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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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64
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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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75
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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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00:11:10 --> 00:11:16
the problem of heat conduction.
I had forgotten whether it was
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on the problem set or I did it
in class, but I am choosing it
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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.
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[LAUGHTER] That is the yoke.
The yolk is contained in a
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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
203
00:12:22 --> 00:12:28
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
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variables instead of just the
one variable for the overall
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00:12:47 --> 00:12:53
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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00:13:01 --> 00:13:07
supposed to be bad for you.
It certainly has different
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00:13:06 --> 00:13:12
conducting.
It is liquid,
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00:13:07 --> 00:13:13
at the beginning at any rate,
but it certainly has different
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00:13:11 --> 00:13:17
constants of conductivity than
the egg white would.
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00:13:14 --> 00:13:20
And the condition of heat
through the shell of the egg
217
00:13:17 --> 00:13:23
would be different from the
conduction of heat through the
218
00:13:20 --> 00:13:26
membrane that keeps the yoke
together.
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00:13:22 --> 00:13:28
So it is quite reasonable to
consider that the white and the
220
00:13:26 --> 00:13:32
yolk will be at different
temperatures and will have
221
00:13:29 --> 00:13:35
different conductivity
properties.
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I am going to use Newton's laws
but with this further
223
00:13:36 --> 00:13:42
refinement.
In other words,
224
00:13:38 --> 00:13:44
introducing two temperatures.
Whereas, before we only had one
225
00:13:43 --> 00:13:49
temperature.
But let's use Newton's law.
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00:13:46 --> 00:13:52
Let's see.
The question is how does T1,
227
00:13:49 --> 00:13:55
the temperature of the yolk,
vary with time?
228
00:13:52 --> 00:13:58
Well, the yolk is getting all
its heat from the white.
229
00:13:57 --> 00:14:03
Therefore, Newton's law of
conduction will be some constant
230
00:14:01 --> 00:14:07
of conductivity for the yolk
times T2 minus T1.
231
00:14:08 --> 00:14:14
The yolk does not know anything
about the external temperature
232
00:14:12 --> 00:14:18
of the water bath.
It is completely surrounded,
233
00:14:15 --> 00:14:21
snug and secure within itself.
But how about the temperature
234
00:14:20 --> 00:14:26
of the egg white?
That gets heat and gives heat
235
00:14:23 --> 00:14:29
to two sources,
from the external water and
236
00:14:26 --> 00:14:32
also from the internal yolk
inside.
237
00:14:30 --> 00:14:36
So you have to take into
account both of those.
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00:14:33 --> 00:14:39
Its conduction of the heat
through that membrane,
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00:14:36 --> 00:14:42
we will use the same a,
which is going to be a times T1
240
00:14:40 --> 00:14:46
minus T2.
Remember the order in which you
241
00:14:44 --> 00:14:50
have to write these is governed
by the yolk outside to the
242
00:14:48 --> 00:14:54
white.
Therefore, that has to come
243
00:14:51 --> 00:14:57
first when I write it in order
that a be a positive constant.
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00:14:55 --> 00:15:01
But it is also getting heat
from the water bath.
245
00:15:00 --> 00:15:06
And, presumably,
the conductivity through the
246
00:15:03 --> 00:15:09
shell is different from what it
is through this membrane around
247
00:15:08 --> 00:15:14
the yolk.
So I am going to call that by a
248
00:15:11 --> 00:15:17
different constant.
This is the conductivity
249
00:15:14 --> 00:15:20
through the shell into the
white.
250
00:15:17 --> 00:15:23
And that is going to be T,
the external temperature minus
251
00:15:21 --> 00:15:27
the temperature of the egg
white.
252
00:15:24 --> 00:15:30
Here I have a system of
equations because I want to make
253
00:15:28 --> 00:15:34
two dependent variables by
refining the original problem.
254
00:15:34 --> 00:15:40
Now, you always have to write a
system in standard form to solve
255
00:15:39 --> 00:15:45
it.
You will see that the left-hand
256
00:15:42 --> 00:15:48
side will give the dependent
variables in a certain order.
257
00:15:47 --> 00:15:53
In this case,
the temperature of the yolk and
258
00:15:51 --> 00:15:57
then the temperature of the
white.
259
00:15:54 --> 00:16:00
The law is that in order not to
make mistakes --
260
00:16:00 --> 00:16:06
And it's a very frequent source
of error so learn from the
261
00:16:03 --> 00:16:09
beginning not to do this.
You must write the variables on
262
00:16:07 --> 00:16:13
the right-hand side in the same
order left to right in which
263
00:16:11 --> 00:16:17
they occur top to bottom here.
In other words,
264
00:16:14 --> 00:16:20
this is not a good way to leave
that.
265
00:16:16 --> 00:16:22
This is the first attempt in
writing this system,
266
00:16:20 --> 00:16:26
but the final version should
like this.
267
00:16:22 --> 00:16:28
T1 prime,
I won't bother writing dT / dt,
268
00:16:25 --> 00:16:31
is equal to --
T1 must come first,
269
00:16:29 --> 00:16:35
so minus a times T1 plus a
times T2.
270
00:16:34 --> 00:16:40
And the same law for the second
one.
271
00:16:38 --> 00:16:44
It must come in the same order.
Now, the coefficient of T1,
272
00:16:44 --> 00:16:50
that is easy.
That's a times T1.
273
00:16:47 --> 00:16:53
The coefficient of T2 is minus
a minus b,
274
00:16:52 --> 00:16:58
so minus (a plus b) times T2.
275
00:16:56 --> 00:17:02
But I am not done yet.
There is still this external
276
00:17:02 --> 00:17:08
temperature I must put into the
equation.
277
00:17:06 --> 00:17:12
Now, that is not a variable.
This is some given function of
278
00:17:11 --> 00:17:17
t.
And what the function of t is,
279
00:17:14 --> 00:17:20
of course, depends upon what
the problem is.
280
00:17:18 --> 00:17:24
So that, for example,
what might be some
281
00:17:22 --> 00:17:28
possibilities,
well, suppose the problem was I
282
00:17:26 --> 00:17:32
wanted to coddle the egg.
I think there is a generation
283
00:17:32 --> 00:17:38
gap here.
How many of you know what a
284
00:17:35 --> 00:17:41
coddled egg is?
How many of you don't know?
285
00:17:38 --> 00:17:44
Well, I'm just saying my
daughter didn't know.
286
00:17:42 --> 00:17:48
I mentioned it to her.
I said I think I'm going to do
287
00:17:46 --> 00:17:52
a coddled egg tomorrow in class.
And she said what is that?
288
00:17:51 --> 00:17:57
And so I said a cuddled egg?
She said why would someone
289
00:17:55 --> 00:18:01
cuddle an egg?
I said coddle.
290
00:17:59 --> 00:18:05
And she said,
oh, you mean like a person,
291
00:18:02 --> 00:18:08
like what you do to somebody
you like or don't like or I
292
00:18:07 --> 00:18:13
don't know.
Whatever.
293
00:18:09 --> 00:18:15
I thought a while and said,
yeah, more like that.
294
00:18:13 --> 00:18:19
[LAUGHTER] Anyway,
for the enrichment of your
295
00:18:17 --> 00:18:23
cooking skills,
to coddle an egg,
296
00:18:20 --> 00:18:26
it is considered to produce a
better quality product than
297
00:18:25 --> 00:18:31
boiling an egg.
That is why people do it.
298
00:18:30 --> 00:18:36
You heat up the water to
boiling, the egg should be at
299
00:18:34 --> 00:18:40
room temperature,
and then you carefully lower
300
00:18:37 --> 00:18:43
the egg into the water.
And you turn off the heat so
301
00:18:41 --> 00:18:47
the water bath cools
exponentially while the egg
302
00:18:45 --> 00:18:51
inside is rising in temperature.
And then you wait four minutes
303
00:18:50 --> 00:18:56
or six minutes or whatever and
take it out.
304
00:18:53 --> 00:18:59
You have a perfect egg.
So for coddling,
305
00:18:56 --> 00:19:02
spelled so, what will the
external temperature be?
306
00:19:02 --> 00:19:08
Well, it starts out at time
zero at 100 degrees centigrade
307
00:19:06 --> 00:19:12
because the water is supposed to
be boiling.
308
00:19:09 --> 00:19:15
The reason you have it boiling
is for calibration so that you
309
00:19:13 --> 00:19:19
can know what temperature it is
without having to use a
310
00:19:17 --> 00:19:23
thermometer, unless you're on
Pike's Peak or some place.
311
00:19:20 --> 00:19:26
It starts out at 100 degrees.
And after that,
312
00:19:24 --> 00:19:30
since the light is off,
it cools exponential because
313
00:19:27 --> 00:19:33
that is another law.
You only have to know what K is
314
00:19:32 --> 00:19:38
for your particular pot and you
will be able to solve the
315
00:19:37 --> 00:19:43
coddled egg problem.
In other words,
316
00:19:40 --> 00:19:46
you will then be able to solve
these equations and know how the
317
00:19:45 --> 00:19:51
temperature rises.
I am going to solve a different
318
00:19:49 --> 00:19:55
problem because I don't want to
have to deal with this
319
00:19:54 --> 00:20:00
inhomogeneous term.
Let's use, as a different
320
00:19:58 --> 00:20:04
problem, a person cooks an egg.
Coddles the egg by the first
321
00:20:04 --> 00:20:10
process, decides the egg is
done, let's say hardboiled,
322
00:20:09 --> 00:20:15
and then you are supposed to
drop a hardboiled egg into cold
323
00:20:14 --> 00:20:20
water.
Not just to cool it but also
324
00:20:17 --> 00:20:23
because I think it prevents that
dark thing from forming that
325
00:20:23 --> 00:20:29
looks sort of unattractive.
Let's ice bath.
326
00:20:28 --> 00:20:34
The only reason for dropping
the egg into an ice bath is so
327
00:20:32 --> 00:20:38
that you could have a homogenous
equation to solve.
328
00:20:36 --> 00:20:42
And since this a first system
we are going to solve,
329
00:20:40 --> 00:20:46
let's make life easy for
ourselves.
330
00:20:43 --> 00:20:49
Now, all my work in preparing
this example,
331
00:20:47 --> 00:20:53
and it took considerably longer
time than actually solving the
332
00:20:52 --> 00:20:58
problem, was in picking values
for a and b which would make
333
00:20:56 --> 00:21:02
everything come out nice.
It's harder than it looks.
334
00:21:02 --> 00:21:08
The values that we are going to
use, which make no physical
335
00:21:07 --> 00:21:13
sense whatsoever,
but a equals 2 and b
336
00:21:11 --> 00:21:17
equals 3.
These are called nice numbers.
337
00:21:15 --> 00:21:21
What is the equation?
What is the system?
338
00:21:18 --> 00:21:24
Can somebody read it off for
me?
339
00:21:21 --> 00:21:27
It is T1 prime equals, what
is it, minus 2T1 plus 2T2.
340
00:21:26 --> 00:21:32
That's good.
341
00:21:30 --> 00:21:36
Minus 2T1 plus 2T2.
342
00:21:34 --> 00:21:40
343
00:21:40 --> 00:21:46
T2 prime is,
what is it?
344
00:21:42 --> 00:21:48
I think this is 2T1.
And the other one is minus a
345
00:21:48 --> 00:21:54
plus b, so minus 5.
346
00:21:51 --> 00:21:57
This is a system.
Now, on Wednesday I will teach
347
00:21:57 --> 00:22:03
you a fancy way of solving this.
But, to be honest,
348
00:22:03 --> 00:22:09
the fancy way will take roughly
about as long as the way I am
349
00:22:07 --> 00:22:13
going to do it now.
The main reason for doing it is
350
00:22:10 --> 00:22:16
that it introduces new
vocabulary which everyone wants
351
00:22:14 --> 00:22:20
you to have.
And also, more important
352
00:22:16 --> 00:22:22
reasons, it gives more insight
into the solution than this
353
00:22:20 --> 00:22:26
method.
This method just produces the
354
00:22:22 --> 00:22:28
answer, but you want insight,
also.
355
00:22:24 --> 00:22:30
And that is just as important.
But for now,
356
00:22:28 --> 00:22:34
let's use a method which always
works and which in 40 years,
357
00:22:33 --> 00:22:39
after you have forgotten all
other fancy methods,
358
00:22:36 --> 00:22:42
will still be available to you
because it is method you can
359
00:22:40 --> 00:22:46
figure out yourself.
You don't have to remember
360
00:22:43 --> 00:22:49
anything.
The method is to eliminate one
361
00:22:46 --> 00:22:52
of the dependent variables.
It is just the way you solve
362
00:22:50 --> 00:22:56
systems of linear equations in
general if you aren't doing
363
00:22:54 --> 00:23:00
something fancy with
determinants and matrices.
364
00:22:59 --> 00:23:05
If you just eliminate
variables.
365
00:23:01 --> 00:23:07
We are going to eliminate one
of these variables.
366
00:23:05 --> 00:23:11
Let's eliminate T2.
You could also eliminate T1.
367
00:23:08 --> 00:23:14
The main thing is eliminate one
of them so you will have just
368
00:23:13 --> 00:23:19
one left to work with.
How do I eliminate T2?
369
00:23:16 --> 00:23:22
Beg your pardon?
Is something wrong?
370
00:23:19 --> 00:23:25
If somebody thinks something is
wrong raise his hand.
371
00:23:23 --> 00:23:29
No?
372
00:23:25 --> 00:23:31
373
00:23:30 --> 00:23:36
Why do I want to get rid of T1?
Well, I can add them.
374
00:23:33 --> 00:23:39
But, on the left-hand side,
I will have T1 prime plus T2
375
00:23:36 --> 00:23:42
prime. What good is that?
376
00:23:39 --> 00:23:45
[LAUGHTER]
377
00:23:40 --> 00:23:46
378
00:23:48 --> 00:23:54
I think you will want to do it
my way.
379
00:23:49 --> 00:23:55
[APPLAUSE]
380
00:23:50 --> 00:23:56
381
00:24:03 --> 00:24:09
Solve for T2 in terms of T1.
That is going to be T1 prime
382
00:24:08 --> 00:24:14
plus 2T1 divided by 2.
383
00:24:12 --> 00:24:18
Now, take that and substitute
it into the second equation.
384
00:24:18 --> 00:24:24
Wherever you see a T2,
put that in,
385
00:24:21 --> 00:24:27
and what you will be left with
is something just in T1.
386
00:24:28 --> 00:24:34
To be honest,
I don't know any other good way
387
00:24:31 --> 00:24:37
of doing this.
There is a fancy method that I
388
00:24:34 --> 00:24:40
think is talked about in your
book, which leads to extraneous
389
00:24:39 --> 00:24:45
solutions and so on,
but you don't want to know
390
00:24:43 --> 00:24:49
about that.
This will work for a simple
391
00:24:46 --> 00:24:52
linear equation with constant
coefficients,
392
00:24:49 --> 00:24:55
always.
Substitute in.
393
00:24:51 --> 00:24:57
What do I do?
Now, here I do not advise doing
394
00:24:54 --> 00:25:00
this mentally.
It is just too easy to make a
395
00:24:57 --> 00:25:03
mistake.
Here, I will do it carefully,
396
00:25:04 --> 00:25:10
writing everything out just as
you would.
397
00:25:10 --> 00:25:16
T1 prime plus 2T1 over 2,
prime, equals 2T1 minus 5 time
398
00:25:18 --> 00:25:24
T1 prime plus 2T1 over two.
399
00:25:27 --> 00:25:33
I took that and substituted
400
00:25:32 --> 00:25:38
into this equation.
Now, I don't like those two's.
401
00:25:38 --> 00:25:44
Let's get rid of them by
multiplying.
402
00:25:42 --> 00:25:48
This will become 4.
403
00:25:45 --> 00:25:51
404
00:25:52 --> 00:25:58
And now write this out.
What is this when you look at
405
00:25:57 --> 00:26:03
it?
This is an equation just in T1.
406
00:26:00 --> 00:26:06
It has constant coefficients.
And what is its order?
407
00:26:05 --> 00:26:11
Its order is two because T1
prime primed.
408
00:26:10 --> 00:26:16
In other words,
I can eliminate T2 okay,
409
00:26:13 --> 00:26:19
but the equation I am going to
get is no longer a first-order.
410
00:26:19 --> 00:26:25
It becomes a second-order
differential equation.
411
00:26:24 --> 00:26:30
And that's a basic law.
Even if you have a system of
412
00:26:30 --> 00:26:36
more equations,
three or four or whatever,
413
00:26:33 --> 00:26:39
the law is that after you do
the elimination successfully and
414
00:26:37 --> 00:26:43
end up with a single equation,
normally the order of that
415
00:26:42 --> 00:26:48
equation will be the sum of the
orders of the things you started
416
00:26:46 --> 00:26:52
with.
So two first-order equations
417
00:26:49 --> 00:26:55
will always produce a
second-order equation in just
418
00:26:53 --> 00:26:59
one dependent variable,
three will produce a third
419
00:26:56 --> 00:27:02
order equation and so on.
So you trade one complexity for
420
00:27:02 --> 00:27:08
another.
You trade the complexity of
421
00:27:04 --> 00:27:10
having to deal with two
equations simultaneously instead
422
00:27:09 --> 00:27:15
of just one for the complexity
of having to deal with a single
423
00:27:13 --> 00:27:19
higher order equation which is
more trouble to solve.
424
00:27:17 --> 00:27:23
It is like all mathematical
problems.
425
00:27:20 --> 00:27:26
Unless you are very lucky,
if you push them down one way,
426
00:27:24 --> 00:27:30
they are really simple now,
they just pop up some place
427
00:27:28 --> 00:27:34
else.
You say, oh,
428
00:27:30 --> 00:27:36
I didn't save anything after
all.
429
00:27:32 --> 00:27:38
That is the law of conservation
of mathematical difficulty.
430
00:27:36 --> 00:27:42
[LAUGHTER] You saw that even
with the Laplace transform.
431
00:27:40 --> 00:27:46
In the beginning it looks
great, you've got these tables,
432
00:27:44 --> 00:27:50
take the equation,
horrible to solve.
433
00:27:46 --> 00:27:52
Take some transform,
trivial to solve for capital Y.
434
00:27:50 --> 00:27:56
Now I have to find the inverse
Laplace transform.
435
00:27:53 --> 00:27:59
And suddenly all the work is
there, partial fractions,
436
00:27:57 --> 00:28:03
funny formulas and so on.
It is very hard in mathematics
437
00:28:02 --> 00:28:08
to get away with something.
It happens now and then and
438
00:28:06 --> 00:28:12
everybody cheers.
Let's write this out now in the
439
00:28:09 --> 00:28:15
form in which it looks like an
equation we can actually solve.
440
00:28:13 --> 00:28:19
Just be careful.
Now it is all right to use the
441
00:28:17 --> 00:28:23
method by which you collect
terms.
442
00:28:19 --> 00:28:25
There is only one term
involving T1 double prime.
443
00:28:23 --> 00:28:29
It's the one that comes from
here.
444
00:28:25 --> 00:28:31
How about the terms in T1
prime?
445
00:28:27 --> 00:28:33
There is a 2.
Here, there is minus 5 T1
446
00:28:33 --> 00:28:39
prime.
If I put it on the other side
447
00:28:37 --> 00:28:43
it makes plus 5 T1 prime plus
this two makes 7 T1 prime.
448
00:28:44 --> 00:28:50
And how many T1's are there?
Well, none on the left-hand
449
00:28:51 --> 00:28:57
side.
On the right-hand side I have 4
450
00:28:55 --> 00:29:01
here minus 10.
4 minus 10 is negative 6.
451
00:29:02 --> 00:29:08
Negative 6 T1 put on this
left-hand side the way we want
452
00:29:06 --> 00:29:12
to do makes plus 6 T1.
453
00:29:09 --> 00:29:15
454
00:29:15 --> 00:29:21
There are no inhomogeneous
terms, so that is equal to zero.
455
00:29:18 --> 00:29:24
If I had gotten a negative
number for one of these
456
00:29:22 --> 00:29:28
coefficients,
I would instantly know if I had
457
00:29:25 --> 00:29:31
made a mistake.
Why?
458
00:29:26 --> 00:29:32
Why must those numbers come out
to be positive?
459
00:29:30 --> 00:29:36
It is because the system must
be, the system must be,
460
00:29:33 --> 00:29:39
fill in with one word,
stable.
461
00:29:36 --> 00:29:42
And why must this system be
stable?
462
00:29:38 --> 00:29:44
In other words,
the long-term solutions must be
463
00:29:42 --> 00:29:48
zero, must all go to zero,
whatever they are.
464
00:29:45 --> 00:29:51
Why is that?
Well, because you are putting
465
00:29:48 --> 00:29:54
the egg into an ice bath.
Or, because we know it was
466
00:29:52 --> 00:29:58
living but after being
hardboiled it is dead and,
467
00:29:56 --> 00:30:02
therefore, dead systems are
stable.
468
00:30:00 --> 00:30:06
That's not a good reason but it
is, so to speak,
469
00:30:03 --> 00:30:09
the real one.
It's clear anyway that all
470
00:30:05 --> 00:30:11
solutions must tend to zero
physically.
471
00:30:08 --> 00:30:14
That's obvious.
And, therefore,
472
00:30:10 --> 00:30:16
the differential equation must
have the same property,
473
00:30:14 --> 00:30:20
and that means that its
coefficients must be positive.
474
00:30:17 --> 00:30:23
All its coefficients must be
positive.
475
00:30:20 --> 00:30:26
If this weren't there,
I would get oscillating
476
00:30:23 --> 00:30:29
solutions, which wouldn't go to
zero.
477
00:30:25 --> 00:30:31
That is physical impossible for
this egg.
478
00:30:30 --> 00:30:36
Now the rest is just solving.
The characteristic equation,
479
00:30:34 --> 00:30:40
if you can remember way,
way back in prehistoric times
480
00:30:39 --> 00:30:45
when we were solving these
equations, is this.
481
00:30:43 --> 00:30:49
And what you want to do is
factor it.
482
00:30:46 --> 00:30:52
This is where all the work was,
getting those numbers so that
483
00:30:51 --> 00:30:57
this would factor. So it's
r plus 1 times r plus 6
484
00:30:56 --> 00:31:02
485
00:30:59 --> 00:31:05
486
00:31:04 --> 00:31:10
And so the solutions are,
the roots are r equals
487
00:31:07 --> 00:31:13
negative 1.
I am just making marks on the
488
00:31:10 --> 00:31:16
board, but you have done this
often enough,
489
00:31:13 --> 00:31:19
you know what I am talking
about.
490
00:31:15 --> 00:31:21
So the characteristic roots are
those two numbers.
491
00:31:18 --> 00:31:24
And, therefore,
the solution is,
492
00:31:20 --> 00:31:26
I could write down immediately
with its arbitrary constant as
493
00:31:24 --> 00:31:30
c1 times e to the negative t
plus c2 times e to the negative
494
00:31:28 --> 00:31:34
6t. Now, I have got to get T2.
495
00:31:34 --> 00:31:40
Here the first worry is T2 is
going to give me two more
496
00:31:39 --> 00:31:45
arbitrary constants.
It better not.
497
00:31:42 --> 00:31:48
The system is only allowed to
have two arbitrary constants in
498
00:31:47 --> 00:31:53
its solution because that is the
initial conditions we are giving
499
00:31:52 --> 00:31:58
it.
By the way, I forgot to give
500
00:31:55 --> 00:32:01
initial conditions.
Let's give initial conditions.
501
00:32:01 --> 00:32:07
Let's say the initial
temperature of the yolk,
502
00:32:05 --> 00:32:11
when it is put in the ice bath,
is 40 degrees centigrade,
503
00:32:10 --> 00:32:16
Celsius.
And T2, let's say the white
504
00:32:13 --> 00:32:19
ought to be a little hotter than
the yolk is always cooler than
505
00:32:18 --> 00:32:24
the white for a soft boiled egg,
I don't know,
506
00:32:22 --> 00:32:28
or a hardboiled egg if it
hasn't been chilled too long.
507
00:32:27 --> 00:32:33
Let's make this 45.
Realistic numbers.
508
00:32:32 --> 00:32:38
Now, the thing not to do is to
say, hey, I found T1.
509
00:32:35 --> 00:32:41
Okay, I will find T2 by the
same procedure.
510
00:32:39 --> 00:32:45
I will go through the whole
thing.
511
00:32:41 --> 00:32:47
I will eliminate T1 instead.
Then I will end up with an
512
00:32:45 --> 00:32:51
equation T2 and I will solve
that and get T2 equals blah,
513
00:32:50 --> 00:32:56
blah, blah.
That is no good,
514
00:32:52 --> 00:32:58
A, because you are working too
hard and, B, because you are
515
00:32:56 --> 00:33:02
going to get two more arbitrary
constants unrelated to these
516
00:33:01 --> 00:33:07
two.
And that is no good.
517
00:33:04 --> 00:33:10
Because the correct solution
only has two constants in it.
518
00:33:09 --> 00:33:15
Not four.
So that procedure is wrong.
519
00:33:12 --> 00:33:18
You must calculate T2 from the
T1 that you found,
520
00:33:15 --> 00:33:21
and that is the equation which
does it.
521
00:33:18 --> 00:33:24
That's the one we have to have.
Where is the chalk?
522
00:33:22 --> 00:33:28
Yes.
Maybe I can have a little thing
523
00:33:25 --> 00:33:31
so I can just carry this around
with me.
524
00:33:30 --> 00:33:36
525
00:33:37 --> 00:33:43
That is the relation between T2
and T1.
526
00:33:40 --> 00:33:46
Or, if you don't like it,
either one of these equations
527
00:33:44 --> 00:33:50
will express T2 in terms of T1
for you.
528
00:33:47 --> 00:33:53
It doesn't matter.
Whichever one you use,
529
00:33:50 --> 00:33:56
however you do it,
that's the way you must
530
00:33:53 --> 00:33:59
calculate T2.
So what is it?
531
00:33:56 --> 00:34:02
T2 is calculated from that pink
box.
532
00:34:00 --> 00:34:06
It is one-half of T1 prime plus
T1.
533
00:34:05 --> 00:34:11
Now, if I take the derivative
of this, I get minus c1 times
534
00:34:11 --> 00:34:17
the exponential.
The coefficient is minus c1,
535
00:34:16 --> 00:34:22
take half of that,
that is minus a half c1
536
00:34:21 --> 00:34:27
and add it to T1.
Minus one-half c1 plus c1 gives
537
00:34:26 --> 00:34:32
me one-half c1.
538
00:34:32 --> 00:34:38
And here I take the derivative,
it is minus 6 c2.
539
00:34:38 --> 00:34:44
Take half of that,
minus 3 c2 and add this c2 to
540
00:34:44 --> 00:34:50
it, minus 3 plus 1 makes minus
2.
541
00:34:48 --> 00:34:54
That is T2.
And notice it uses the same
542
00:34:53 --> 00:34:59
arbitrary constants that T1
uses.
543
00:34:59 --> 00:35:05
So we end up with just two
because we calculated T2 from
544
00:35:02 --> 00:35:08
that formula or from the
equation which is equivalent to
545
00:35:06 --> 00:35:12
it, not from scratch.
We haven't put in the initial
546
00:35:09 --> 00:35:15
conditions yet,
but that is easy to do.
547
00:35:11 --> 00:35:17
Everybody, when working with
exponentials,
548
00:35:14 --> 00:35:20
of course, you always want the
initial conditions to be when T
549
00:35:18 --> 00:35:24
is equal to zero
because that makes all the
550
00:35:21 --> 00:35:27
exponentials one and you don't
have to worry about them.
551
00:35:25 --> 00:35:31
But this you know.
If I put in the initial
552
00:35:27 --> 00:35:33
conditions, at time zero,
T1 has the value 40.
553
00:35:32 --> 00:35:38
So 40 should be equal to c1 +
c2.
554
00:35:38 --> 00:35:44
And the other equation will say
that 45 is equal to one-half c1
555
00:35:45 --> 00:35:51
minus 2 c2.
Now we are supposed to
556
00:35:52 --> 00:35:58
solve these.
Well, this is called solving
557
00:35:57 --> 00:36:03
simultaneous linear equations.
We could use Kramer's rule,
558
00:36:05 --> 00:36:11
inverse matrices,
but why don't we just
559
00:36:09 --> 00:36:15
eliminate.
Let me see.
560
00:36:12 --> 00:36:18
If I multiply by,
45, so multiply by two,
561
00:36:17 --> 00:36:23
you get 90 equals c1
minus 4 c2.
562
00:36:23 --> 00:36:29
Then subtract this guy from
that guy.
563
00:36:27 --> 00:36:33
So, 40 taken from 90 makes 50.
And c1 taken from c1,
564
00:36:35 --> 00:36:41
because I multiplied by two,
makes zero.
565
00:36:40 --> 00:36:46
And c2 taken from minus 4 c2,
that makes minus 5 c2,
566
00:36:47 --> 00:36:53
I guess.
567
00:36:49 --> 00:36:55
I seem to get c2 is equal to
negative 10.
568
00:36:56 --> 00:37:02
And if c2 is negative 10,
then c1 must be 50.
569
00:37:04 --> 00:37:10
There are two ways of checking
the answer.
570
00:37:07 --> 00:37:13
One is to plug it into the
equations, and the other is to
571
00:37:13 --> 00:37:19
peak.
Yes, that's right.
572
00:37:15 --> 00:37:21
[LAUGHTER]
573
00:37:17 --> 00:37:23
574
00:37:25 --> 00:37:31
The final answer is,
in other words,
575
00:37:27 --> 00:37:33
you put a 50 here,
25 there, negative 10 here,
576
00:37:30 --> 00:37:36
and positive 20 there.
That gives the answer to the
577
00:37:34 --> 00:37:40
problem.
It tells you,
578
00:37:35 --> 00:37:41
in other words,
how the temperature of the yolk
579
00:37:39 --> 00:37:45
varies with time and how the
temperature of the white varies
580
00:37:43 --> 00:37:49
with time.
As I said, we are going to
581
00:37:46 --> 00:37:52
learn a slick way of doing this
problem, or at least a very
582
00:37:51 --> 00:37:57
different way of doing the same
problem next time,
583
00:37:54 --> 00:38:00
but let's put that on ice for
the moment.
584
00:37:57 --> 00:38:03
And instead I would like to
spend the rest of the period
585
00:38:01 --> 00:38:07
doing for first order systems
the same thing that I did for
586
00:38:05 --> 00:38:11
you the very first day of the
term.
587
00:38:09 --> 00:38:15
Remember, I walked in assuming
that you knew how to separate
588
00:38:13 --> 00:38:19
variables the first day of the
term, and I did not talk to you
589
00:38:17 --> 00:38:23
about how to solve fancier
equations by fancier methods.
590
00:38:21 --> 00:38:27
I instead talked to you about
the geometric significance,
591
00:38:25 --> 00:38:31
what the geometric meaning of a
single first order equation was
592
00:38:29 --> 00:38:35
and how that geometric meaning
enabled you to solve it
593
00:38:33 --> 00:38:39
numerically.
And we spent a little while
594
00:38:36 --> 00:38:42
working on such problems because
nowadays with computers it is
595
00:38:40 --> 00:38:46
really important that you get a
feeling for what these things
596
00:38:44 --> 00:38:50
mean as opposed to just
algorithms for solving them.
597
00:38:47 --> 00:38:53
As I say, most differential
equations, especially systems,
598
00:38:50 --> 00:38:56
are likely to be solved by a
computer anyway.
599
00:38:54 --> 00:39:00
You have to be the guiding
genius that interprets the
600
00:38:57 --> 00:39:03
answers and can see when
mistakes are being made,
601
00:39:01 --> 00:39:07
stuff like that.
The problem is,
602
00:39:04 --> 00:39:10
therefore, what is the meaning
of this system?
603
00:39:08 --> 00:39:14
604
00:39:15 --> 00:39:21
Well, you are not going to get
anywhere interpreting it
605
00:39:18 --> 00:39:24
geometrically,
unless you get rid of that t on
606
00:39:21 --> 00:39:27
the right-hand side.
And the only way of getting rid
607
00:39:25 --> 00:39:31
of the t is to declare it is not
there.
608
00:39:28 --> 00:39:34
So I hereby declare that I will
only consider,
609
00:39:31 --> 00:39:37
for the rest of the period,
that is only ten minutes,
610
00:39:34 --> 00:39:40
systems in which no t appears
explicitly on the right-hand
611
00:39:38 --> 00:39:44
side.
Because I don't know what to do
612
00:39:42 --> 00:39:48
if it does up here.
We have a word for these.
613
00:39:45 --> 00:39:51
Remember what the first order
word was?
614
00:39:48 --> 00:39:54
A first order equation where
there was no t explicitly on the
615
00:39:53 --> 00:39:59
right-hand side,
we called it,
616
00:39:55 --> 00:40:01
anybody remember?
Just curious.
617
00:39:57 --> 00:40:03
Autonomous, right.
618
00:40:00 --> 00:40:06
619
00:40:05 --> 00:40:11
This is an autonomous system.
It is not a linear system
620
00:40:08 --> 00:40:14
because these are messy
functions.
621
00:40:10 --> 00:40:16
This could be x times y
or x squared minus 3y squared
622
00:40:14 --> 00:40:20
divided by sine of x plus y.
623
00:40:18 --> 00:40:24
It could be a mess.
Definitely not linear.
624
00:40:21 --> 00:40:27
But autonomous means no t.
t means the independent
625
00:40:24 --> 00:40:30
variable appears on the
right-hand side.
626
00:40:27 --> 00:40:33
Of course, it is there.
It is buried in the dx/dt and
627
00:40:30 --> 00:40:36
dy/dt.
But it is not on the right-hand
628
00:40:33 --> 00:40:39
side.
No t appears on the right-hand
629
00:40:35 --> 00:40:41
side.
630
00:40:36 --> 00:40:42
631
00:40:41 --> 00:40:47
Because no t appears on the
right-hand side,
632
00:40:44 --> 00:40:50
I can now draw a picture of
this.
633
00:40:47 --> 00:40:53
But, let's see,
what does a solution look like?
634
00:40:52 --> 00:40:58
I never even talked about what
a solution was,
635
00:40:56 --> 00:41:02
did I?
Well, pretend that immediately
636
00:40:59 --> 00:41:05
after I talked about that,
I talked about this.
637
00:41:05 --> 00:41:11
What is the solution?
Well, the solution,
638
00:41:07 --> 00:41:13
maybe you took it for granted,
is a pair of functions,
639
00:41:10 --> 00:41:16
x of t, y of t if when
you plug it in
640
00:41:13 --> 00:41:19
it satisfies the equation.
And so what else is new?
641
00:41:16 --> 00:41:22
The solution is x
equals x of t,
642
00:41:19 --> 00:41:25
y equals y of t.
643
00:41:22 --> 00:41:28
644
00:41:27 --> 00:41:33
If I draw a picture of that
what would it look like?
645
00:41:30 --> 00:41:36
This is where your previous
knowledge of physics above all
646
00:41:35 --> 00:41:41
18.02, maybe 18.01 if you
learned this in high school,
647
00:41:39 --> 00:41:45
what is x equals x of t and
y equals y of t?
648
00:41:44 --> 00:41:50
How do you draw a picture of
649
00:41:47 --> 00:41:53
that?
What does it represent?
650
00:41:49 --> 00:41:55
A curve.
And what will be the title of
651
00:41:52 --> 00:41:58
the chapter of the calculus book
in which that is discussed?
652
00:41:56 --> 00:42:02
Parametric equations.
This is a parameterized curve.
653
00:42:02 --> 00:42:08
654
00:42:12 --> 00:42:18
So we know what the solution
looks like.
655
00:42:15 --> 00:42:21
Our solution is a parameterized
curve.
656
00:42:18 --> 00:42:24
And what does a parameterized
curve look like?
657
00:42:21 --> 00:42:27
Well, it travels,
and in a certain direction.
658
00:42:26 --> 00:42:32
659
00:42:34 --> 00:42:40
Okay.
That's enough.
660
00:42:35 --> 00:42:41
Why do I have several of those
curves?
661
00:42:38 --> 00:42:44
Well, because I have several
solutions.
662
00:42:40 --> 00:42:46
In fact, given any initial
starting point,
663
00:42:43 --> 00:42:49
there is a solution that goes
through it.
664
00:42:46 --> 00:42:52
I will put in possible starting
points.
665
00:42:49 --> 00:42:55
And you can do this on the
computer screen with a little
666
00:42:53 --> 00:42:59
program you will have,
one of the visuals you'll have.
667
00:42:56 --> 00:43:02
It's being made right now.
You put down starter point,
668
00:43:01 --> 00:43:07
put down a click,
and then it just draws the
669
00:43:04 --> 00:43:10
curve passing through that
point.
670
00:43:06 --> 00:43:12
Didn't we do this early in the
term?
671
00:43:09 --> 00:43:15
Yes.
But there is a difference now
672
00:43:11 --> 00:43:17
which I will explain.
These are various possible
673
00:43:14 --> 00:43:20
starting points at time zero for
this solution,
674
00:43:17 --> 00:43:23
and then you see what happens
to it afterwards.
675
00:43:20 --> 00:43:26
In fact, through every point in
the plane will pass a solution
676
00:43:25 --> 00:43:31
curve, parameterized curve.
Now, what is then the
677
00:43:29 --> 00:43:35
representation of this?
Well, what is the meaning of x
678
00:43:32 --> 00:43:38
prime of t and y prime of t?
679
00:43:36 --> 00:43:42
680
00:43:40 --> 00:43:46
I am not going to worry for the
moment about the right-hand
681
00:43:44 --> 00:43:50
side.
What does this mean by itself?
682
00:43:47 --> 00:43:53
If this is the curve,
the parameterized motion,
683
00:43:50 --> 00:43:56
then this represents its
velocity vector.
684
00:43:53 --> 00:43:59
It is the velocity of the
solution at time t.
685
00:43:58 --> 00:44:04
If I think of the solution as
being a parameterized motion.
686
00:44:03 --> 00:44:09
All I have drawn here is the
trace, the path of the motion.
687
00:44:08 --> 00:44:14
This hasn't indicated how fast
it was going.
688
00:44:11 --> 00:44:17
One solution might go whoosh
and another one might go rah.
689
00:44:16 --> 00:44:22
That is a velocity,
and that velocity changes from
690
00:44:20 --> 00:44:26
point to point.
It changes direction.
691
00:44:23 --> 00:44:29
Well, we know its direction at
each point.
692
00:44:27 --> 00:44:33
That's tangent.
What I cannot tell is the
693
00:44:31 --> 00:44:37
speed.
From this picture,
694
00:44:33 --> 00:44:39
I cannot tell what the speed
was.
695
00:44:36 --> 00:44:42
Too bad.
Now, what is then the meaning
696
00:44:39 --> 00:44:45
of the system?
What the system does,
697
00:44:41 --> 00:44:47
it prescribes at each point the
velocity vector.
698
00:44:45 --> 00:44:51
If you tell me what the point
(x, y) is in the plane then
699
00:44:50 --> 00:44:56
these equations give you the
velocity vector at that point.
700
00:44:54 --> 00:45:00
And, therefore,
what I end up with,
701
00:44:57 --> 00:45:03
the system is what you call in
physics and what you call in
702
00:45:01 --> 00:45:07
18.02 a velocity field.
So at each point there is a
703
00:45:06 --> 00:45:12
certain vector.
The vector is always tangent to
704
00:45:09 --> 00:45:15
the solution curve through
there, but I cannot predict from
705
00:45:13 --> 00:45:19
just this picture what its
length will be because at some
706
00:45:17 --> 00:45:23
points, it might be going slow.
The solution might be going
707
00:45:21 --> 00:45:27
slowly.
In other words,
708
00:45:22 --> 00:45:28
the plane is filled up with
these guys.
709
00:45:26 --> 00:45:32
710
00:45:33 --> 00:45:39
Stop me.
Not enough here.
711
00:45:37 --> 00:45:43
So on and so on.
We can say a system of first
712
00:45:44 --> 00:45:50
order equations,
ODEs of first order equations,
713
00:45:52 --> 00:45:58
autonomous because there must
be no t on the right-hand side,
714
00:46:03 --> 00:46:09
is equal to a velocity field.
A field of velocity.
715
00:46:12 --> 00:46:18
The plane covered with velocity
vectors.
716
00:46:18 --> 00:46:24
And a solution is a
parameterized curve with the
717
00:46:25 --> 00:46:31
right velocity everywhere.
718
00:46:30 --> 00:46:36
719
00:46:38 --> 00:46:44
Now, there obviously must be a
connection between that and the
720
00:47:39 --> 00:47:45
direction fields we studied at
the beginning of the term.
721
00:48:36 --> 00:48:42
And there is.
It is a very important
722
00:49:11 --> 00:49:17
connection.
It is too important to talk
723
00:49:49 --> 00:49:55
about in minus one minute.
When we need it,
724
00:50:32 --> 00:50:38
I will have to spend some time
talking about it then.