WEBVTT
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This time, we started solving
differential equations.
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This is the third
lecture of the term,
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and I have yet to solve
a single differential
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equation in this class.
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Well, that will be
rectified from now
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until the end of the term.
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So, once you learn
separation of variables,
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which is the most elementary
method there is, the single,
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I think the single
most important equation
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is the one that's called the
first order linear equation,
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both because it
occurs frequently
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in models because it's solvable,
and-- I think that's enough.
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If you drop the
course after today
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you will still have learned
those two important methods:
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separation of variables, and
first order linear equations.
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So, what does such an
equation look like?
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Well, I'll write it in there.
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There are several
ways of writing it,
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but I think the
most basic is this.
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I'm going to use x as
the independent variable
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because that's what
your book does.
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But in the
applications, it's often
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t, time, that is the
independent variable.
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And, I'll try to give you
examples which show that.
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So, the equation
looks like this.
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I'll find some
function of x times y
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prime plus some other
function of x times y
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is equal to yet
another function of x.
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Obviously, the x doesn't
have the same status
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here that y does, so y is
extremely limited in how
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it can appear in the equation.
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But, x can be pretty much
arbitrary in those places.
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So, that's the equation
we are talking about,
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and I'll put it up.
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This is the first version of
it, and we'll call them purple.
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Now, why is that called
the linear equation?
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The word linear is a
very heavily used word
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in mathematics, science,
and engineering.
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For the moment, the
best simple answer
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is because it's linear in y
and y prime, the variables
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y and y prime.
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Well, y prime is not a variable.
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Well, you will learn,
in a certain sense,
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it helps to think of it as
one, not right now perhaps,
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but think of it as linear.
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The most closely analogous thing
would be a linear equation,
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a real linear equation, the
kind you studied in high school,
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which would look like this.
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It would have two
variables, and, I guess,
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constant coefficients, equal c.
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Now, that's a linear equation.
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And that's the sense in
which this is linear.
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It's linear in y
prime and y, which
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are the analogs of the
variables y1 and y2.
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A little bit of terminology,
if c is equal to zero,
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it's called homogeneous, the
same way this equation is
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called homogeneous, as you know
from 18.02, if the right hand
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side is zero.
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So, c of x I should
write here, but I won't.
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That's called homogeneous.
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Now, this is a common
form for the equation,
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but it's not what it's
called standard form.
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The standard form for the
equation, and since this
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is going to be a prime course
of confusion, which is probably
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completely correct, but a
prime source of confusion
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is what I meant.
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The standard linear
form, and I'll
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underline linear is the
first co efficient of y prime
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is taken to be one.
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So, you can always convert
that to a standard form
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by simply dividing
through by it.
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And if I do that, the equation
will look like y prime plus,
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now, it's common to not call
it b anymore, the coefficient,
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because it's really b over a.
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And, therefore, it's
common to adopt, yet,
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a new letter for it.
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And, the standard one
that many people use is p.
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How about the right hand side?
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We needed a letter
for that, too.
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It's c over a, but
we'll call it q.
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So, when I talk about
the standard linear form
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for a linear first
order equation,
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it's absolutely that
that I'm talking about.
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Now, you immediately
see that there
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is a potential
for confusion here
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because what did I call the
standard form for a first order
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equation?
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So, I'm going to say, not this.
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The standard first order
form, what would that be?
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Well, it would be y prime
equals, and everything else
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on the left hand side.
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So, it would be y prime.
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And now, if I turn this into
the standard first order form,
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it would be negative
p of x y plus q of x.
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But, of course, nobody
would write negative p of x.
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So, now, I explicitly
want to say
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that this is a form which I will
never use for this equation,
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although half the
books of the world do.
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In short, this poor little
first order equation
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belongs to two ethnic groups.
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It's both a first
order equation,
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and therefore, its standard
form should be written this way,
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but it's also a linear equation,
and therefore its standard form
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should be used this way.
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Well, it has to decide,
and I have decided for it.
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It is, above all, a
linear equation, not just
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a first order equation.
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And, in this course, this will
always be the standard form.
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Now, well, what on
earth is the difference?
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If you don't do it that
way, the difference
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is entirely in the sin(p).
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But, if you get the sign
of p wrong in the answers,
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it is just a disaster
from that point on.
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A trivial little change
of sign in the answer
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produces solutions and
functions which have
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totally different behavior.
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And, you are going to be
really lost in this course.
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So, maybe I should
draw a line through it
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to indicate, please
don't pay any attention
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to this whatsoever, except that
we are not going to do that.
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Okay, well, what's so
important about this equation?
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Well, number one, it
can always be solved.
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That's a very, very big thing
in differential equations.
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But, it's also
the equation which
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arises in a variety of models.
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Now, I'm just going
to list a few of them.
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All of them I think you will
need either in part one or part
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two of problem sets over these
first couple of problem sets,
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or second and third maybe.
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But, of them, I'm going to put
at the very top of the list
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of what I'll call here,
I'll give it two names:
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the temperature
diffusion model, well, it
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would be better to call it
temperature concentration
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by analogy, temperature
concentration model.
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There's the mixing model,
which is hardly less important.
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In other words, it's
almost as important.
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You have that in
your problem set.
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And then, there are other,
slightly less important models.
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There is the model
of radioactive decay.
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There's the model of a bank
interest, bank account,
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various motion models, you
know, Newton's Law type problems
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if you can figure
out a way of getting
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rid of the second derivative,
some motion problems.
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A classic example is the motion
of a rocket being fired off,
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etc., etc., etc.
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Now, today I have
to pick a model.
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And, the one I'm going to
pick is this temperature
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concentration model.
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So, this is going
to be today's model.
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Tomorrow's model
in the recitation,
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I'm asking the recitations
to, among other things,
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make sure they do a mixing
problem, A) to show you
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how to do it, and B) because
it's on the problem sets.
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That's not a good reason,
but it's not a bad one.
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The others are
either in part one
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or we will take them
up later in the term.
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This is not going to be the only
lecture on the linear equation.
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There will be another one
next week of equal importance.
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But, we can't do
everything today.
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So, let's talk about the
temperature concentration
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model, except I'm going
to change its name.
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I'm going to change its name to
the conduction diffusion model.
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I'll put conduction over
there, and diffusion over here,
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let's say, since, as you
will see, the similarities,
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they are practically
the same model.
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All that's changed
from one to the other
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is the name of the ideas.
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In one case, you
call it temperature,
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and the other, you should
call it concentration.
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But, the actual mathematics
isn't identical.
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So, let's begin with conduction.
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All right, so, I need a
simple physical situation
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that I'm modeling.
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So, imagine a tank
of some liquid.
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Water will do as
well as anything.
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And, in the inside
is a suspended,
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somehow, is a chamber.
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A metal cube will
do, and let's suppose
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that its walls are partly
insulated, not so much
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that no heat can get through.
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There is no such thing
as perfect insulation
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anyway, except maybe an
absolute perfect vacuum.
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Now, inside, so here on
the outside is liquid.
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Okay, on the inside
is, what I'm interested
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in is the temperature
of this thing.
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I'll call that T. Now, that's
different from the temperature
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of the external water bath.
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So, I'll call that T
sub e, T for temperature
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measured in Celsius, let's say,
for the sake of definiteness.
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But, this is the
external temperature.
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So, I'll indicate it with an e.
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Now, what is the model?
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Well, in other words, how do I
set up a differential equation
00:12:43.328 --> 00:12:46.000
to model the situation?
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Well, it's based on a physical
law, which I think you know,
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you've had simple
examples like this,
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the so called Newton's
Law of cooling, -- --
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which says that
the rate of change,
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the temperature of the heat goes
from the outside to the inside
00:13:13.000 --> 00:13:14.500
by conduction only.
00:13:14.500 --> 00:13:18.140
Heat, of course, can
travel in various ways,
00:13:18.140 --> 00:13:23.000
by convection, by conduction,
as here, or by radiation,
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are the three most common.
00:13:25.270 --> 00:13:31.332
Of these, I only want one,
namely transmission of heat
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by conduction.
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And, that's the way it's
probably a little better
00:13:36.267 --> 00:13:41.000
to call it the conduction model,
rather than the temperature
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model, which might involve
other ways for the heat
00:13:45.086 --> 00:13:46.600
to be traveling.
00:13:46.600 --> 00:13:49.333
So, dt, the
independent variable,
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is not going to be x,
as it was over there.
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It's going to be t for time.
00:13:55.688 --> 00:13:59.000
So, maybe I should write
that down. t equals time.
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Capital T equals temperature
in degrees Celsius.
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So, you can put in the
degrees Celsius if you want.
00:14:08.000 --> 00:14:12.375
So, it's proportional to
the temperature difference
00:14:12.375 --> 00:14:13.750
between these two.
00:14:13.750 --> 00:14:16.428
Now, how shall I
write the difference?
00:14:16.428 --> 00:14:21.000
Write it this way because if you
don't you will be in trouble.
00:14:21.000 --> 00:14:24.000
Now, why do I write it that way?
00:14:24.000 --> 00:14:26.664
Well, I write it
that way because I
00:14:26.664 --> 00:14:32.000
want this constant to be
positive, a positive constant.
00:14:32.000 --> 00:14:35.800
In general, any constant, so,
parameters which are physical,
00:14:35.800 --> 00:14:39.726
have some physical
significance, one always
00:14:39.726 --> 00:14:44.142
wants to arrange the equation so
that they are positive numbers,
00:14:44.142 --> 00:14:47.855
the way people normally
think of these things.
00:14:47.855 --> 00:14:49.600
This is called the conductivity.
00:14:49.600 --> 00:14:52.000
The conductivity of what?
00:14:52.000 --> 00:14:54.800
Well, I don't
know, of the system
00:14:54.800 --> 00:14:58.080
of the situation, the
conductivity of the wall,
00:14:58.080 --> 00:15:01.999
or the wall if the metal
were just by itself.
00:15:01.999 --> 00:15:04.332
At any rate, it's a constant.
00:15:04.332 --> 00:15:07.000
It's thought of as a constant.
00:15:07.000 --> 00:15:11.142
And, why positive, well, because
if the external temperature is
00:15:11.142 --> 00:15:14.000
bigger than the
internal temperature,
00:15:14.000 --> 00:15:18.666
I expect T to rise, the
internal temperature to rise.
00:15:18.666 --> 00:15:24.220
That means dT / dt, its
slope, should be positive.
00:15:24.220 --> 00:15:27.500
So, in other words, if
Te is bigger than T,
00:15:27.500 --> 00:15:29.375
I expect this number
to be positive.
00:15:29.375 --> 00:15:33.452
And, that tells you that k
must be a positive constant.
00:15:33.452 --> 00:15:37.000
If I had turned it the
other way, expressed
00:15:37.000 --> 00:15:40.000
the difference in
the reverse order,
00:15:40.000 --> 00:15:44.000
K would then be negative,
have to be negative in order
00:15:44.000 --> 00:15:47.000
that this turn out to be
positive in that situation I
00:15:47.000 --> 00:15:47.600
described.
00:15:47.600 --> 00:15:51.200
And, since nobody wants
negative values of k,
00:15:51.200 --> 00:15:53.999
you have to write the
equation in this form
00:15:53.999 --> 00:15:56.000
rather than the
other way around.
00:15:56.000 --> 00:15:59.000
So, there's our
differential equation.
00:15:59.000 --> 00:16:02.000
It will probably have
an initial condition.
00:16:02.000 --> 00:16:05.663
So, it could be the temperature
at the starting time should
00:16:05.663 --> 00:16:10.000
be some given number, T zero.
00:16:10.000 --> 00:16:13.000
But, the condition could
be given in other ways.
00:16:13.000 --> 00:16:15.000
One can ask, what's
the temperature as time
00:16:15.000 --> 00:16:17.000
goes to infinity, for example?
00:16:17.000 --> 00:16:21.000
There are different ways of
getting that initial condition.
00:16:21.000 --> 00:16:23.000
Okay, that's the
conduction model.
00:16:23.000 --> 00:16:25.000
What would the
diffusion model be?
00:16:25.000 --> 00:16:28.142
The diffusion model,
mathematically, would be,
00:16:28.142 --> 00:16:31.000
word for word, the same.
00:16:31.000 --> 00:16:33.912
The only difference
is that now, what
00:16:33.912 --> 00:16:38.270
I imagine is I'll draw
the picture the same way,
00:16:38.270 --> 00:16:43.541
except now I'm going to put,
label the inside not with a T
00:16:43.541 --> 00:16:46.665
but with a C, C
for concentration.
00:16:46.665 --> 00:16:50.856
It's in an external
water bath, let's say.
00:16:50.856 --> 00:16:53.571
So, there is an
external concentration.
00:16:53.571 --> 00:16:57.888
And, what I'm talking
about is some chemical,
00:16:57.888 --> 00:17:02.250
let's say salt will do
as well as anything.
00:17:02.250 --> 00:17:07.500
So, C is equal to salt
concentration inside,
00:17:07.500 --> 00:17:13.714
and Ce would be the salt
concentration outside,
00:17:13.714 --> 00:17:18.000
outside in the water bath.
00:17:18.000 --> 00:17:26.000
Now, I imagine some mechanism,
so this is a salt solution.
00:17:26.000 --> 00:17:28.664
That's a salt solution.
00:17:28.664 --> 00:17:34.178
And, I imagine some mechanism
by which the salt can diffuse,
00:17:34.178 --> 00:17:36.999
it's a diffusion model
now, diffuse from here
00:17:36.999 --> 00:17:40.000
into the air or possibly
out the other way.
00:17:40.000 --> 00:17:43.500
And that's usually done
by vaguely referring
00:17:43.500 --> 00:17:47.140
to the outside as a
semi permeable membrane,
00:17:47.140 --> 00:17:50.080
semi permeable, so
that the salt will
00:17:50.080 --> 00:17:53.750
have a little hard time
getting through but permeable,
00:17:53.750 --> 00:17:56.800
so that it won't be
blocked completely.
00:17:56.800 --> 00:18:00.000
So, there's a membrane.
00:18:00.000 --> 00:18:06.000
You write the semi
permeable membrane outside,
00:18:06.000 --> 00:18:07.713
outside the inside.
00:18:07.713 --> 00:18:10.000
Well, I give up.
00:18:10.000 --> 00:18:14.000
You know, membrane somewhere.
00:18:14.000 --> 00:18:17.000
Sorry, membrane wall.
00:18:17.000 --> 00:18:18.332
How's that?
00:18:18.332 --> 00:18:21.000
Now, what's the equation?
00:18:21.000 --> 00:18:26.600
Well, the equation is
the same, except it's
00:18:26.600 --> 00:18:29.500
called the diffusion equation.
00:18:29.500 --> 00:18:35.284
I don't think Newton
got his name on this.
00:18:35.284 --> 00:18:40.178
The diffusion equation says
that the rate at which the salt
00:18:40.178 --> 00:18:42.384
diffuses across
the membrane, which
00:18:42.384 --> 00:18:47.000
is the same up to a constant
as the rate at which
00:18:47.000 --> 00:18:51.000
the concentration inside
changes, is some constant,
00:18:51.000 --> 00:18:54.000
usually called k still, okay.
00:18:54.000 --> 00:18:55.332
Do I contradict?
00:18:55.332 --> 00:18:58.000
Okay, let's keep calling it k1.
00:18:58.000 --> 00:19:01.000
Now it's different,
times Ce minus C.
00:19:01.000 --> 00:19:03.800
And, for the same
reason as before,
00:19:03.800 --> 00:19:07.142
if the external
concentration is bigger
00:19:07.142 --> 00:19:12.724
than the internal concentration,
we expect salt to flow in.
00:19:12.724 --> 00:19:15.000
That will make C rise.
00:19:15.000 --> 00:19:18.307
It will make this
positive, and therefore, we
00:19:18.307 --> 00:19:21.070
want k to be positive,
just k1 to be
00:19:21.070 --> 00:19:25.000
positive for the same reason
it had to be positive before.
00:19:25.000 --> 00:19:28.600
So, in each case, the model
that I'm talking about
00:19:28.600 --> 00:19:31.000
is the differential equation.
00:19:31.000 --> 00:19:34.500
So, maybe I should, let's
put that, make that clear.
00:19:34.500 --> 00:19:39.713
Or, I would say that this first
order differential equation
00:19:39.713 --> 00:19:43.332
models this physical
situation, and the same thing
00:19:43.332 --> 00:19:46.000
is true on the other
side over here.
00:19:46.000 --> 00:19:48.800
This is the diffusion
equation, and this
00:19:48.800 --> 00:19:50.375
is the conduction equation.
00:19:50.375 --> 00:19:55.220
Now, if you are in any doubt
about the power of differential
00:19:55.220 --> 00:19:58.842
equations, the point is,
when I talk about this thing,
00:19:58.842 --> 00:20:02.875
I don't have to say which
of these I'm following.
00:20:02.875 --> 00:20:07.284
I'll use neutral
variables like Y and X
00:20:07.284 --> 00:20:09.000
to solve these equations.
00:20:09.000 --> 00:20:12.600
But, with a single
stroke, I will be handling
00:20:12.600 --> 00:20:13.750
those situations together.
00:20:13.750 --> 00:20:16.500
And, that's the
power of the method.
00:20:16.500 --> 00:20:20.200
Now, you obviously must
be wondering, look,
00:20:20.200 --> 00:20:22.666
these look very, very special.
00:20:22.666 --> 00:20:27.140
He said he was going to talk
about the first, general first
00:20:27.140 --> 00:20:28.000
order equation.
00:20:28.000 --> 00:20:31.000
But, these look
rather special to me.
00:20:31.000 --> 00:20:33.220
Well, not too special.
00:20:33.220 --> 00:20:36.000
How should we write it?
00:20:36.000 --> 00:20:41.332
Suppose I write, let's take
the temperature equation just
00:20:41.332 --> 00:20:44.000
to have something definite.
00:20:44.000 --> 00:20:48.000
Notice that it's in a form
corresponding to Newton's Law.
00:20:48.000 --> 00:20:52.000
But it is not in the
standard linear form.
00:20:52.000 --> 00:20:54.688
Let's put it in
standard linear form,
00:20:54.688 --> 00:20:59.080
so at least you could see
that it's a linear equation.
00:20:59.080 --> 00:21:02.384
So, if I put it
in standard form,
00:21:02.384 --> 00:21:06.224
it's going to look like
DTDTD little t plus KT
00:21:06.224 --> 00:21:09.220
is equal to K times TE.
00:21:09.220 --> 00:21:13.200
Now, compare that with
the general, the way
00:21:13.200 --> 00:21:16.000
the general equation
is supposed to look,
00:21:16.000 --> 00:21:20.000
the yellow box over there,
the standard linear form.
00:21:20.000 --> 00:21:21.998
How are they going to compare?
00:21:21.998 --> 00:21:24.500
Well, this is a pretty
general function.
00:21:24.500 --> 00:21:26.000
This is general.
00:21:26.000 --> 00:21:29.840
This is a general function
of T because I can
00:21:29.840 --> 00:21:31.428
make the external temperature.
00:21:31.428 --> 00:21:37.000
I could suppose it behaves in
anyway I like, steadily rising,
00:21:37.000 --> 00:21:39.000
decaying exponentially,
maybe oscillating
00:21:39.000 --> 00:21:42.180
back and forth for some reason.
00:21:42.180 --> 00:21:46.500
The only way in which
it's not general
00:21:46.500 --> 00:21:50.000
is that this K is a constant.
00:21:50.000 --> 00:21:53.000
So, I will ask you
to be generous.
00:21:53.000 --> 00:21:57.000
Let's imagine the conductivity
is changing over time.
00:21:57.000 --> 00:22:00.815
So, this is usually
constant, but there's
00:22:00.815 --> 00:22:05.220
no law which says it has to be.
00:22:05.220 --> 00:22:09.142
How could a conductivity
change over time?
00:22:09.142 --> 00:22:12.888
Well, we could
suppose that this wall
00:22:12.888 --> 00:22:17.142
was made of slowly congealing
Jell O, for instance.
00:22:17.142 --> 00:22:23.000
It starts out as liquid,
and then it gets solid.
00:22:23.000 --> 00:22:26.444
And, Jell O doesn't
transmit heat,
00:22:26.444 --> 00:22:32.180
I believe, quite as well as
liquid does, as a liquid would.
00:22:32.180 --> 00:22:36.000
Is Jell O a solid or liquid?
00:22:36.000 --> 00:22:37.713
I don't know.
00:22:37.713 --> 00:22:40.000
Let's forget about that.
00:22:40.000 --> 00:22:46.000
So, with this understanding,
so let's say not necessarily
00:22:46.000 --> 00:22:51.500
here, but not necessarily, I
can think of this, therefore,
00:22:51.500 --> 00:22:55.000
by allowing K to vary with time.
00:22:55.000 --> 00:23:00.000
And the external temperature
to vary with time.
00:23:00.000 --> 00:23:07.000
I can think of it as a
general, linear equation.
00:23:07.000 --> 00:23:09.000
So, these models
are not special.
00:23:09.000 --> 00:23:10.452
They are fairly general.
00:23:10.452 --> 00:23:14.284
Well, I did promise you I
would solve an equation,
00:23:14.284 --> 00:23:18.000
and that this lecture, I still
have not solved any equations.
00:23:18.000 --> 00:23:21.000
OK, time to stop
temporizing and solve.
00:23:21.000 --> 00:23:24.000
So, I'm going to, in order
not to play favorites
00:23:24.000 --> 00:23:27.000
with these two models,
I'll go back to,
00:23:27.000 --> 00:23:31.000
and to get you used to thinking
of the variables all the time,
00:23:31.000 --> 00:23:35.444
that is, you know, be eclectic
switching from one variable
00:23:35.444 --> 00:23:38.552
to another according to
which particular lecture
00:23:38.552 --> 00:23:42.635
you happened to be sitting in.
00:23:42.635 --> 00:23:52.712
So, let's take our equation in
the form, Y prime plus P of XY,
00:23:52.712 --> 00:23:58.000
the general form using
the old variables
00:23:58.000 --> 00:24:04.000
equals Q of X. Solve me.
00:24:04.000 --> 00:24:06.928
Well, there are different ways
of describing the solution
00:24:06.928 --> 00:24:07.428
process.
00:24:07.428 --> 00:24:10.614
No matter how you
do it, it amounts
00:24:10.614 --> 00:24:13.684
to the same amount of
work and there is always
00:24:13.684 --> 00:24:16.331
a trick involved
at each one of them
00:24:16.331 --> 00:24:19.125
since you can't suppress
a trick by doing
00:24:19.125 --> 00:24:21.000
the problem some other way.
00:24:21.000 --> 00:24:24.000
The way I'm going to do
it, I think, is the best.
00:24:24.000 --> 00:24:26.000
That's why I'm giving it to you.
00:24:26.000 --> 00:24:27.815
It's the easiest to remember.
00:24:27.815 --> 00:24:31.000
It leads to the
least work, but I
00:24:31.000 --> 00:24:36.000
have colleagues who would
fight with me about that point.
00:24:36.000 --> 00:24:39.330
So, since they are not
here to fight with me
00:24:39.330 --> 00:24:42.332
I am free to do whatever I like.
00:24:42.332 --> 00:24:45.400
One of the main
reasons for doing
00:24:45.400 --> 00:24:49.400
it the way I'm going
to do is because I
00:24:49.400 --> 00:24:55.555
want you to get what our word
into your consciousness, two
00:24:55.555 --> 00:24:57.220
words, integrating factor.
00:24:57.220 --> 00:25:02.500
I'm going to solve this equation
by finding and integrating
00:25:02.500 --> 00:25:09.000
factor of the form U of X.
What's an integrating factor?
00:25:09.000 --> 00:25:13.000
Well, I'll show you not by
writing an elaborate definition
00:25:13.000 --> 00:25:16.666
on the board, but showing
you what its function is.
00:25:16.666 --> 00:25:19.332
It's a certain
function, U of X, I
00:25:19.332 --> 00:25:23.200
don't know what it is, but
here's what I wanted to do.
00:25:23.200 --> 00:25:26.763
I want to multiply, I'm going
to drop the X's a just so
00:25:26.763 --> 00:25:29.110
that the thing looks
less complicated.
00:25:29.110 --> 00:25:34.125
So, what I want to do is
multiply this equation
00:25:34.125 --> 00:25:36.000
through by U of X.
00:25:36.000 --> 00:25:40.360
That's why it's called
a factor because you're
00:25:40.360 --> 00:25:43.816
going to multiply
everything through by it.
00:25:43.816 --> 00:25:49.220
So, it's going to look like
UY prime plus PUY equals QU,
00:25:49.220 --> 00:25:52.776
and now, so far,
it's just a factor.
00:25:52.776 --> 00:25:55.375
What makes it an
integrating factor
00:25:55.375 --> 00:26:00.766
is that this, after I do
that, I want this to turn out
00:26:00.766 --> 00:26:04.500
to be the derivative of
something with respect
00:26:04.500 --> 00:26:08.250
to X. You see the
motivation for that.
00:26:08.250 --> 00:26:11.125
If this turns out to be the
derivative of something,
00:26:11.125 --> 00:26:13.363
because I've chosen
U so cleverly,
00:26:13.363 --> 00:26:17.000
then I will be able to solve
the equation immediately
00:26:17.000 --> 00:26:19.000
just by integrating
this with respect to X,
00:26:19.000 --> 00:26:21.000
and integrating that
with respect to X.
00:26:21.000 --> 00:26:24.750
You just, then, integrate
both sides with respect to X,
00:26:24.750 --> 00:26:26.000
and the equation is solved.
00:26:26.000 --> 00:26:31.000
Now, the only question is,
what should I choose for U?
00:26:31.000 --> 00:26:34.000
Well, if you think of
the product formula,
00:26:34.000 --> 00:26:37.000
there might be many
things to try here.
00:26:37.000 --> 00:26:40.000
But there's only one
reasonable thing to try.
00:26:40.000 --> 00:26:45.000
Try to pick U so that it's
the derivative of U times Y.
00:26:45.000 --> 00:26:46.665
See how reasonable that is?
00:26:46.665 --> 00:26:49.333
If I use the product
rule on this,
00:26:49.333 --> 00:26:52.000
the first term is
U times Y prime.
00:26:52.000 --> 00:26:56.000
The second term would
be U prime times Y.
00:26:56.000 --> 00:26:59.000
Well, I've got the Y there.
00:26:59.000 --> 00:27:01.000
So, this will work.
00:27:01.000 --> 00:27:05.888
It works if, what's the
condition that you must satisfy
00:27:05.888 --> 00:27:09.000
in order for that to be true?
00:27:09.000 --> 00:27:15.000
Well, it must be that after
it to the differentiation,
00:27:15.000 --> 00:27:19.000
U prime turns out
to be P times U.
00:27:19.000 --> 00:27:20.816
So, is it clear?
00:27:20.816 --> 00:27:26.625
This is something we want to be
equal to, and the thing I will
00:27:26.625 --> 00:27:32.331
try to do it is by
choosing U in such a way
00:27:32.331 --> 00:27:37.000
that this equality
will take place.
00:27:37.000 --> 00:27:40.000
And then I will be able
to solve the equation.
00:27:40.000 --> 00:27:43.000
And so, here's what my
U prime must satisfy.
00:27:43.000 --> 00:27:44.815
Hey, we can solve that.
00:27:44.815 --> 00:27:47.222
But please don't
forget that P is
00:27:47.222 --> 00:27:49.000
P of X. It's a function of X.
00:27:49.000 --> 00:27:53.000
So, if you separate variables,
I'm going to do this.
00:27:53.000 --> 00:27:56.600
So, what is it, DU over
U equals P of X times DX.
00:27:56.600 --> 00:28:00.500
If I integrate that,
so, separate variables,
00:28:00.500 --> 00:28:04.665
integrate, and you're going
to get DU over U integrates
00:28:04.665 --> 00:28:08.428
to the be the log of
U, and the other side
00:28:08.428 --> 00:28:12.000
integrates to be the
integral of P of X DX.
00:28:12.000 --> 00:28:16.000
Now, you can put an
arbitrary constant there,
00:28:16.000 --> 00:28:18.448
or you can think of
it as already implied
00:28:18.448 --> 00:28:20.142
by the indefinite integral.
00:28:20.142 --> 00:28:24.500
Well, that doesn't tell
us, yet, what U is.
00:28:24.500 --> 00:28:26.000
What should U be?
00:28:26.000 --> 00:28:30.250
Notice, I don't have to find
every possible U, which works.
00:28:30.250 --> 00:28:34.000
All I'm looking for is one.
00:28:34.000 --> 00:28:38.000
All I want is a single view
which satisfies that equation.
00:28:38.000 --> 00:28:42.000
Well, U equals the integral,
E to the integral of PDX.
00:28:42.000 --> 00:28:44.500
That's not too
beautiful looking,
00:28:44.500 --> 00:28:46.726
but by differential
equations, things
00:28:46.726 --> 00:28:50.454
can get so complicated
that in a week or two,
00:28:50.454 --> 00:28:55.000
you will think of this as
an extremely simple formula.
00:28:55.000 --> 00:29:00.000
So, there is a formula for
our integrating factor.
00:29:00.000 --> 00:29:01.500
We found it.
00:29:01.500 --> 00:29:07.250
We will always be able to
write an integrating factor.
00:29:07.250 --> 00:29:14.750
Don't worry about the arbitrary
constant because you only need
00:29:14.750 --> 00:29:17.000
one such U.
00:29:17.000 --> 00:29:23.000
So: no arbitrary constant
since only one U needed.
00:29:23.000 --> 00:29:26.600
And, that's the
solution, the way
00:29:26.600 --> 00:29:30.200
we solve the linear equation.
00:29:30.200 --> 00:29:37.544
OK, let's take over,
and actually do it.
00:29:37.544 --> 00:29:43.714
I think it would be
better to summarize it
00:29:43.714 --> 00:29:48.000
as a clear cut method.
00:29:48.000 --> 00:29:51.000
So, let's do that.
00:29:51.000 --> 00:29:54.000
So, what's our method?
00:29:54.000 --> 00:30:04.000
It's the method for solving
Y prime plus PY equals Q.
00:30:04.000 --> 00:30:07.800
Well, the first place, make sure
it's in standard linear form.
00:30:07.800 --> 00:30:11.331
If it isn't, you must
put it in that form.
00:30:11.331 --> 00:30:14.712
Notice, the formula for the
integrating factor, the formula
00:30:14.712 --> 00:30:16.856
for the integrating
factor involves
00:30:16.856 --> 00:30:19.000
P, the integral of PDX.
00:30:19.000 --> 00:30:21.000
So, you'd better
get the right P.
00:30:21.000 --> 00:30:22.600
Otherwise, you are sunk.
00:30:22.600 --> 00:30:25.750
OK, so put it in
standard linear form.
00:30:25.750 --> 00:30:29.332
That way, you will have
the right P. Notice
00:30:29.332 --> 00:30:32.544
that if you wrote
it in that form,
00:30:32.544 --> 00:30:35.375
and all you remembered
was E to the integral PDX,
00:30:35.375 --> 00:30:38.000
the P would have the wrong sign.
00:30:38.000 --> 00:30:41.070
If you're going to write, that
P should have a negative sign
00:30:41.070 --> 00:30:41.570
there.
00:30:41.570 --> 00:30:44.500
So, do it this way,
and no other way.
00:30:44.500 --> 00:30:47.665
Otherwise, you will get
confused and get wrong signs.
00:30:47.665 --> 00:30:51.331
And, as I say, that will
produce wrong answers, and not
00:30:51.331 --> 00:30:54.000
just slightly wrong
answers, but disastrously
00:30:54.000 --> 00:30:57.664
wrong answers from the point
of view of the modeling
00:30:57.664 --> 00:31:02.108
if you really want answers
to physical problems.
00:31:02.108 --> 00:31:06.600
So, here's a
standard linear form.
00:31:06.600 --> 00:31:09.666
Then, find the
integrating factor.
00:31:09.666 --> 00:31:17.000
So, calculate E to the integral,
PDX, the integrating factor,
00:31:17.000 --> 00:31:22.000
and that multiply both,
I'm putting this as both,
00:31:22.000 --> 00:31:31.000
underlined that as many times
as you have room in your notes.
00:31:31.000 --> 00:31:38.362
Multiply both sides by this
integrating factor by E
00:31:38.362 --> 00:31:42.400
to the integral PDX.
00:31:42.400 --> 00:31:46.000
And then, integrate.
00:31:46.000 --> 00:31:51.000
OK, let's take a simple example.
00:31:51.000 --> 00:31:57.400
Suppose we started with
the equation XY prime
00:31:57.400 --> 00:32:06.000
minus Y equals, I had X2,
X3, something like that,
00:32:06.000 --> 00:32:12.000
X3, I think, yeah, X2.
00:32:12.000 --> 00:32:16.000
OK, what's the
first thing to do?
00:32:16.000 --> 00:32:18.915
Put it in standard form.
00:32:18.915 --> 00:32:25.915
So, step zero will be to
write it as Y prime minus
00:32:25.915 --> 00:32:30.000
one over X times Y equals X2.
00:32:30.000 --> 00:32:34.833
Let's do the work first, and
then I'll talk about mistakes.
00:32:34.833 --> 00:32:39.888
Well, we now calculate
the integrating factor.
00:32:39.888 --> 00:32:43.000
So, I would do it in steps.
00:32:43.000 --> 00:32:48.000
You can integrate negative
one over X, right?
00:32:48.000 --> 00:32:51.500
That integrates to
minus log X. So,
00:32:51.500 --> 00:32:57.000
the integrating factor is E
to the integral of this, DX.
00:32:57.000 --> 00:33:02.000
So, it's E to the
negative log X.
00:33:02.000 --> 00:33:06.500
Now, in real life, that's
not the way to leave that.
00:33:06.500 --> 00:33:10.333
What is E to the negative log X?
00:33:10.333 --> 00:33:16.000
Well, think of it as E to
the log X to the minus one.
00:33:16.000 --> 00:33:21.570
Or, in other words, it is
E to the log X is X. So,
00:33:21.570 --> 00:33:23.000
it's one over X.
00:33:23.000 --> 00:33:27.000
So, the integrating
factor is one over X.
00:33:27.000 --> 00:33:34.000
OK, multiply both sides
by the integrating factor.
00:33:34.000 --> 00:33:35.500
Both sides of what?
00:33:35.500 --> 00:33:41.000
Both sides of this: the equation
written in standard form,
00:33:41.000 --> 00:33:42.200
and both sides.
00:33:42.200 --> 00:33:47.800
So, it's going to be one over
XY prime minus one over X2 Y
00:33:47.800 --> 00:33:52.815
is equal to X2 times one over
X, which is simply X. Now,
00:33:52.815 --> 00:33:55.545
if you have done
the work correctly,
00:33:55.545 --> 00:34:01.000
you should be able, now,
to integrate the left hand
00:34:01.000 --> 00:34:02.110
side directly.
00:34:02.110 --> 00:34:06.428
So, I'm going to
write it this way.
00:34:06.428 --> 00:34:10.815
I always recommend that you
put it as extra step, well,
00:34:10.815 --> 00:34:14.142
put it as an extra
step the reason
00:34:14.142 --> 00:34:17.000
for using that
integrating factor,
00:34:17.000 --> 00:34:21.200
in other words, that the left
hand side is supposed to be,
00:34:21.200 --> 00:34:24.000
now, one over X times Y prime.
00:34:24.000 --> 00:34:27.328
I always put it that
because there's always
00:34:27.328 --> 00:34:31.775
a chance you made a mistake
or forgot something.
00:34:31.775 --> 00:34:34.555
Look at it, mentally
differentiated
00:34:34.555 --> 00:34:39.855
using the product rule just
to check that, in fact, it
00:34:39.855 --> 00:34:43.000
turns out to be the same
as the left hand side.
00:34:43.000 --> 00:34:44.535
So, what do we get?
00:34:44.535 --> 00:34:48.500
One over X times Y prime
plus Y times the derivative
00:34:48.500 --> 00:34:53.800
of one over X, which indeed
is negative one over X2.
00:34:53.800 --> 00:34:58.000
And now, finally, that's 3A,
continue, do the integration.
00:34:58.000 --> 00:35:00.149
So, you're going
to get, let's see
00:35:00.149 --> 00:35:04.904
if we can do it all on one
board, one over X times Y
00:35:04.904 --> 00:35:07.904
is equal to X plus a
constant, X, sorry, X2
00:35:07.904 --> 00:35:09.857
over two plus a constant.
00:35:09.857 --> 00:35:16.332
And, the final step
will be, therefore, now
00:35:16.332 --> 00:35:21.000
I want to isolate Y by itself.
00:35:21.000 --> 00:35:26.000
So, Y will be equal to
multiply through by X.
00:35:26.000 --> 00:35:34.000
X3 over two plus C times X.
And, that's the solution.
00:35:34.000 --> 00:35:40.000
OK, let's do one a little
slightly more complicated.
00:35:40.000 --> 00:35:41.816
Let's try this one.
00:35:41.816 --> 00:35:45.545
Now, my equation
is going to be one,
00:35:45.545 --> 00:35:51.000
I'll still keep two, Y
and X, as the variables.
00:35:51.000 --> 00:35:57.000
I'll use T and F
for a minute or two.
00:35:57.000 --> 00:36:01.360
One plus cosine X,
so, I'm not going
00:36:01.360 --> 00:36:07.000
to give you this one in
standard form either.
00:36:07.000 --> 00:36:09.332
It's a trick question.
00:36:09.332 --> 00:36:17.332
Y prime minus sine X times Y is
equal to anything reasonable,
00:36:17.332 --> 00:36:19.000
I guess.
00:36:19.000 --> 00:36:24.000
I think X, 2X, make
it more exciting.
00:36:24.000 --> 00:36:28.920
OK, now, I think
I should warn you
00:36:28.920 --> 00:36:38.000
where the mistakes are just so
that you can make all of them.
00:36:38.000 --> 00:36:41.000
So, this is mistake number one.
00:36:41.000 --> 00:36:44.750
You don't put it
in standard form.
00:36:44.750 --> 00:36:51.000
Mistake number two: generally
people can do step one fine.
00:36:51.000 --> 00:36:57.000
Mistake number two is, this
is my most common mistake,
00:36:57.000 --> 00:37:00.000
so I'm very sensitive to it.
00:37:00.000 --> 00:37:03.750
But that doesn't
mean if you make it,
00:37:03.750 --> 00:37:06.000
you'll get any sympathy from me.
00:37:06.000 --> 00:37:08.000
I don't give sympathy to myself.
00:37:08.000 --> 00:37:10.400
You are so intense,
so happy at having
00:37:10.400 --> 00:37:12.332
found the integrating
factor, you
00:37:12.332 --> 00:37:16.332
forget to multiply Q by the
integrating factor also.
00:37:16.332 --> 00:37:20.332
You just handle the left
hand side of the equation,
00:37:20.332 --> 00:37:22.999
if you forget about
the right hand side.
00:37:22.999 --> 00:37:27.220
So, the emphasis on the
both here is the right hand,
00:37:27.220 --> 00:37:29.000
please include the Q.
00:37:29.000 --> 00:37:33.000
Please include the
right hand side.
00:37:33.000 --> 00:37:34.332
Any other mistakes?
00:37:34.332 --> 00:37:37.250
Well, nothing that
I can think of.
00:37:37.250 --> 00:37:39.664
Well, maybe only,
anyway, we are not
00:37:39.664 --> 00:37:43.570
going to make any mistakes
the rest of this lecture.
00:37:43.570 --> 00:37:45.000
So, what do we do?
00:37:45.000 --> 00:37:47.304
We write this in standard form.
00:37:47.304 --> 00:37:51.600
So, it's going to look
like Y prime minus sine X,
00:37:51.600 --> 00:37:55.816
sine X divided by one
plus cosine X times Y
00:37:55.816 --> 00:37:59.000
equals, my heart
sinks because I know
00:37:59.000 --> 00:38:04.000
I'm supposed to integrate
something like this.
00:38:04.000 --> 00:38:07.000
And, boy, that's going
to give me problems.
00:38:07.000 --> 00:38:08.713
Well, not yet.
00:38:08.713 --> 00:38:11.000
With the integrating factor?
00:38:11.000 --> 00:38:14.000
The integrating
factor is, well, we
00:38:14.000 --> 00:38:18.220
want to calculate the
integral of negative sine X
00:38:18.220 --> 00:38:20.000
over one plus cosine.
00:38:20.000 --> 00:38:21.875
That's the integral of PDX.
00:38:21.875 --> 00:38:25.500
And, after that, we
have to exponentiate it.
00:38:25.500 --> 00:38:28.000
Well, can you do this?
00:38:28.000 --> 00:38:31.000
Yeah, but if you stare
at it a little while,
00:38:31.000 --> 00:38:38.000
you can see that the top is
the derivative of the bottom.
00:38:38.000 --> 00:38:39.332
That is great.
00:38:39.332 --> 00:38:43.600
That means it integrates
to be the log of one
00:38:43.600 --> 00:38:47.362
plus cosine X. Is that
right, one over one
00:38:47.362 --> 00:38:51.625
plus cosine X times the
derivative of this, which
00:38:51.625 --> 00:38:56.375
is negative cosine X. Therefore,
the integrating factor
00:38:56.375 --> 00:38:57.875
is E to that.
00:38:57.875 --> 00:39:02.270
In other words, it
is one plus cosine X.
00:39:02.270 --> 00:39:05.500
Therefore, so this
was step zero.
00:39:05.500 --> 00:39:09.332
Step one, we found the
integrating factor.
00:39:09.332 --> 00:39:17.000
And now, step two, we multiply
through the integrating factor.
00:39:17.000 --> 00:39:19.220
And what do we get?
00:39:19.220 --> 00:39:23.142
We multiply through the
standard for equation
00:39:23.142 --> 00:39:29.000
by the integrating factor, if
you do that, what you get is,
00:39:29.000 --> 00:39:35.000
well, Y prime gets the
coefficient one plus cosine X,
00:39:35.000 --> 00:39:38.885
Y prime minus sign X equals 2X.
00:39:38.885 --> 00:39:40.000
Oh, dear.
00:39:40.000 --> 00:39:45.000
Well, I hope somebody
would giggle at this point.
00:39:45.000 --> 00:39:47.270
What's giggle able about it?
00:39:47.270 --> 00:39:51.000
Well, that all this was
totally wasted work.
00:39:51.000 --> 00:39:53.500
It's called spinning
your wheels.
00:39:53.500 --> 00:39:56.500
No, it's not
spinning your wheels.
00:39:56.500 --> 00:40:00.000
It's doing what
you're supposed to do,
00:40:00.000 --> 00:40:05.000
and finding out that you
wasted the entire time doing
00:40:05.000 --> 00:40:08.000
what you were supposed to do.
00:40:08.000 --> 00:40:12.140
Well, in other words,
that net effect of this
00:40:12.140 --> 00:40:16.856
is to end up with the same
equation we started with.
00:40:16.856 --> 00:40:19.000
But, what is the point?
00:40:19.000 --> 00:40:22.178
The point of having
done all this
00:40:22.178 --> 00:40:26.775
was because now the left
hand side is exactly
00:40:26.775 --> 00:40:32.270
the derivative of something,
and the left hand side should
00:40:32.270 --> 00:40:35.000
be the derivative of what?
00:40:35.000 --> 00:40:37.904
Well, it should be
the derivative of one
00:40:37.904 --> 00:40:40.600
plus cosine X
times Y, all prime.
00:40:40.600 --> 00:40:44.776
Now, you can check that
that's in fact the case.
00:40:44.776 --> 00:40:48.815
It's one plus cosine X,
Y prime, plus minus sine
00:40:48.815 --> 00:40:51.856
X, the derivative of
this side times Y.
00:40:51.856 --> 00:40:56.220
So, if you had thought, in
looking at the equation,
00:40:56.220 --> 00:41:00.724
to say to yourself, this
is a derivative of that,
00:41:00.724 --> 00:41:04.248
maybe I'll just check
right away to see
00:41:04.248 --> 00:41:08.571
if it's the derivative of
one plus cosine X sine.
00:41:08.571 --> 00:41:12.000
You would have saved that work.
00:41:12.000 --> 00:41:16.000
Well, you don't have to
be brilliant or clever,
00:41:16.000 --> 00:41:17.332
or anything like that.
00:41:17.332 --> 00:41:20.200
You can follow your
nose, and it's just,
00:41:20.200 --> 00:41:24.875
I want to give you a positive
experience in solving
00:41:24.875 --> 00:41:28.000
linear equations,
not too negative.
00:41:28.000 --> 00:41:31.000
Anyway, so we got to this point.
00:41:31.000 --> 00:41:37.000
So, now this is 2X, and
now we are ready to solve
00:41:37.000 --> 00:41:43.500
the equation, which is the
solution now will be one plus
00:41:43.500 --> 00:41:49.000
cosine X times Y is equal
to X2 plus a constant,
00:41:49.000 --> 00:41:55.524
and so Y is equal to X2 divided
by X2 plus a constant divided
00:41:55.524 --> 00:42:01.400
by one plus cosine X. Suppose
I have given you an initial
00:42:01.400 --> 00:42:03.000
condition, which I didn't.
00:42:03.000 --> 00:42:07.000
But, suppose the initial
condition said that Y of zero
00:42:07.000 --> 00:42:08.332
were one, for instance.
00:42:08.332 --> 00:42:11.665
Then, the solution would
be, so, this is an if,
00:42:11.665 --> 00:42:15.498
I'm throwing in at the end just
to make it a little bit more
00:42:15.498 --> 00:42:17.400
of a problem, how
would I put, then
00:42:17.400 --> 00:42:21.400
I could evaluate the constant
by using the initial condition.
00:42:21.400 --> 00:42:23.000
What would it be?
00:42:23.000 --> 00:42:25.454
This would be, on the
left hand side, one,
00:42:25.454 --> 00:42:30.000
on the right hand side
would be C over two.
00:42:30.000 --> 00:42:34.000
So, I would get one
equals C over two.
00:42:34.000 --> 00:42:35.875
Is that correct?
00:42:35.875 --> 00:42:42.180
Cosine of zero is one,
so that's two down below.
00:42:42.180 --> 00:42:48.856
Therefore, C is equal to
two, and that would then
00:42:48.856 --> 00:42:51.000
complete the solution.
00:42:51.000 --> 00:42:57.000
We would be X2 plus two
over one plus cosine X.
00:42:57.000 --> 00:43:03.000
Now, you can do this
in general, of course,
00:43:03.000 --> 00:43:06.180
and get a general formula.
00:43:06.180 --> 00:43:11.712
And, we will have occasion
to use that next week.
00:43:11.712 --> 00:43:15.220
But for now, why
don't we concentrate
00:43:15.220 --> 00:43:18.555
on the most interesting
case, namely
00:43:18.555 --> 00:43:21.885
that of the most
linear equation,
00:43:21.885 --> 00:43:24.332
with constant
coefficient, that is,
00:43:24.332 --> 00:43:27.714
so let's look at
the linear equation
00:43:27.714 --> 00:43:31.284
with constant coefficient,
because that's
00:43:31.284 --> 00:43:37.000
the one that most closely models
the conduction and diffusion
00:43:37.000 --> 00:43:37.666
equations.
00:43:37.666 --> 00:43:43.000
So, what I'm interested in, is
since this is the, of them all,
00:43:43.000 --> 00:43:45.400
probably it's the
most important case
00:43:45.400 --> 00:43:48.500
is the one where P
is a constant because
00:43:48.500 --> 00:43:50.000
of its application to that.
00:43:50.000 --> 00:43:53.500
And, many of the other, the
bank account, for example,
00:43:53.500 --> 00:43:55.666
all of those will use
a constant coefficient.
00:43:55.666 --> 00:43:58.428
So, how is the
thing going to look?
00:43:58.428 --> 00:44:01.000
Well, I will use the cooling.
00:44:01.000 --> 00:44:05.000
Let's use the temperature
model, for example.
00:44:05.000 --> 00:44:07.220
The temperature
model, the equation
00:44:07.220 --> 00:44:10.600
will be DTDT plus
KT is equal to.
00:44:10.600 --> 00:44:14.800
Now, notice on the right hand
side, this is a common error.
00:44:14.800 --> 00:44:16.000
You don't put TE.
00:44:16.000 --> 00:44:21.000
You have to put KTE because
that's what the equation says.
00:44:21.000 --> 00:44:25.000
If you think units, you
won't have any trouble.
00:44:25.000 --> 00:44:30.000
Units have to be compatible on
both sides of a differential
00:44:30.000 --> 00:44:30.666
equation.
00:44:30.666 --> 00:44:34.331
And therefore, whatever the
units were for capital KT,
00:44:34.331 --> 00:44:38.541
I'd have to have the same
units on the right hand side,
00:44:38.541 --> 00:44:42.664
which indicates I cannot have KT
on the left of the differential
00:44:42.664 --> 00:44:44.998
equation, and just
T on the right,
00:44:44.998 --> 00:44:47.500
and expect the units
to be compatible.
00:44:47.500 --> 00:44:49.000
That's not possible.
00:44:49.000 --> 00:44:51.100
So, that's a good
way of remembering
00:44:51.100 --> 00:44:54.000
that if you're modeling
temperature or concentration,
00:44:54.000 --> 00:44:57.000
you have to have
the K on both sides.
00:44:57.000 --> 00:45:02.080
OK, let's do, now, a lot of this
we are going to do in our head
00:45:02.080 --> 00:45:05.000
now because this
is really too easy.
00:45:05.000 --> 00:45:07.220
What's the integrating factor?
00:45:07.220 --> 00:45:13.227
Well, the integrating factor is
going to be the integral of K,
00:45:13.227 --> 00:45:16.000
the coefficient now is just K.
00:45:16.000 --> 00:45:21.284
P is a constant, K, and if
I integrate KDT, I get KT,
00:45:21.284 --> 00:45:23.000
and I exponentiate that.
00:45:23.000 --> 00:45:28.000
So, the integrating
factor is E to the KT.
00:45:28.000 --> 00:45:34.000
I multiply through both sides,
multiply by E to the KT,
00:45:34.000 --> 00:45:38.000
and what's the
resulting equation?
00:45:38.000 --> 00:45:44.333
Well, it's going to be , I'll
write it in the compact form.
00:45:44.333 --> 00:45:50.000
It's going to be E to the
KT times T, all prime.
00:45:50.000 --> 00:45:55.284
The differentiation is now,
of course, with respect
00:45:55.284 --> 00:45:57.000
to the time.
00:45:57.000 --> 00:46:00.750
And, that's equal
to KTE, whatever
00:46:00.750 --> 00:46:04.725
that is, times E to the KT.
00:46:04.725 --> 00:46:09.000
This is a function
of T, of course,
00:46:09.000 --> 00:46:13.500
the function of little
time, sorry, little T time.
00:46:13.500 --> 00:46:18.000
OK, and now, finally, we
are going to integrate.
00:46:18.000 --> 00:46:19.332
What's the answer?
00:46:19.332 --> 00:46:26.224
Well, it is E to the, so, are we
going to get E to the KT times
00:46:26.224 --> 00:46:31.750
T is, sorry, K little t, K
times time times the temperature
00:46:31.750 --> 00:46:37.000
is equal to the integral of KTE.
00:46:37.000 --> 00:46:40.840
I'll put the fact that
it's a function of T
00:46:40.840 --> 00:46:43.998
inside just to remind
you, E to the KT,
00:46:43.998 --> 00:46:46.500
and now I'll put the
arbitrary constant.
00:46:46.500 --> 00:46:50.142
Let's put in the arbitrary
constant explicitly.
00:46:50.142 --> 00:46:53.000
So, what will T be?
00:46:53.000 --> 00:46:56.000
OK, T will look
like this, finally.
00:46:56.000 --> 00:46:59.000
It will be E to the negative KT.
00:46:59.000 --> 00:47:01.500
That's on the outside.
00:47:01.500 --> 00:47:04.000
Then, you will integrate.
00:47:04.000 --> 00:47:07.552
Of course, the difficulty
of doing this integral
00:47:07.552 --> 00:47:12.000
depends entirely upon how this
external temperature varies.
00:47:12.000 --> 00:47:16.726
But anyways, it's going to be
K times that function, which
00:47:16.726 --> 00:47:20.000
I haven't specified,
E to the KT plus C
00:47:20.000 --> 00:47:22.664
times E to the negative KT.
00:47:22.664 --> 00:47:27.000
Now, some people, many, in
fact, that almost always,
00:47:27.000 --> 00:47:30.750
in the engineering
literature, almost never
00:47:30.750 --> 00:47:36.285
write indefinite integrals
because an indefinite integral
00:47:36.285 --> 00:47:38.000
is indefinite.
00:47:38.000 --> 00:47:40.400
In other words, this covers
not just one function,
00:47:40.400 --> 00:47:42.800
but a whole multitude
of functions
00:47:42.800 --> 00:47:46.000
which differ from each other
by an arbitrary constant.
00:47:46.000 --> 00:47:49.000
So, in a formula like this,
there's a certain vagueness,
00:47:49.000 --> 00:47:51.541
and it's further
compounded by the fact
00:47:51.541 --> 00:47:55.000
that I don't know whether the
arbitrary constant is here.
00:47:55.000 --> 00:47:58.571
I seem to have put it explicitly
on the outside the way
00:47:58.571 --> 00:48:02.000
you're used to
doing from calculus.
00:48:02.000 --> 00:48:04.570
Many people, therefore,
prefer, and I
00:48:04.570 --> 00:48:06.565
think you should
learn this, to do
00:48:06.565 --> 00:48:11.885
what is done in the very first
section of the notes called
00:48:11.885 --> 00:48:13.500
definite integral solutions.
00:48:13.500 --> 00:48:16.400
If there's an initial
condition saying
00:48:16.400 --> 00:48:19.200
that the internal
temperature at time zero
00:48:19.200 --> 00:48:22.912
is some given value,
what they like to do
00:48:22.912 --> 00:48:26.248
is make this thing definite
by integrating here
00:48:26.248 --> 00:48:30.625
from zero to T, and making
this a dummy variable.
00:48:30.625 --> 00:48:36.272
You see, what that
does is it gives you
00:48:36.272 --> 00:48:39.452
a particular
function, whereas, I'm
00:48:39.452 --> 00:48:44.766
sorry I didn't put in
the DT one minus two.
00:48:44.766 --> 00:48:49.554
What it does is that
when time is zero,
00:48:49.554 --> 00:48:52.662
all this automatically
disappears,
00:48:52.662 --> 00:49:00.000
and the arbitrary constant
will then be, it's T.
00:49:00.000 --> 00:49:03.125
So, in other words, C
times this, which is one,
00:49:03.125 --> 00:49:05.000
is that equal to [T?].
00:49:05.000 --> 00:49:07.000
In other words, if
I make this zero,
00:49:07.000 --> 00:49:12.000
that I can write C as equal to
this arbitrary starting value.
00:49:12.000 --> 00:49:15.000
Now, when you do this,
the essential thing,
00:49:15.000 --> 00:49:18.000
and we're going to come
back to this next week,
00:49:18.000 --> 00:49:21.000
but right away,
because K is positive,
00:49:21.000 --> 00:49:25.400
I want to emphasize that so much
at the beginning of the period,
00:49:25.400 --> 00:49:30.428
I want to conclude by showing
you what its significance is.
00:49:30.428 --> 00:49:35.332
This part disappears
because K is positive.
00:49:35.332 --> 00:49:38.000
The conductivity is positive.
00:49:38.000 --> 00:49:41.000
This part disappears
as T goes to zero.
00:49:41.000 --> 00:49:45.000
This goes to zero as
T goes to infinity.
00:49:45.000 --> 00:49:48.000
So, this is a
solution that remains.
00:49:48.000 --> 00:49:52.625
This, therefore, is called
the steady state solution,
00:49:52.625 --> 00:49:57.000
the thing which the
temperature behaves like,
00:49:57.000 --> 00:49:59.000
as T goes to infinity.
00:49:59.000 --> 00:50:01.000
This is called the transient.
00:50:01.000 --> 00:50:07.000
because it disappears
as T goes to infinity.
00:50:07.000 --> 00:50:09.664
It depends on the
initial condition,
00:50:09.664 --> 00:50:12.200
but it disappears,
which shows you,
00:50:12.200 --> 00:50:16.665
then, in the long run
for this type of problem
00:50:16.665 --> 00:50:20.000
the initial condition
makes no difference.
00:50:20.000 --> 00:50:24.550
The function behaves always the
same way as T goes to infinity.