 00:48:42.000
And there is.
It is a very important
00:49:11.000 --> 00:49:17.000
connection.
It is too important to talk
00:49:49.000 --> 00:49:55.000
about in minus one minute.
When we need it,
00:50:32.000 --> 00:50:38.000
I will have to spend some time
talking about it then.