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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
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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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00:12:57,362 --> 00:13:03,000
the so called Newton's
Law of cooling, -- --
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00:13:03,000 --> 00:13:06,545
which says that
the rate of change,
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00:13:06,545 --> 00:13:13,000
the temperature of the heat goes
from the outside to the inside
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00:13:13,000 --> 00:13:14,500
by conduction only.
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00:13:14,500 --> 00:13:18,140
Heat, of course, can
travel in various ways,
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00:13:18,140 --> 00:13:23,000
by convection, by conduction,
as here, or by radiation,
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00:13:23,000 --> 00:13:25,270
are the three most common.
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Of these, I only want one,
namely transmission of heat
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by conduction.
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00:13:33,000 --> 00:13:36,267
And, that's the way it's
probably a little better
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00:13:36,267 --> 00:13:41,000
to call it the conduction model,
rather than the temperature
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00:13:41,000 --> 00:13:45,086
model, which might involve
other ways for the heat
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00:13:45,086 --> 00:13:46,600
to be traveling.
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00:13:46,600 --> 00:13:49,333
So, dt, the
independent variable,
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00:13:49,333 --> 00:13:53,000
is not going to be x,
as it was over there.
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00:13:53,000 --> 00:13:55,688
It's going to be t for time.
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00:13:55,688 --> 00:13:59,000
So, maybe I should write
that down. t equals time.
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00:13:59,000 --> 00:14:05,000
Capital T equals temperature
in degrees Celsius.
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00:14:05,000 --> 00:14:08,000
So, you can put in the
degrees Celsius if you want.
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00:14:08,000 --> 00:14:12,375
So, it's proportional to
the temperature difference
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00:14:12,375 --> 00:14:13,750
between these two.
218
00:14:13,750 --> 00:14:16,428
Now, how shall I
write the difference?
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00:14:16,428 --> 00:14:21,000
Write it this way because if you
don't you will be in trouble.
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00:14:21,000 --> 00:14:24,000
Now, why do I write it that way?
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00:14:24,000 --> 00:14:26,664
Well, I write it
that way because I
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00:14:26,664 --> 00:14:32,000
want this constant to be
positive, a positive constant.
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00:14:32,000 --> 00:14:35,800
In general, any constant, so,
parameters which are physical,
224
00:14:35,800 --> 00:14:39,726
have some physical
significance, one always
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00:14:39,726 --> 00:14:44,142
wants to arrange the equation so
that they are positive numbers,
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00:14:44,142 --> 00:14:47,855
the way people normally
think of these things.
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00:14:47,855 --> 00:14:49,600
This is called the conductivity.
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00:14:49,600 --> 00:14:52,000
The conductivity of what?
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00:14:52,000 --> 00:14:54,800
Well, I don't
know, of the system
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00:14:54,800 --> 00:14:58,080
of the situation, the
conductivity of the wall,
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00:14:58,080 --> 00:15:01,999
or the wall if the metal
were just by itself.
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00:15:01,999 --> 00:15:04,332
At any rate, it's a constant.
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00:15:04,332 --> 00:15:07,000
It's thought of as a constant.
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00:15:07,000 --> 00:15:11,142
And, why positive, well, because
if the external temperature is
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00:15:11,142 --> 00:15:14,000
bigger than the
internal temperature,
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I expect T to rise, the
internal temperature to rise.
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00:15:18,666 --> 00:15:24,220
That means dT / dt, its
slope, should be positive.
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00:15:24,220 --> 00:15:27,500
So, in other words, if
Te is bigger than T,
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00:15:27,500 --> 00:15:29,375
I expect this number
to be positive.
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00:15:29,375 --> 00:15:33,452
And, that tells you that k
must be a positive constant.
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00:15:33,452 --> 00:15:37,000
If I had turned it the
other way, expressed
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00:15:37,000 --> 00:15:40,000
the difference in
the reverse order,
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00:15:40,000 --> 00:15:44,000
K would then be negative,
have to be negative in order
244
00:15:44,000 --> 00:15:47,000
that this turn out to be
positive in that situation I
245
00:15:47,000 --> 00:15:47,600
described.
246
00:15:47,600 --> 00:15:51,200
And, since nobody wants
negative values of k,
247
00:15:51,200 --> 00:15:53,999
you have to write the
equation in this form
248
00:15:53,999 --> 00:15:56,000
rather than the
other way around.
249
00:15:56,000 --> 00:15:59,000
So, there's our
differential equation.
250
00:15:59,000 --> 00:16:02,000
It will probably have
an initial condition.
251
00:16:02,000 --> 00:16:05,663
So, it could be the temperature
at the starting time should
252
00:16:05,663 --> 00:16:10,000
be some given number, T zero.
253
00:16:10,000 --> 00:16:13,000
But, the condition could
be given in other ways.
254
00:16:13,000 --> 00:16:15,000
One can ask, what's
the temperature as time
255
00:16:15,000 --> 00:16:17,000
goes to infinity, for example?
256
00:16:17,000 --> 00:16:21,000
There are different ways of
getting that initial condition.
257
00:16:21,000 --> 00:16:23,000
Okay, that's the
conduction model.
258
00:16:23,000 --> 00:16:25,000
What would the
diffusion model be?
259
00:16:25,000 --> 00:16:28,142
The diffusion model,
mathematically, would be,
260
00:16:28,142 --> 00:16:31,000
word for word, the same.
261
00:16:31,000 --> 00:16:33,912
The only difference
is that now, what
262
00:16:33,912 --> 00:16:38,270
I imagine is I'll draw
the picture the same way,
263
00:16:38,270 --> 00:16:43,541
except now I'm going to put,
label the inside not with a T
264
00:16:43,541 --> 00:16:46,665
but with a C, C
for concentration.
265
00:16:46,665 --> 00:16:50,856
It's in an external
water bath, let's say.
266
00:16:50,856 --> 00:16:53,571
So, there is an
external concentration.
267
00:16:53,571 --> 00:16:57,888
And, what I'm talking
about is some chemical,
268
00:16:57,888 --> 00:17:02,250
let's say salt will do
as well as anything.
269
00:17:02,250 --> 00:17:07,500
So, C is equal to salt
concentration inside,
270
00:17:07,500 --> 00:17:13,714
and Ce would be the salt
concentration outside,
271
00:17:13,714 --> 00:17:18,000
outside in the water bath.
272
00:17:18,000 --> 00:17:26,000
Now, I imagine some mechanism,
so this is a salt solution.
273
00:17:26,000 --> 00:17:28,664
That's a salt solution.
274
00:17:28,664 --> 00:17:34,178
And, I imagine some mechanism
by which the salt can diffuse,
275
00:17:34,178 --> 00:17:36,999
it's a diffusion model
now, diffuse from here
276
00:17:36,999 --> 00:17:40,000
into the air or possibly
out the other way.
277
00:17:40,000 --> 00:17:43,500
And that's usually done
by vaguely referring
278
00:17:43,500 --> 00:17:47,140
to the outside as a
semi permeable membrane,
279
00:17:47,140 --> 00:17:50,080
semi permeable, so
that the salt will
280
00:17:50,080 --> 00:17:53,750
have a little hard time
getting through but permeable,
281
00:17:53,750 --> 00:17:56,800
so that it won't be
blocked completely.
282
00:17:56,800 --> 00:18:00,000
So, there's a membrane.
283
00:18:00,000 --> 00:18:06,000
You write the semi
permeable membrane outside,
284
00:18:06,000 --> 00:18:07,713
outside the inside.
285
00:18:07,713 --> 00:18:10,000
Well, I give up.
286
00:18:10,000 --> 00:18:14,000
You know, membrane somewhere.
287
00:18:14,000 --> 00:18:17,000
Sorry, membrane wall.
288
00:18:17,000 --> 00:18:18,332
How's that?
289
00:18:18,332 --> 00:18:21,000
Now, what's the equation?
290
00:18:21,000 --> 00:18:26,600
Well, the equation is
the same, except it's
291
00:18:26,600 --> 00:18:29,500
called the diffusion equation.
292
00:18:29,500 --> 00:18:35,284
I don't think Newton
got his name on this.
293
00:18:35,284 --> 00:18:40,178
The diffusion equation says
that the rate at which the salt
294
00:18:40,178 --> 00:18:42,384
diffuses across
the membrane, which
295
00:18:42,384 --> 00:18:47,000
is the same up to a constant
as the rate at which
296
00:18:47,000 --> 00:18:51,000
the concentration inside
changes, is some constant,
297
00:18:51,000 --> 00:18:54,000
usually called k still, okay.
298
00:18:54,000 --> 00:18:55,332
Do I contradict?
299
00:18:55,332 --> 00:18:58,000
Okay, let's keep calling it k1.
300
00:18:58,000 --> 00:19:01,000
Now it's different,
times Ce minus C.
301
00:19:01,000 --> 00:19:03,800
And, for the same
reason as before,
302
00:19:03,800 --> 00:19:07,142
if the external
concentration is bigger
303
00:19:07,142 --> 00:19:12,724
than the internal concentration,
we expect salt to flow in.
304
00:19:12,724 --> 00:19:15,000
That will make C rise.
305
00:19:15,000 --> 00:19:18,307
It will make this
positive, and therefore, we
306
00:19:18,307 --> 00:19:21,070
want k to be positive,
just k1 to be
307
00:19:21,070 --> 00:19:25,000
positive for the same reason
it had to be positive before.
308
00:19:25,000 --> 00:19:28,600
So, in each case, the model
that I'm talking about
309
00:19:28,600 --> 00:19:31,000
is the differential equation.
310
00:19:31,000 --> 00:19:34,500
So, maybe I should, let's
put that, make that clear.
311
00:19:34,500 --> 00:19:39,713
Or, I would say that this first
order differential equation
312
00:19:39,713 --> 00:19:43,332
models this physical
situation, and the same thing
313
00:19:43,332 --> 00:19:46,000
is true on the other
side over here.
314
00:19:46,000 --> 00:19:48,800
This is the diffusion
equation, and this
315
00:19:48,800 --> 00:19:50,375
is the conduction equation.
316
00:19:50,375 --> 00:19:55,220
Now, if you are in any doubt
about the power of differential
317
00:19:55,220 --> 00:19:58,842
equations, the point is,
when I talk about this thing,
318
00:19:58,842 --> 00:20:02,875
I don't have to say which
of these I'm following.
319
00:20:02,875 --> 00:20:07,284
I'll use neutral
variables like Y and X
320
00:20:07,284 --> 00:20:09,000
to solve these equations.
321
00:20:09,000 --> 00:20:12,600
But, with a single
stroke, I will be handling
322
00:20:12,600 --> 00:20:13,750
those situations together.
323
00:20:13,750 --> 00:20:16,500
And, that's the
power of the method.
324
00:20:16,500 --> 00:20:20,200
Now, you obviously must
be wondering, look,
325
00:20:20,200 --> 00:20:22,666
these look very, very special.
326
00:20:22,666 --> 00:20:27,140
He said he was going to talk
about the first, general first
327
00:20:27,140 --> 00:20:28,000
order equation.
328
00:20:28,000 --> 00:20:31,000
But, these look
rather special to me.
329
00:20:31,000 --> 00:20:33,220
Well, not too special.
330
00:20:33,220 --> 00:20:36,000
How should we write it?
331
00:20:36,000 --> 00:20:41,332
Suppose I write, let's take
the temperature equation just
332
00:20:41,332 --> 00:20:44,000
to have something definite.
333
00:20:44,000 --> 00:20:48,000
Notice that it's in a form
corresponding to Newton's Law.
334
00:20:48,000 --> 00:20:52,000
But it is not in the
standard linear form.
335
00:20:52,000 --> 00:20:54,688
Let's put it in
standard linear form,
336
00:20:54,688 --> 00:20:59,080
so at least you could see
that it's a linear equation.
337
00:20:59,080 --> 00:21:02,384
So, if I put it
in standard form,
338
00:21:02,384 --> 00:21:06,224
it's going to look like
DTDTD little t plus KT
339
00:21:06,224 --> 00:21:09,220
is equal to K times TE.
340
00:21:09,220 --> 00:21:13,200
Now, compare that with
the general, the way
341
00:21:13,200 --> 00:21:16,000
the general equation
is supposed to look,
342
00:21:16,000 --> 00:21:20,000
the yellow box over there,
the standard linear form.
343
00:21:20,000 --> 00:21:21,998
How are they going to compare?
344
00:21:21,998 --> 00:21:24,500
Well, this is a pretty
general function.
345
00:21:24,500 --> 00:21:26,000
This is general.
346
00:21:26,000 --> 00:21:29,840
This is a general function
of T because I can
347
00:21:29,840 --> 00:21:31,428
make the external temperature.
348
00:21:31,428 --> 00:21:37,000
I could suppose it behaves in
anyway I like, steadily rising,
349
00:21:37,000 --> 00:21:39,000
decaying exponentially,
maybe oscillating
350
00:21:39,000 --> 00:21:42,180
back and forth for some reason.
351
00:21:42,180 --> 00:21:46,500
The only way in which
it's not general
352
00:21:46,500 --> 00:21:50,000
is that this K is a constant.
353
00:21:50,000 --> 00:21:53,000
So, I will ask you
to be generous.
354
00:21:53,000 --> 00:21:57,000
Let's imagine the conductivity
is changing over time.
355
00:21:57,000 --> 00:22:00,815
So, this is usually
constant, but there's
356
00:22:00,815 --> 00:22:05,220
no law which says it has to be.
357
00:22:05,220 --> 00:22:09,142
How could a conductivity
change over time?
358
00:22:09,142 --> 00:22:12,888
Well, we could
suppose that this wall
359
00:22:12,888 --> 00:22:17,142
was made of slowly congealing
Jell O, for instance.
360
00:22:17,142 --> 00:22:23,000
It starts out as liquid,
and then it gets solid.
361
00:22:23,000 --> 00:22:26,444
And, Jell O doesn't
transmit heat,
362
00:22:26,444 --> 00:22:32,180
I believe, quite as well as
liquid does, as a liquid would.
363
00:22:32,180 --> 00:22:36,000
Is Jell O a solid or liquid?
364
00:22:36,000 --> 00:22:37,713
I don't know.
365
00:22:37,713 --> 00:22:40,000
Let's forget about that.
366
00:22:40,000 --> 00:22:46,000
So, with this understanding,
so let's say not necessarily
367
00:22:46,000 --> 00:22:51,500
here, but not necessarily, I
can think of this, therefore,
368
00:22:51,500 --> 00:22:55,000
by allowing K to vary with time.
369
00:22:55,000 --> 00:23:00,000
And the external temperature
to vary with time.
370
00:23:00,000 --> 00:23:07,000
I can think of it as a
general, linear equation.
371
00:23:07,000 --> 00:23:09,000
So, these models
are not special.
372
00:23:09,000 --> 00:23:10,452
They are fairly general.
373
00:23:10,452 --> 00:23:14,284
Well, I did promise you I
would solve an equation,
374
00:23:14,284 --> 00:23:18,000
and that this lecture, I still
have not solved any equations.
375
00:23:18,000 --> 00:23:21,000
OK, time to stop
temporizing and solve.
376
00:23:21,000 --> 00:23:24,000
So, I'm going to, in order
not to play favorites
377
00:23:24,000 --> 00:23:27,000
with these two models,
I'll go back to,
378
00:23:27,000 --> 00:23:31,000
and to get you used to thinking
of the variables all the time,
379
00:23:31,000 --> 00:23:35,444
that is, you know, be eclectic
switching from one variable
380
00:23:35,444 --> 00:23:38,552
to another according to
which particular lecture
381
00:23:38,552 --> 00:23:42,635
you happened to be sitting in.
382
00:23:42,635 --> 00:23:52,712
So, let's take our equation in
the form, Y prime plus P of XY,
383
00:23:52,712 --> 00:23:58,000
the general form using
the old variables
384
00:23:58,000 --> 00:24:04,000
equals Q of X. Solve me.
385
00:24:04,000 --> 00:24:06,928
Well, there are different ways
of describing the solution
386
00:24:06,928 --> 00:24:07,428
process.
387
00:24:07,428 --> 00:24:10,614
No matter how you
do it, it amounts
388
00:24:10,614 --> 00:24:13,684
to the same amount of
work and there is always
389
00:24:13,684 --> 00:24:16,331
a trick involved
at each one of them
390
00:24:16,331 --> 00:24:19,125
since you can't suppress
a trick by doing
391
00:24:19,125 --> 00:24:21,000
the problem some other way.
392
00:24:21,000 --> 00:24:24,000
The way I'm going to do
it, I think, is the best.
393
00:24:24,000 --> 00:24:26,000
That's why I'm giving it to you.
394
00:24:26,000 --> 00:24:27,815
It's the easiest to remember.
395
00:24:27,815 --> 00:24:31,000
It leads to the
least work, but I
396
00:24:31,000 --> 00:24:36,000
have colleagues who would
fight with me about that point.
397
00:24:36,000 --> 00:24:39,330
So, since they are not
here to fight with me
398
00:24:39,330 --> 00:24:42,332
I am free to do whatever I like.
399
00:24:42,332 --> 00:24:45,400
One of the main
reasons for doing
400
00:24:45,400 --> 00:24:49,400
it the way I'm going
to do is because I
401
00:24:49,400 --> 00:24:55,555
want you to get what our word
into your consciousness, two
402
00:24:55,555 --> 00:24:57,220
words, integrating factor.
403
00:24:57,220 --> 00:25:02,500
I'm going to solve this equation
by finding and integrating
404
00:25:02,500 --> 00:25:09,000
factor of the form U of X.
What's an integrating factor?
405
00:25:09,000 --> 00:25:13,000
Well, I'll show you not by
writing an elaborate definition
406
00:25:13,000 --> 00:25:16,666
on the board, but showing
you what its function is.
407
00:25:16,666 --> 00:25:19,332
It's a certain
function, U of X, I
408
00:25:19,332 --> 00:25:23,200
don't know what it is, but
here's what I wanted to do.
409
00:25:23,200 --> 00:25:26,763
I want to multiply, I'm going
to drop the X's a just so
410
00:25:26,763 --> 00:25:29,110
that the thing looks
less complicated.
411
00:25:29,110 --> 00:25:34,125
So, what I want to do is
multiply this equation
412
00:25:34,125 --> 00:25:36,000
through by U of X.
413
00:25:36,000 --> 00:25:40,360
That's why it's called
a factor because you're
414
00:25:40,360 --> 00:25:43,816
going to multiply
everything through by it.
415
00:25:43,816 --> 00:25:49,220
So, it's going to look like
UY prime plus PUY equals QU,
416
00:25:49,220 --> 00:25:52,776
and now, so far,
it's just a factor.
417
00:25:52,776 --> 00:25:55,375
What makes it an
integrating factor
418
00:25:55,375 --> 00:26:00,766
is that this, after I do
that, I want this to turn out
419
00:26:00,766 --> 00:26:04,500
to be the derivative of
something with respect
420
00:26:04,500 --> 00:26:08,250
to X. You see the
motivation for that.
421
00:26:08,250 --> 00:26:11,125
If this turns out to be the
derivative of something,
422
00:26:11,125 --> 00:26:13,363
because I've chosen
U so cleverly,
423
00:26:13,363 --> 00:26:17,000
then I will be able to solve
the equation immediately
424
00:26:17,000 --> 00:26:19,000
just by integrating
this with respect to X,
425
00:26:19,000 --> 00:26:21,000
and integrating that
with respect to X.
426
00:26:21,000 --> 00:26:24,750
You just, then, integrate
both sides with respect to X,
427
00:26:24,750 --> 00:26:26,000
and the equation is solved.
428
00:26:26,000 --> 00:26:31,000
Now, the only question is,
what should I choose for U?
429
00:26:31,000 --> 00:26:34,000
Well, if you think of
the product formula,
430
00:26:34,000 --> 00:26:37,000
there might be many
things to try here.
431
00:26:37,000 --> 00:26:40,000
But there's only one
reasonable thing to try.
432
00:26:40,000 --> 00:26:45,000
Try to pick U so that it's
the derivative of U times Y.
433
00:26:45,000 --> 00:26:46,665
See how reasonable that is?
434
00:26:46,665 --> 00:26:49,333
If I use the product
rule on this,
435
00:26:49,333 --> 00:26:52,000
the first term is
U times Y prime.
436
00:26:52,000 --> 00:26:56,000
The second term would
be U prime times Y.
437
00:26:56,000 --> 00:26:59,000
Well, I've got the Y there.
438
00:26:59,000 --> 00:27:01,000
So, this will work.
439
00:27:01,000 --> 00:27:05,888
It works if, what's the
condition that you must satisfy
440
00:27:05,888 --> 00:27:09,000
in order for that to be true?
441
00:27:09,000 --> 00:27:15,000
Well, it must be that after
it to the differentiation,
442
00:27:15,000 --> 00:27:19,000
U prime turns out
to be P times U.
443
00:27:19,000 --> 00:27:20,816
So, is it clear?
444
00:27:20,816 --> 00:27:26,625
This is something we want to be
equal to, and the thing I will
445
00:27:26,625 --> 00:27:32,331
try to do it is by
choosing U in such a way
446
00:27:32,331 --> 00:27:37,000
that this equality
will take place.
447
00:27:37,000 --> 00:27:40,000
And then I will be able
to solve the equation.
448
00:27:40,000 --> 00:27:43,000
And so, here's what my
U prime must satisfy.
449
00:27:43,000 --> 00:27:44,815
Hey, we can solve that.
450
00:27:44,815 --> 00:27:47,222
But please don't
forget that P is
451
00:27:47,222 --> 00:27:49,000
P of X. It's a function of X.
452
00:27:49,000 --> 00:27:53,000
So, if you separate variables,
I'm going to do this.
453
00:27:53,000 --> 00:27:56,600
So, what is it, DU over
U equals P of X times DX.
454
00:27:56,600 --> 00:28:00,500
If I integrate that,
so, separate variables,
455
00:28:00,500 --> 00:28:04,665
integrate, and you're going
to get DU over U integrates
456
00:28:04,665 --> 00:28:08,428
to the be the log of
U, and the other side
457
00:28:08,428 --> 00:28:12,000
integrates to be the
integral of P of X DX.
458
00:28:12,000 --> 00:28:16,000
Now, you can put an
arbitrary constant there,
459
00:28:16,000 --> 00:28:18,448
or you can think of
it as already implied
460
00:28:18,448 --> 00:28:20,142
by the indefinite integral.
461
00:28:20,142 --> 00:28:24,500
Well, that doesn't tell
us, yet, what U is.
462
00:28:24,500 --> 00:28:26,000
What should U be?
463
00:28:26,000 --> 00:28:30,250
Notice, I don't have to find
every possible U, which works.
464
00:28:30,250 --> 00:28:34,000
All I'm looking for is one.
465
00:28:34,000 --> 00:28:38,000
All I want is a single view
which satisfies that equation.
466
00:28:38,000 --> 00:28:42,000
Well, U equals the integral,
E to the integral of PDX.
467
00:28:42,000 --> 00:28:44,500
That's not too
beautiful looking,
468
00:28:44,500 --> 00:28:46,726
but by differential
equations, things
469
00:28:46,726 --> 00:28:50,454
can get so complicated
that in a week or two,
470
00:28:50,454 --> 00:28:55,000
you will think of this as
an extremely simple formula.
471
00:28:55,000 --> 00:29:00,000
So, there is a formula for
our integrating factor.
472
00:29:00,000 --> 00:29:01,500
We found it.
473
00:29:01,500 --> 00:29:07,250
We will always be able to
write an integrating factor.
474
00:29:07,250 --> 00:29:14,750
Don't worry about the arbitrary
constant because you only need
475
00:29:14,750 --> 00:29:17,000
one such U.
476
00:29:17,000 --> 00:29:23,000
So: no arbitrary constant
since only one U needed.
477
00:29:23,000 --> 00:29:26,600
And, that's the
solution, the way
478
00:29:26,600 --> 00:29:30,200
we solve the linear equation.
479
00:29:30,200 --> 00:29:37,544
OK, let's take over,
and actually do it.
480
00:29:37,544 --> 00:29:43,714
I think it would be
better to summarize it
481
00:29:43,714 --> 00:29:48,000
as a clear cut method.
482
00:29:48,000 --> 00:29:51,000
So, let's do that.
483
00:29:51,000 --> 00:29:54,000
So, what's our method?
484
00:29:54,000 --> 00:30:04,000
It's the method for solving
Y prime plus PY equals Q.
485
00:30:04,000 --> 00:30:07,800
Well, the first place, make sure
it's in standard linear form.
486
00:30:07,800 --> 00:30:11,331
If it isn't, you must
put it in that form.
487
00:30:11,331 --> 00:30:14,712
Notice, the formula for the
integrating factor, the formula
488
00:30:14,712 --> 00:30:16,856
for the integrating
factor involves
489
00:30:16,856 --> 00:30:19,000
P, the integral of PDX.
490
00:30:19,000 --> 00:30:21,000
So, you'd better
get the right P.
491
00:30:21,000 --> 00:30:22,600
Otherwise, you are sunk.
492
00:30:22,600 --> 00:30:25,750
OK, so put it in
standard linear form.
493
00:30:25,750 --> 00:30:29,332
That way, you will have
the right P. Notice
494
00:30:29,332 --> 00:30:32,544
that if you wrote
it in that form,
495
00:30:32,544 --> 00:30:35,375
and all you remembered
was E to the integral PDX,
496
00:30:35,375 --> 00:30:38,000
the P would have the wrong sign.
497
00:30:38,000 --> 00:30:41,070
If you're going to write, that
P should have a negative sign
498
00:30:41,070 --> 00:30:41,570
there.
499
00:30:41,570 --> 00:30:44,500
So, do it this way,
and no other way.
500
00:30:44,500 --> 00:30:47,665
Otherwise, you will get
confused and get wrong signs.
501
00:30:47,665 --> 00:30:51,331
And, as I say, that will
produce wrong answers, and not
502
00:30:51,331 --> 00:30:54,000
just slightly wrong
answers, but disastrously
503
00:30:54,000 --> 00:30:57,664
wrong answers from the point
of view of the modeling
504
00:30:57,664 --> 00:31:02,108
if you really want answers
to physical problems.
505
00:31:02,108 --> 00:31:06,600
So, here's a
standard linear form.
506
00:31:06,600 --> 00:31:09,666
Then, find the
integrating factor.
507
00:31:09,666 --> 00:31:17,000
So, calculate E to the integral,
PDX, the integrating factor,
508
00:31:17,000 --> 00:31:22,000
and that multiply both,
I'm putting this as both,
509
00:31:22,000 --> 00:31:31,000
underlined that as many times
as you have room in your notes.
510
00:31:31,000 --> 00:31:38,362
Multiply both sides by this
integrating factor by E
511
00:31:38,362 --> 00:31:42,400
to the integral PDX.
512
00:31:42,400 --> 00:31:46,000
And then, integrate.
513
00:31:46,000 --> 00:31:51,000
OK, let's take a simple example.
514
00:31:51,000 --> 00:31:57,400
Suppose we started with
the equation XY prime
515
00:31:57,400 --> 00:32:06,000
minus Y equals, I had X2,
X3, something like that,
516
00:32:06,000 --> 00:32:12,000
X3, I think, yeah, X2.
517
00:32:12,000 --> 00:32:16,000
OK, what's the
first thing to do?
518
00:32:16,000 --> 00:32:18,915
Put it in standard form.
519
00:32:18,915 --> 00:32:25,915
So, step zero will be to
write it as Y prime minus
520
00:32:25,915 --> 00:32:30,000
one over X times Y equals X2.
521
00:32:30,000 --> 00:32:34,833
Let's do the work first, and
then I'll talk about mistakes.
522
00:32:34,833 --> 00:32:39,888
Well, we now calculate
the integrating factor.
523
00:32:39,888 --> 00:32:43,000
So, I would do it in steps.
524
00:32:43,000 --> 00:32:48,000
You can integrate negative
one over X, right?
525
00:32:48,000 --> 00:32:51,500
That integrates to
minus log X. So,
526
00:32:51,500 --> 00:32:57,000
the integrating factor is E
to the integral of this, DX.
527
00:32:57,000 --> 00:33:02,000
So, it's E to the
negative log X.
528
00:33:02,000 --> 00:33:06,500
Now, in real life, that's
not the way to leave that.
529
00:33:06,500 --> 00:33:10,333
What is E to the negative log X?
530
00:33:10,333 --> 00:33:16,000
Well, think of it as E to
the log X to the minus one.
531
00:33:16,000 --> 00:33:21,570
Or, in other words, it is
E to the log X is X. So,
532
00:33:21,570 --> 00:33:23,000
it's one over X.
533
00:33:23,000 --> 00:33:27,000
So, the integrating
factor is one over X.
534
00:33:27,000 --> 00:33:34,000
OK, multiply both sides
by the integrating factor.
535
00:33:34,000 --> 00:33:35,500
Both sides of what?
536
00:33:35,500 --> 00:33:41,000
Both sides of this: the equation
written in standard form,
537
00:33:41,000 --> 00:33:42,200
and both sides.
538
00:33:42,200 --> 00:33:47,800
So, it's going to be one over
XY prime minus one over X2 Y
539
00:33:47,800 --> 00:33:52,815
is equal to X2 times one over
X, which is simply X. Now,
540
00:33:52,815 --> 00:33:55,545
if you have done
the work correctly,
541
00:33:55,545 --> 00:34:01,000
you should be able, now,
to integrate the left hand
542
00:34:01,000 --> 00:34:02,110
side directly.
543
00:34:02,110 --> 00:34:06,428
So, I'm going to
write it this way.
544
00:34:06,428 --> 00:34:10,815
I always recommend that you
put it as extra step, well,
545
00:34:10,815 --> 00:34:14,142
put it as an extra
step the reason
546
00:34:14,142 --> 00:34:17,000
for using that
integrating factor,
547
00:34:17,000 --> 00:34:21,200
in other words, that the left
hand side is supposed to be,
548
00:34:21,200 --> 00:34:24,000
now, one over X times Y prime.
549
00:34:24,000 --> 00:34:27,328
I always put it that
because there's always
550
00:34:27,328 --> 00:34:31,775
a chance you made a mistake
or forgot something.
551
00:34:31,775 --> 00:34:34,555
Look at it, mentally
differentiated
552
00:34:34,555 --> 00:34:39,855
using the product rule just
to check that, in fact, it
553
00:34:39,855 --> 00:34:43,000
turns out to be the same
as the left hand side.
554
00:34:43,000 --> 00:34:44,535
So, what do we get?
555
00:34:44,535 --> 00:34:48,500
One over X times Y prime
plus Y times the derivative
556
00:34:48,500 --> 00:34:53,800
of one over X, which indeed
is negative one over X2.
557
00:34:53,800 --> 00:34:58,000
And now, finally, that's 3A,
continue, do the integration.
558
00:34:58,000 --> 00:35:00,149
So, you're going
to get, let's see
559
00:35:00,149 --> 00:35:04,904
if we can do it all on one
board, one over X times Y
560
00:35:04,904 --> 00:35:07,904
is equal to X plus a
constant, X, sorry, X2
561
00:35:07,904 --> 00:35:09,857
over two plus a constant.
562
00:35:09,857 --> 00:35:16,332
And, the final step
will be, therefore, now
563
00:35:16,332 --> 00:35:21,000
I want to isolate Y by itself.
564
00:35:21,000 --> 00:35:26,000
So, Y will be equal to
multiply through by X.
565
00:35:26,000 --> 00:35:34,000
X3 over two plus C times X.
And, that's the solution.
566
00:35:34,000 --> 00:35:40,000
OK, let's do one a little
slightly more complicated.
567
00:35:40,000 --> 00:35:41,816
Let's try this one.
568
00:35:41,816 --> 00:35:45,545
Now, my equation
is going to be one,
569
00:35:45,545 --> 00:35:51,000
I'll still keep two, Y
and X, as the variables.
570
00:35:51,000 --> 00:35:57,000
I'll use T and F
for a minute or two.
571
00:35:57,000 --> 00:36:01,360
One plus cosine X,
so, I'm not going
572
00:36:01,360 --> 00:36:07,000
to give you this one in
standard form either.
573
00:36:07,000 --> 00:36:09,332
It's a trick question.
574
00:36:09,332 --> 00:36:17,332
Y prime minus sine X times Y is
equal to anything reasonable,
575
00:36:17,332 --> 00:36:19,000
I guess.
576
00:36:19,000 --> 00:36:24,000
I think X, 2X, make
it more exciting.
577
00:36:24,000 --> 00:36:28,920
OK, now, I think
I should warn you
578
00:36:28,920 --> 00:36:38,000
where the mistakes are just so
that you can make all of them.
579
00:36:38,000 --> 00:36:41,000
So, this is mistake number one.
580
00:36:41,000 --> 00:36:44,750
You don't put it
in standard form.
581
00:36:44,750 --> 00:36:51,000
Mistake number two: generally
people can do step one fine.
582
00:36:51,000 --> 00:36:57,000
Mistake number two is, this
is my most common mistake,
583
00:36:57,000 --> 00:37:00,000
so I'm very sensitive to it.
584
00:37:00,000 --> 00:37:03,750
But that doesn't
mean if you make it,
585
00:37:03,750 --> 00:37:06,000
you'll get any sympathy from me.
586
00:37:06,000 --> 00:37:08,000
I don't give sympathy to myself.
587
00:37:08,000 --> 00:37:10,400
You are so intense,
so happy at having
588
00:37:10,400 --> 00:37:12,332
found the integrating
factor, you
589
00:37:12,332 --> 00:37:16,332
forget to multiply Q by the
integrating factor also.
590
00:37:16,332 --> 00:37:20,332
You just handle the left
hand side of the equation,
591
00:37:20,332 --> 00:37:22,999
if you forget about
the right hand side.
592
00:37:22,999 --> 00:37:27,220
So, the emphasis on the
both here is the right hand,
593
00:37:27,220 --> 00:37:29,000
please include the Q.
594
00:37:29,000 --> 00:37:33,000
Please include the
right hand side.
595
00:37:33,000 --> 00:37:34,332
Any other mistakes?
596
00:37:34,332 --> 00:37:37,250
Well, nothing that
I can think of.
597
00:37:37,250 --> 00:37:39,664
Well, maybe only,
anyway, we are not
598
00:37:39,664 --> 00:37:43,570
going to make any mistakes
the rest of this lecture.
599
00:37:43,570 --> 00:37:45,000
So, what do we do?
600
00:37:45,000 --> 00:37:47,304
We write this in standard form.
601
00:37:47,304 --> 00:37:51,600
So, it's going to look
like Y prime minus sine X,
602
00:37:51,600 --> 00:37:55,816
sine X divided by one
plus cosine X times Y
603
00:37:55,816 --> 00:37:59,000
equals, my heart
sinks because I know
604
00:37:59,000 --> 00:38:04,000
I'm supposed to integrate
something like this.
605
00:38:04,000 --> 00:38:07,000
And, boy, that's going
to give me problems.
606
00:38:07,000 --> 00:38:08,713
Well, not yet.
607
00:38:08,713 --> 00:38:11,000
With the integrating factor?
608
00:38:11,000 --> 00:38:14,000
The integrating
factor is, well, we
609
00:38:14,000 --> 00:38:18,220
want to calculate the
integral of negative sine X
610
00:38:18,220 --> 00:38:20,000
over one plus cosine.
611
00:38:20,000 --> 00:38:21,875
That's the integral of PDX.
612
00:38:21,875 --> 00:38:25,500
And, after that, we
have to exponentiate it.
613
00:38:25,500 --> 00:38:28,000
Well, can you do this?
614
00:38:28,000 --> 00:38:31,000
Yeah, but if you stare
at it a little while,
615
00:38:31,000 --> 00:38:38,000
you can see that the top is
the derivative of the bottom.
616
00:38:38,000 --> 00:38:39,332
That is great.
617
00:38:39,332 --> 00:38:43,600
That means it integrates
to be the log of one
618
00:38:43,600 --> 00:38:47,362
plus cosine X. Is that
right, one over one
619
00:38:47,362 --> 00:38:51,625
plus cosine X times the
derivative of this, which
620
00:38:51,625 --> 00:38:56,375
is negative cosine X. Therefore,
the integrating factor
621
00:38:56,375 --> 00:38:57,875
is E to that.
622
00:38:57,875 --> 00:39:02,270
In other words, it
is one plus cosine X.
623
00:39:02,270 --> 00:39:05,500
Therefore, so this
was step zero.
624
00:39:05,500 --> 00:39:09,332
Step one, we found the
integrating factor.
625
00:39:09,332 --> 00:39:17,000
And now, step two, we multiply
through the integrating factor.
626
00:39:17,000 --> 00:39:19,220
And what do we get?
627
00:39:19,220 --> 00:39:23,142
We multiply through the
standard for equation
628
00:39:23,142 --> 00:39:29,000
by the integrating factor, if
you do that, what you get is,
629
00:39:29,000 --> 00:39:35,000
well, Y prime gets the
coefficient one plus cosine X,
630
00:39:35,000 --> 00:39:38,885
Y prime minus sign X equals 2X.
631
00:39:38,885 --> 00:39:40,000
Oh, dear.
632
00:39:40,000 --> 00:39:45,000
Well, I hope somebody
would giggle at this point.
633
00:39:45,000 --> 00:39:47,270
What's giggle able about it?
634
00:39:47,270 --> 00:39:51,000
Well, that all this was
totally wasted work.
635
00:39:51,000 --> 00:39:53,500
It's called spinning
your wheels.
636
00:39:53,500 --> 00:39:56,500
No, it's not
spinning your wheels.
637
00:39:56,500 --> 00:40:00,000
It's doing what
you're supposed to do,
638
00:40:00,000 --> 00:40:05,000
and finding out that you
wasted the entire time doing
639
00:40:05,000 --> 00:40:08,000
what you were supposed to do.
640
00:40:08,000 --> 00:40:12,140
Well, in other words,
that net effect of this
641
00:40:12,140 --> 00:40:16,856
is to end up with the same
equation we started with.
642
00:40:16,856 --> 00:40:19,000
But, what is the point?
643
00:40:19,000 --> 00:40:22,178
The point of having
done all this
644
00:40:22,178 --> 00:40:26,775
was because now the left
hand side is exactly
645
00:40:26,775 --> 00:40:32,270
the derivative of something,
and the left hand side should
646
00:40:32,270 --> 00:40:35,000
be the derivative of what?
647
00:40:35,000 --> 00:40:37,904
Well, it should be
the derivative of one
648
00:40:37,904 --> 00:40:40,600
plus cosine X
times Y, all prime.
649
00:40:40,600 --> 00:40:44,776
Now, you can check that
that's in fact the case.
650
00:40:44,776 --> 00:40:48,815
It's one plus cosine X,
Y prime, plus minus sine
651
00:40:48,815 --> 00:40:51,856
X, the derivative of
this side times Y.
652
00:40:51,856 --> 00:40:56,220
So, if you had thought, in
looking at the equation,
653
00:40:56,220 --> 00:41:00,724
to say to yourself, this
is a derivative of that,
654
00:41:00,724 --> 00:41:04,248
maybe I'll just check
right away to see
655
00:41:04,248 --> 00:41:08,571
if it's the derivative of
one plus cosine X sine.
656
00:41:08,571 --> 00:41:12,000
You would have saved that work.
657
00:41:12,000 --> 00:41:16,000
Well, you don't have to
be brilliant or clever,
658
00:41:16,000 --> 00:41:17,332
or anything like that.
659
00:41:17,332 --> 00:41:20,200
You can follow your
nose, and it's just,
660
00:41:20,200 --> 00:41:24,875
I want to give you a positive
experience in solving
661
00:41:24,875 --> 00:41:28,000
linear equations,
not too negative.
662
00:41:28,000 --> 00:41:31,000
Anyway, so we got to this point.
663
00:41:31,000 --> 00:41:37,000
So, now this is 2X, and
now we are ready to solve
664
00:41:37,000 --> 00:41:43,500
the equation, which is the
solution now will be one plus
665
00:41:43,500 --> 00:41:49,000
cosine X times Y is equal
to X2 plus a constant,
666
00:41:49,000 --> 00:41:55,524
and so Y is equal to X2 divided
by X2 plus a constant divided
667
00:41:55,524 --> 00:42:01,400
by one plus cosine X. Suppose
I have given you an initial
668
00:42:01,400 --> 00:42:03,000
condition, which I didn't.
669
00:42:03,000 --> 00:42:07,000
But, suppose the initial
condition said that Y of zero
670
00:42:07,000 --> 00:42:08,332
were one, for instance.
671
00:42:08,332 --> 00:42:11,665
Then, the solution would
be, so, this is an if,
672
00:42:11,665 --> 00:42:15,498
I'm throwing in at the end just
to make it a little bit more
673
00:42:15,498 --> 00:42:17,400
of a problem, how
would I put, then
674
00:42:17,400 --> 00:42:21,400
I could evaluate the constant
by using the initial condition.
675
00:42:21,400 --> 00:42:23,000
What would it be?
676
00:42:23,000 --> 00:42:25,454
This would be, on the
left hand side, one,
677
00:42:25,454 --> 00:42:30,000
on the right hand side
would be C over two.
678
00:42:30,000 --> 00:42:34,000
So, I would get one
equals C over two.
679
00:42:34,000 --> 00:42:35,875
Is that correct?
680
00:42:35,875 --> 00:42:42,180
Cosine of zero is one,
so that's two down below.
681
00:42:42,180 --> 00:42:48,856
Therefore, C is equal to
two, and that would then
682
00:42:48,856 --> 00:42:51,000
complete the solution.
683
00:42:51,000 --> 00:42:57,000
We would be X2 plus two
over one plus cosine X.
684
00:42:57,000 --> 00:43:03,000
Now, you can do this
in general, of course,
685
00:43:03,000 --> 00:43:06,180
and get a general formula.
686
00:43:06,180 --> 00:43:11,712
And, we will have occasion
to use that next week.
687
00:43:11,712 --> 00:43:15,220
But for now, why
don't we concentrate
688
00:43:15,220 --> 00:43:18,555
on the most interesting
case, namely
689
00:43:18,555 --> 00:43:21,885
that of the most
linear equation,
690
00:43:21,885 --> 00:43:24,332
with constant
coefficient, that is,
691
00:43:24,332 --> 00:43:27,714
so let's look at
the linear equation
692
00:43:27,714 --> 00:43:31,284
with constant coefficient,
because that's
693
00:43:31,284 --> 00:43:37,000
the one that most closely models
the conduction and diffusion
694
00:43:37,000 --> 00:43:37,666
equations.
695
00:43:37,666 --> 00:43:43,000
So, what I'm interested in, is
since this is the, of them all,
696
00:43:43,000 --> 00:43:45,400
probably it's the
most important case
697
00:43:45,400 --> 00:43:48,500
is the one where P
is a constant because
698
00:43:48,500 --> 00:43:50,000
of its application to that.
699
00:43:50,000 --> 00:43:53,500
And, many of the other, the
bank account, for example,
700
00:43:53,500 --> 00:43:55,666
all of those will use
a constant coefficient.
701
00:43:55,666 --> 00:43:58,428
So, how is the
thing going to look?
702
00:43:58,428 --> 00:44:01,000
Well, I will use the cooling.
703
00:44:01,000 --> 00:44:05,000
Let's use the temperature
model, for example.
704
00:44:05,000 --> 00:44:07,220
The temperature
model, the equation
705
00:44:07,220 --> 00:44:10,600
will be DTDT plus
KT is equal to.
706
00:44:10,600 --> 00:44:14,800
Now, notice on the right hand
side, this is a common error.
707
00:44:14,800 --> 00:44:16,000
You don't put TE.
708
00:44:16,000 --> 00:44:21,000
You have to put KTE because
that's what the equation says.
709
00:44:21,000 --> 00:44:25,000
If you think units, you
won't have any trouble.
710
00:44:25,000 --> 00:44:30,000
Units have to be compatible on
both sides of a differential
711
00:44:30,000 --> 00:44:30,666
equation.
712
00:44:30,666 --> 00:44:34,331
And therefore, whatever the
units were for capital KT,
713
00:44:34,331 --> 00:44:38,541
I'd have to have the same
units on the right hand side,
714
00:44:38,541 --> 00:44:42,664
which indicates I cannot have KT
on the left of the differential
715
00:44:42,664 --> 00:44:44,998
equation, and just
T on the right,
716
00:44:44,998 --> 00:44:47,500
and expect the units
to be compatible.
717
00:44:47,500 --> 00:44:49,000
That's not possible.
718
00:44:49,000 --> 00:44:51,100
So, that's a good
way of remembering
719
00:44:51,100 --> 00:44:54,000
that if you're modeling
temperature or concentration,
720
00:44:54,000 --> 00:44:57,000
you have to have
the K on both sides.
721
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
722
00:45:02,080 --> 00:45:05,000
now because this
is really too easy.
723
00:45:05,000 --> 00:45:07,220
What's the integrating factor?
724
00:45:07,220 --> 00:45:13,227
Well, the integrating factor is
going to be the integral of K,
725
00:45:13,227 --> 00:45:16,000
the coefficient now is just K.
726
00:45:16,000 --> 00:45:21,284
P is a constant, K, and if
I integrate KDT, I get KT,
727
00:45:21,284 --> 00:45:23,000
and I exponentiate that.
728
00:45:23,000 --> 00:45:28,000
So, the integrating
factor is E to the KT.
729
00:45:28,000 --> 00:45:34,000
I multiply through both sides,
multiply by E to the KT,
730
00:45:34,000 --> 00:45:38,000
and what's the
resulting equation?
731
00:45:38,000 --> 00:45:44,333
Well, it's going to be , I'll
write it in the compact form.
732
00:45:44,333 --> 00:45:50,000
It's going to be E to the
KT times T, all prime.
733
00:45:50,000 --> 00:45:55,284
The differentiation is now,
of course, with respect
734
00:45:55,284 --> 00:45:57,000
to the time.
735
00:45:57,000 --> 00:46:00,750
And, that's equal
to KTE, whatever
736
00:46:00,750 --> 00:46:04,725
that is, times E to the KT.
737
00:46:04,725 --> 00:46:09,000
This is a function
of T, of course,
738
00:46:09,000 --> 00:46:13,500
the function of little
time, sorry, little T time.
739
00:46:13,500 --> 00:46:18,000
OK, and now, finally, we
are going to integrate.
740
00:46:18,000 --> 00:46:19,332
What's the answer?
741
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
742
00:46:26,224 --> 00:46:31,750
T is, sorry, K little t, K
times time times the temperature
743
00:46:31,750 --> 00:46:37,000
is equal to the integral of KTE.
744
00:46:37,000 --> 00:46:40,840
I'll put the fact that
it's a function of T
745
00:46:40,840 --> 00:46:43,998
inside just to remind
you, E to the KT,
746
00:46:43,998 --> 00:46:46,500
and now I'll put the
arbitrary constant.
747
00:46:46,500 --> 00:46:50,142
Let's put in the arbitrary
constant explicitly.
748
00:46:50,142 --> 00:46:53,000
So, what will T be?
749
00:46:53,000 --> 00:46:56,000
OK, T will look
like this, finally.
750
00:46:56,000 --> 00:46:59,000
It will be E to the negative KT.
751
00:46:59,000 --> 00:47:01,500
That's on the outside.
752
00:47:01,500 --> 00:47:04,000
Then, you will integrate.
753
00:47:04,000 --> 00:47:07,552
Of course, the difficulty
of doing this integral
754
00:47:07,552 --> 00:47:12,000
depends entirely upon how this
external temperature varies.
755
00:47:12,000 --> 00:47:16,726
But anyways, it's going to be
K times that function, which
756
00:47:16,726 --> 00:47:20,000
I haven't specified,
E to the KT plus C
757
00:47:20,000 --> 00:47:22,664
times E to the negative KT.
758
00:47:22,664 --> 00:47:27,000
Now, some people, many, in
fact, that almost always,
759
00:47:27,000 --> 00:47:30,750
in the engineering
literature, almost never
760
00:47:30,750 --> 00:47:36,285
write indefinite integrals
because an indefinite integral
761
00:47:36,285 --> 00:47:38,000
is indefinite.
762
00:47:38,000 --> 00:47:40,400
In other words, this covers
not just one function,
763
00:47:40,400 --> 00:47:42,800
but a whole multitude
of functions
764
00:47:42,800 --> 00:47:46,000
which differ from each other
by an arbitrary constant.
765
00:47:46,000 --> 00:47:49,000
So, in a formula like this,
there's a certain vagueness,
766
00:47:49,000 --> 00:47:51,541
and it's further
compounded by the fact
767
00:47:51,541 --> 00:47:55,000
that I don't know whether the
arbitrary constant is here.
768
00:47:55,000 --> 00:47:58,571
I seem to have put it explicitly
on the outside the way
769
00:47:58,571 --> 00:48:02,000
you're used to
doing from calculus.
770
00:48:02,000 --> 00:48:04,570
Many people, therefore,
prefer, and I
771
00:48:04,570 --> 00:48:06,565
think you should
learn this, to do
772
00:48:06,565 --> 00:48:11,885
what is done in the very first
section of the notes called
773
00:48:11,885 --> 00:48:13,500
definite integral solutions.
774
00:48:13,500 --> 00:48:16,400
If there's an initial
condition saying
775
00:48:16,400 --> 00:48:19,200
that the internal
temperature at time zero
776
00:48:19,200 --> 00:48:22,912
is some given value,
what they like to do
777
00:48:22,912 --> 00:48:26,248
is make this thing definite
by integrating here
778
00:48:26,248 --> 00:48:30,625
from zero to T, and making
this a dummy variable.
779
00:48:30,625 --> 00:48:36,272
You see, what that
does is it gives you
780
00:48:36,272 --> 00:48:39,452
a particular
function, whereas, I'm
781
00:48:39,452 --> 00:48:44,766
sorry I didn't put in
the DT one minus two.
782
00:48:44,766 --> 00:48:49,554
What it does is that
when time is zero,
783
00:48:49,554 --> 00:48:52,662
all this automatically
disappears,
784
00:48:52,662 --> 00:49:00,000
and the arbitrary constant
will then be, it's T.
785
00:49:00,000 --> 00:49:03,125
So, in other words, C
times this, which is one,
786
00:49:03,125 --> 00:49:05,000
is that equal to [T?].
787
00:49:05,000 --> 00:49:07,000
In other words, if
I make this zero,
788
00:49:07,000 --> 00:49:12,000
that I can write C as equal to
this arbitrary starting value.
789
00:49:12,000 --> 00:49:15,000
Now, when you do this,
the essential thing,
790
00:49:15,000 --> 00:49:18,000
and we're going to come
back to this next week,
791
00:49:18,000 --> 00:49:21,000
but right away,
because K is positive,
792
00:49:21,000 --> 00:49:25,400
I want to emphasize that so much
at the beginning of the period,
793
00:49:25,400 --> 00:49:30,428
I want to conclude by showing
you what its significance is.
794
00:49:30,428 --> 00:49:35,332
This part disappears
because K is positive.
795
00:49:35,332 --> 00:49:38,000
The conductivity is positive.
796
00:49:38,000 --> 00:49:41,000
This part disappears
as T goes to zero.
797
00:49:41,000 --> 00:49:45,000
This goes to zero as
T goes to infinity.
798
00:49:45,000 --> 00:49:48,000
So, this is a
solution that remains.
799
00:49:48,000 --> 00:49:52,625
This, therefore, is called
the steady state solution,
800
00:49:52,625 --> 00:49:57,000
the thing which the
temperature behaves like,
801
00:49:57,000 --> 00:49:59,000
as T goes to infinity.
802
00:49:59,000 --> 00:50:01,000
This is called the transient.
803
00:50:01,000 --> 00:50:07,000
because it disappears
as T goes to infinity.
804
00:50:07,000 --> 00:50:09,664
It depends on the
initial condition,
805
00:50:09,664 --> 00:50:12,200
but it disappears,
which shows you,
806
00:50:12,200 --> 00:50:16,665
then, in the long run
for this type of problem
807
00:50:16,665 --> 00:50:20,000
the initial condition
makes no difference.
808
00:50:20,000 --> 00:50:24,550
The function behaves always the
same way as T goes to infinity.