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This time, we started solving
differential equations.
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This is the third lecture of
the term, and I have yet to
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solve a single differential
equation in this class.
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Well, that will be rectified
from now until the end of the
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term.
So, once you learn separation
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of variables,
which is the most elementary
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method there is,
the single, I think the single
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most important equation is the
one that's called the first
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order linear equation,
both because it occurs
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frequently in models because
it's solvable,
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and-- I think that's enough.
If you drop the course after
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today you will still have
learned those two important
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methods: separation of
variables, and first order
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linear equations.
So, what does such an equation
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look like?
Well, I'll write it in there.
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There are several ways of
writing it, but I think the most
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basic is this.
I'm going to use x as the
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independent variable because
that's what your book does.
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But in the applications,
it's often t,
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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 prime plus some other
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function of x times y is equal
to yet another function of x.
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Obviously, the x doesn't have
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the same status here that y
does, so y is extremely limited
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in how 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.
This is the first version of
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it, and we'll call them purple.
Now, why is that called the
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linear equation?
The word linear is a very
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heavily used word in
mathematics, science,
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and engineering.
For the moment,
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the best simple answer is
because it's linear in y and y
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prime,
the variables 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.
The most closely analogous
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thing would be a linear
equation, a real linear
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equation, the kind you studied
in high school,
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which would look like this.
It would have two variables,
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and, I guess,
constant coefficients,
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equal c.
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 are the analogs of the
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variables y1 and y2.
A little bit of terminology,
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if c is equal to zero,
it's called homogeneous,
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the same way this equation is
called homogeneous,
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as you know from 18.02,
if the right-hand 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.
Now, this is a common form for
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the equation,
but it's not what it's called
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standard form.
The standard form for the
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equation, and since this is
going to be a prime course of
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confusion, which is probably
completely correct,
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but a prime source of confusion
is what I meant.
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The standard linear form,
and I'll underline linear is
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the first co-efficient of y
prime is taken to be one.
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So, you can always convert that
to a standard form by simply
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dividing through by it.
And if I do that,
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the equation will look like y
prime plus, now,
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it's common to not call it b
anymore, the coefficient,
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because it's really b over a.
And, therefore,
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it's common to adopt,
yet, 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?
We needed a letter for that,
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too.
It's c over a,
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but we'll call it q.
So, when I talk about the
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standard linear form 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 is a potential for
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confusion here because what did
I call the standard form for a
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first-order equation?
So, I'm going to say,
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not this.
The standard first order form,
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what would that be?
Well, it would be y prime
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equals, and everything else on
the left-hand side.
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So, it would be y prime.
And now, if I turn this into
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the standard first-order form,
it would be negative p of x y
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plus q of x.
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But, of course,
nobody would write negative p
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of x.
So, now, I explicitly want to
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say that this is a form which I
will never use for this
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equation, although half the
books of the world do.
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In short, this poor little
first-order equation belongs to
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two ethnic groups.
It's both a first order
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equation, and therefore,
its standard form should be
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written this way,
but it's also a linear
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equation, and therefore its
standard form should be used
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this way.
Well, it has to decide,
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and I have decided for it.
It is, above all,
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a linear equation,
not just a first-order
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equation.
And, in this course,
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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 is entirely in
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the sin(p).
But, if you get the sign of p
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wrong in the answers,
it is just a disaster from that
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point on.
A trivial little change of sign
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in the answer produces solutions
and functions which have totally
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different behavior.
And, you are going to be really
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lost in this course.
So, maybe I should draw a line
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through it to indicate,
please don't pay any attention
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to this whatsoever,
except that we are not going to
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do that.
Okay, well, what's so important
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about this equation?
Well, number one,
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it can always be solved.
That's a very,
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very big thing in differential
equations.
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But, it's also the equation
which arises in a variety of
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models.
Now, I'm just going to list a
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few of them.
All of them I think you will
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need either in part one or part
two of problem sets over these
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first couple of problem sets,
or second and third maybe.
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But, of them,
I'm going to put at the very
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top of the list of what I'll
call here, I'll give it two
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names: the temperature diffusion
model, well, it would be better
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to call it temperature
concentration by analogy,
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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
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problems if you can figure out a
way of getting rid of the second
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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.
Now, today I have to pick a
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model.
And, the one I'm going to pick
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is this temperature
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, I'm asking the
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recitations to,
among other things,
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make sure they do a mixing
problem, A) to show you how to
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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 or we will take them up
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later in the term.
This is not going to be the
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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 model,
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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,
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the similarities,
they are practically the same
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model.
All that's changed from one to
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the other 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
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concentration.
But, the actual mathematics
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isn't identical.
So, let's begin with
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conduction.
All right, so,
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I need a simple physical
situation 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, somehow,
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is a chamber.
A metal cube will do,
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and let's suppose that its
walls are partly insulated,
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not so much that no heat can
get through.
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There is no such thing as
perfect insulation anyway,
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except maybe an absolute
perfect vacuum.
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Now, inside,
so here on the outside is
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liquid.
Okay, on the inside is,
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what I'm interested in is the
temperature of this thing.
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I'll call that T.
Now, that's different from the
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temperature 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
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definiteness.
But, this is the external
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temperature.
So, I'll indicate it with an e.
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Now, what is the model?
Well, in other words,
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how do I set up a differential
equation 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, the so-called Newton's Law
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of cooling, --
-- which says that the rate of
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change, the temperature of the
heat goes from the outside to
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the inside by conduction only.
Heat, of course,
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can travel in various ways,
by convection,
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by conduction,
as here, or by radiation,
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are the three most common.
Of these, I only want one,
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namely transmission of heat by
conduction.
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And, that's the way it's
probably a little better to call
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it the conduction model,
rather than the temperature
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model, which might involve other
ways for the heat to be
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traveling.
So, dt, the independent
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variable, is not going to be x,
as it was over there.
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It's going to be t for time.
So, maybe I should write that
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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.
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So, it's proportional to the
temperature difference between
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these two.
Now, how shall I write the
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difference?
Write it this way because if
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you don't you will be in
trouble.
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Now, why do I write it that
way?
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Well, I write it that way
because I want this constant to
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be positive, a positive
constant.
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In general, any constant,
so, parameters which are
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physical, have some physical
significance,
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one always wants to arrange the
equation so that they are
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positive numbers,
the way people normally think
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of these things.
This is called the
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conductivity.
The conductivity of what?
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Well, I don't know,
of the system of the situation,
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the conductivity of the wall,
or the wall if the metal were
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just by itself.
At any rate,
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it's a constant.
It's thought of as a constant.
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And, y positive,
well, because if the external
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temperature is bigger than the
internal temperature,
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I expect T to rise,
the internal temperature to
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rise.
That means dT / dt,
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its slope, should be positive.
So, in other words,
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if Te is bigger than T,
I expect this number to be
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positive.
And, that tells you that k must
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be a positive constant.
If I had turned it the other
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way, expressed the difference in
the reverse order,
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K would then be negative,
have to be negative in order
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that this turn out to be
positive in that situation I
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described.
And, since nobody wants
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negative values of k,
you have to write the equation
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in this form rather than the
other way around.
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So, there's our differential
equation.
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It will probably have an
initial condition.
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So, it could be the temperature
at the starting time should be
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some given number, T zero.
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But, the condition could be
given in other ways.
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One can ask,
what's the temperature as time
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goes to infinity,
for example?
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There are different ways of
getting that initial condition.
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Okay, that's the conduction
model.
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What would the diffusion model
be?
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The diffusion model,
mathematically,
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would be, word for word,
the same.
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The only difference is that
now, what I imagine is I'll draw
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the picture the same way,
except now I'm going to put,
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label the inside not with a T
but with a C,
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C for concentration.
It's in an external water bath,
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let's say.
So, there is an external
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concentration.
And, what I'm talking about is
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some chemical,
let's say salt will do as well
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as anything.
So, C is equal to salt
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concentration inside,
and Ce would be the salt
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concentration outside,
outside in the water bath.
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Now, I imagine some mechanism,
so this is a salt solution.
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That's a salt solution.
And, I imagine some mechanism
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by which the salt can diffuse,
it's a diffusion model now,
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diffuse from here into the air
or possibly out the other way.
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And that's usually done by
vaguely referring to the outside
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as a semi-permeable membrane,
semi-permeable,
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so that the salt will have a
little hard time getting through
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but permeable,
so that it won't be blocked
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completely.
So, there's a membrane.
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You write the semi-permeable
membrane outside,
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outside the inside.
Well, I give up.
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You know, membrane somewhere.
Sorry, membrane wall.
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How's that?
Now, what's the equation?
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Well, the equation is the same,
except it's called the
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diffusion equation.
I don't think Newton got his
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name on this.
The diffusion equation says
265
00:18:37 --> 00:18:43
that the rate at which the salt
diffuses across the membrane,
266
00:18:42 --> 00:18:48
which is the same up to a
constant as the rate at which
267
00:18:47 --> 00:18:53
the concentration inside
changes, is some constant,
268
00:18:51 --> 00:18:57
usually called k still,
okay.
269
00:18:54 --> 00:19:00
Do I contradict?
Okay, let's keep calling it k1.
270
00:18:58 --> 00:19:04
Now it's different,
times Ce minus C.
271
00:19:01 --> 00:19:07
And, for the same reason as
before, if the external
272
00:19:05 --> 00:19:11
concentration is bigger than the
internal concentration,
273
00:19:10 --> 00:19:16
we expect salt to flow in.
That will make C rise.
274
00:19:15 --> 00:19:21
It will make this positive,
and therefore,
275
00:19:18 --> 00:19:24
we want k to be positive,
just k1 to be positive for the
276
00:19:22 --> 00:19:28
same reason it had to be
positive before.
277
00:19:25 --> 00:19:31
So, in each case,
the model that I'm talking
278
00:19:28 --> 00:19:34
about is the differential
equation.
279
00:19:31 --> 00:19:37
So, maybe I should,
let's put that,
280
00:19:33 --> 00:19:39
make that clear.
Or, I would say that this first
281
00:19:38 --> 00:19:44
order differential equation
models this physical situation,
282
00:19:42 --> 00:19:48
and the same thing is true on
the other side over here.
283
00:19:46 --> 00:19:52
This is the diffusion equation,
and this is the conduction
284
00:19:50 --> 00:19:56
equation.
Now, if you are in any doubt
285
00:19:53 --> 00:19:59
about the power of differential
equations, the point is,
286
00:19:57 --> 00:20:03
when I talk about this thing,
I don't have to say which of
287
00:20:01 --> 00:20:07
these I'm following.
I'll use neutral variables like
288
00:20:06 --> 00:20:12
Y and X to solve these
equations.
289
00:20:09 --> 00:20:15
But, with a single stroke,
I will be handling those
290
00:20:13 --> 00:20:19
situations together.
And, that's the power of the
291
00:20:16 --> 00:20:22
method.
Now, you obviously must be
292
00:20:19 --> 00:20:25
wondering, look,
these look very,
293
00:20:22 --> 00:20:28
very special.
He said he was going to talk
294
00:20:25 --> 00:20:31
about the first,
general first-order equation.
295
00:20:28 --> 00:20:34
But, these look rather special
to me.
296
00:20:31 --> 00:20:37
Well, not too special.
How should we write it?
297
00:20:36 --> 00:20:42
Suppose I write,
let's take the temperature
298
00:20:40 --> 00:20:46
equation just to have something
definite.
299
00:20:44 --> 00:20:50
Notice that it's in a form
corresponding to Newton's Law.
300
00:20:48 --> 00:20:54
But it is not in the standard
linear form.
301
00:20:52 --> 00:20:58
Let's put it in standard linear
form, so at least you could see
302
00:20:57 --> 00:21:03
that it's a linear equation.
So, if I put it in standard
303
00:21:02 --> 00:21:08
form, it's going to look like
DTDTD little t plus KT is equal
304
00:21:07 --> 00:21:13
to K times TE.
Now, compare that with the
305
00:21:12 --> 00:21:18
general, the way the general
equation is supposed to look,
306
00:21:16 --> 00:21:22
the yellow box over there,
the standard linear form.
307
00:21:20 --> 00:21:26
How are they going to compare?
Well, this is a pretty general
308
00:21:24 --> 00:21:30
function.
This is general.
309
00:21:26 --> 00:21:32
This is a general function of T
because I can make the external
310
00:21:31 --> 00:21:37
temperature.
I could suppose it behaves in
311
00:21:34 --> 00:21:40
anyway I like,
steadily rising,
312
00:21:37 --> 00:21:43
decaying exponentially,
maybe oscillating back and
313
00:21:40 --> 00:21:46
forth for some reason.
The only way in which it's not
314
00:21:46 --> 00:21:52
general is that this K is a
constant.
315
00:21:50 --> 00:21:56
So, I will ask you to be
generous.
316
00:21:53 --> 00:21:59
Let's imagine the conductivity
is changing over time.
317
00:21:57 --> 00:22:03
So, this is usually constant,
but there's no law which says
318
00:22:03 --> 00:22:09
it has to be.
How could a conductivity change
319
00:22:08 --> 00:22:14
over time?
Well, we could suppose that
320
00:22:12 --> 00:22:18
this wall was made of slowly
congealing Jell-O,
321
00:22:16 --> 00:22:22
for instance.
It starts out as liquid,
322
00:22:20 --> 00:22:26
and then it gets solid.
And, Jell-O doesn't transmit
323
00:22:26 --> 00:22:32
heat, I believe,
quite as well as liquid does,
324
00:22:30 --> 00:22:36
as a liquid would.
Is Jell-O a solid or liquid?
325
00:22:36 --> 00:22:42
I don't know.
Let's forget about that.
326
00:22:40 --> 00:22:46
So, with this understanding,
so let's say not necessarily
327
00:22:46 --> 00:22:52
here, but not necessarily,
I can think of this,
328
00:22:51 --> 00:22:57
therefore, by allowing K to
vary with time.
329
00:22:55 --> 00:23:01
And the external temperature to
vary with time.
330
00:23:00 --> 00:23:06
I can think of it as a general,
linear equation.
331
00:23:07 --> 00:23:13
So, these models are not
special.
332
00:23:09 --> 00:23:15
They are fairly general.
Well, I did promise you I would
333
00:23:13 --> 00:23:19
solve an equation,
and that this lecture,
334
00:23:16 --> 00:23:22
I still have not solved any
equations.
335
00:23:18 --> 00:23:24
OK, time to stop temporizing
and solve.
336
00:23:21 --> 00:23:27
So, I'm going to,
in order not to play favorites
337
00:23:24 --> 00:23:30
with these two models,
I'll go back to,
338
00:23:27 --> 00:23:33
and to get you used to thinking
of the variables all the time,
339
00:23:31 --> 00:23:37
that is, you know,
be eclectic switching from one
340
00:23:35 --> 00:23:41
variable to another according to
which particular lecture you
341
00:23:39 --> 00:23:45
happened to be sitting in.
So, let's take our equation in
342
00:23:47 --> 00:23:53
the form, Y prime plus P of XY,
the general form using the old
343
00:23:57 --> 00:24:03
variables equals Q of X.
Solve me.
344
00:24:04 --> 00:24:10
Well, there are different ways
of describing the solution
345
00:24:07 --> 00:24:13
process.
No matter how you do it,
346
00:24:10 --> 00:24:16
it amounts to the same amount
of work and there is always a
347
00:24:14 --> 00:24:20
trick involved at each one of
them since you can't suppress a
348
00:24:18 --> 00:24:24
trick by doing the problem some
other way.
349
00:24:21 --> 00:24:27
The way I'm going to do it,
I think, is the best.
350
00:24:24 --> 00:24:30
That's why I'm giving it to
you.
351
00:24:26 --> 00:24:32
It's the easiest to remember.
It leads to the least work,
352
00:24:30 --> 00:24:36
but I have colleagues who would
fight with me about that point.
353
00:24:36 --> 00:24:42
So, since they are not here to
fight with me I am free to do
354
00:24:41 --> 00:24:47
whatever I like.
One of the main reasons for
355
00:24:45 --> 00:24:51
doing it the way I'm going to do
is because I want you to get
356
00:24:51 --> 00:24:57
what our word into your
consciousness,
357
00:24:55 --> 00:25:01
two words, integrating factor.
I'm going to solve this
358
00:25:00 --> 00:25:06
equation by finding and
integrating factor of the form U
359
00:25:05 --> 00:25:11
of X.
What's an integrating factor?
360
00:25:09 --> 00:25:15
Well, I'll show you not by
writing an elaborate definition
361
00:25:13 --> 00:25:19
on the board,
but showing you what its
362
00:25:16 --> 00:25:22
function is.
It's a certain function,
363
00:25:18 --> 00:25:24
U of X, I don't know what it
is, but here's what I wanted to
364
00:25:23 --> 00:25:29
do.
I want to multiply,
365
00:25:24 --> 00:25:30
I'm going to drop the X's a
just so that the thing looks
366
00:25:28 --> 00:25:34
less complicated.
So, what I want to do is
367
00:25:33 --> 00:25:39
multiply this equation through
by U of X.
368
00:25:36 --> 00:25:42
That's why it's called a factor
because you're going to multiply
369
00:25:42 --> 00:25:48
everything through by it.
So, it's going to look like UY
370
00:25:47 --> 00:25:53
prime plus PUY equals QU,
and now, so far,
371
00:25:51 --> 00:25:57
it's just a factor.
What makes it an integrating
372
00:25:55 --> 00:26:01
factor is that this,
after I do that,
373
00:25:58 --> 00:26:04
I want this to turn out to be
the derivative of something with
374
00:26:04 --> 00:26:10
respect to X.
You see the motivation for
375
00:26:08 --> 00:26:14
that.
If this turns out to be the
376
00:26:10 --> 00:26:16
derivative of something,
because I've chosen U so
377
00:26:13 --> 00:26:19
cleverly, then I will be able to
solve the equation immediately
378
00:26:17 --> 00:26:23
just by integrating this with
respect to X,
379
00:26:19 --> 00:26:25
and integrating that with
respect to X.
380
00:26:21 --> 00:26:27
You just, then,
integrate both sides with
381
00:26:24 --> 00:26:30
respect to X,
and the equation is solved.
382
00:26:26 --> 00:26:32
Now, the only question is,
what should I choose for U?
383
00:26:31 --> 00:26:37
Well, if you think of the
product formula,
384
00:26:34 --> 00:26:40
there might be many things to
try here.
385
00:26:37 --> 00:26:43
But there's only one reasonable
thing to try.
386
00:26:40 --> 00:26:46
Try to pick U so that it's the
derivative of U times Y.
387
00:26:45 --> 00:26:51
See how reasonable that is?
If I use the product rule on
388
00:26:49 --> 00:26:55
this, the first term is U times
Y prime.
389
00:26:52 --> 00:26:58
The second term would be U
prime times Y.
390
00:26:56 --> 00:27:02
Well, I've got the Y there.
So, this will work.
391
00:27:01 --> 00:27:07
It works if,
what's the condition that you
392
00:27:05 --> 00:27:11
must satisfy in order for that
to be true?
393
00:27:09 --> 00:27:15
Well, it must be that after it
to the differentiation,
394
00:27:15 --> 00:27:21
U prime turns out to be P times
U.
395
00:27:19 --> 00:27:25
So, is it clear?
This is something we want to be
396
00:27:24 --> 00:27:30
equal to, and the thing I will
try to do it is by choosing U in
397
00:27:30 --> 00:27:36
such a way that this equality
will take place.
398
00:27:37 --> 00:27:43
And then I will be able to
solve the equation.
399
00:27:40 --> 00:27:46
And so, here's what my U prime
must satisfy.
400
00:27:43 --> 00:27:49
Hey, we can solve that.
But please don't forget that P
401
00:27:47 --> 00:27:53
is P of X.
It's a function of X.
402
00:27:49 --> 00:27:55
So, if you separate variables,
I'm going to do this.
403
00:27:53 --> 00:27:59
So, what is it,
DU over U equals P of X times
404
00:27:56 --> 00:28:02
DX.
If I integrate that,
405
00:27:59 --> 00:28:05
so, separate variables,
integrate, and you're going to
406
00:28:03 --> 00:28:09
get DU over U integrates to the
be the log of U,
407
00:28:07 --> 00:28:13
and the other side integrates
to be the integral of P of X DX.
408
00:28:12 --> 00:28:18
Now, you can put an arbitrary
constant there,
409
00:28:16 --> 00:28:22
or you can think of it as
already implied by the
410
00:28:19 --> 00:28:25
indefinite integral.
Well, that doesn't tell us,
411
00:28:23 --> 00:28:29
yet, what U is.
What should U be?
412
00:28:26 --> 00:28:32
Notice, I don't have to find
every possible U,
413
00:28:29 --> 00:28:35
which works.
All I'm looking for is one.
414
00:28:34 --> 00:28:40
All I want is a single view
which satisfies that equation.
415
00:28:38 --> 00:28:44
Well, U equals the integral,
E to the integral of PDX.
416
00:28:42 --> 00:28:48
That's not too beautiful
looking, but by differential
417
00:28:46 --> 00:28:52
equations, things can get so
complicated that in a week or
418
00:28:50 --> 00:28:56
two, you will think of this as
an extremely simple formula.
419
00:28:55 --> 00:29:01
So, there is a formula for our
integrating factor.
420
00:29:00 --> 00:29:06
We found it.
We will always be able to write
421
00:29:05 --> 00:29:11
an integrating factor.
Don't worry about the arbitrary
422
00:29:11 --> 00:29:17
constant because you only need
one such U.
423
00:29:17 --> 00:29:23
So: no arbitrary constant since
only one U needed.
424
00:29:23 --> 00:29:29
And, that's the solution,
the way we solve the linear
425
00:29:29 --> 00:29:35
equation.
OK, let's take over,
426
00:29:35 --> 00:29:41
and actually do it.
I think it would be better to
427
00:29:42 --> 00:29:48
summarize it as a clear-cut
method.
428
00:29:48 --> 00:29:54
So, let's do that.
So, what's our method?
429
00:29:54 --> 00:30:00
It's the method for solving Y
prime plus PY equals Q.
430
00:30:04 --> 00:30:10
Well, the first place,
make sure it's in standard
431
00:30:07 --> 00:30:13
linear form.
If it isn't,
432
00:30:09 --> 00:30:15
you must put it in that form.
Notice, the formula for the
433
00:30:13 --> 00:30:19
integrating factor,
the formula for the integrating
434
00:30:16 --> 00:30:22
factor involves P,
the integral of PDX.
435
00:30:19 --> 00:30:25
So, you'd better get the right
P.
436
00:30:21 --> 00:30:27
Otherwise, you are sunk.
OK, so put it in standard
437
00:30:25 --> 00:30:31
linear form.
That way, you will have the
438
00:30:28 --> 00:30:34
right P.
Notice that if you wrote it in
439
00:30:32 --> 00:30:38
that form, and all you
remembered was E to the integral
440
00:30:35 --> 00:30:41
PDX, the P would have the wrong
sign.
441
00:30:38 --> 00:30:44
If you're going to write,
that P should have a negative
442
00:30:41 --> 00:30:47
sign there.
So, do it this way,
443
00:30:43 --> 00:30:49
and no other way.
Otherwise, you will get
444
00:30:46 --> 00:30:52
confused and get wrong signs.
And, as I say,
445
00:30:49 --> 00:30:55
that will produce wrong
answers, and not just slightly
446
00:30:52 --> 00:30:58
wrong answers,
but disastrously wrong answers
447
00:30:55 --> 00:31:01
from the point of view of the
modeling if you really want
448
00:30:59 --> 00:31:05
answers to physical problems.
So, here's a standard linear
449
00:31:06 --> 00:31:12
form.
Then, find the integrating
450
00:31:09 --> 00:31:15
factor.
So, calculate E to the
451
00:31:13 --> 00:31:19
integral, PDX,
the integrating factor,
452
00:31:17 --> 00:31:23
and that multiply both,
I'm putting this as both,
453
00:31:22 --> 00:31:28
underlined that as many times
as you have room in your notes.
454
00:31:31 --> 00:31:37
Multiply both sides by this
integrating factor by E to the
455
00:31:40 --> 00:31:46
integral PDX.
And then, integrate.
456
00:31:46 --> 00:31:52
OK, let's take a simple
example.
457
00:31:51 --> 00:31:57
Suppose we started with the
equation XY prime minus Y
458
00:31:59 --> 00:32:05
equals, I had X2,
X3, something like that,
459
00:32:06 --> 00:32:12
X3, I think,
yeah, X2.
460
00:32:12 --> 00:32:18
OK, what's the first thing to
do?
461
00:32:16 --> 00:32:22
Put it in standard form.
So, step zero will be to write
462
00:32:23 --> 00:32:29
it as Y prime minus one over X
times Y equals X2.
463
00:32:30 --> 00:32:36
Let's do the work first,
and then I'll talk about
464
00:32:34 --> 00:32:40
mistakes.
Well, we now calculate the
465
00:32:39 --> 00:32:45
integrating factor.
So, I would do it in steps.
466
00:32:43 --> 00:32:49
You can integrate negative one
over X, right?
467
00:32:48 --> 00:32:54
That integrates to minus log X.
So, the integrating factor is E
468
00:32:54 --> 00:33:00
to the integral of this,
DX.
469
00:32:57 --> 00:33:03
So, it's E to the negative log
X.
470
00:33:02 --> 00:33:08
Now, in real life,
that's not the way to leave
471
00:33:06 --> 00:33:12
that.
What is E to the negative log
472
00:33:10 --> 00:33:16
X?
Well, think of it as E to the
473
00:33:13 --> 00:33:19
log X to the minus one.
Or, in other words,
474
00:33:18 --> 00:33:24
it is E to the log X is X.
So, it's one over X.
475
00:33:23 --> 00:33:29
So, the integrating factor is
one over X.
476
00:33:27 --> 00:33:33
OK, multiply both sides by the
integrating factor.
477
00:33:34 --> 00:33:40
Both sides of what?
Both sides of this:
478
00:33:37 --> 00:33:43
the equation written in
standard form,
479
00:33:41 --> 00:33:47
and both sides.
So, it's going to be one over
480
00:33:45 --> 00:33:51
XY prime minus one over X2 Y is
equal to X2 times one over X,
481
00:33:51 --> 00:33:57
which is simply X.
Now, if you have done the work
482
00:33:55 --> 00:34:01
correctly, you should be able,
now, to integrate the left-hand
483
00:34:01 --> 00:34:07
side directly.
So, I'm going to write it this
484
00:34:06 --> 00:34:12
way.
I always recommend that you put
485
00:34:09 --> 00:34:15
it as extra step,
well, put it as an extra step
486
00:34:13 --> 00:34:19
the reason for using that
integrating factor,
487
00:34:17 --> 00:34:23
in other words,
that the left-hand side is
488
00:34:20 --> 00:34:26
supposed to be,
now, one over X times Y prime.
489
00:34:24 --> 00:34:30
I always put it that because
there's always a chance you made
490
00:34:29 --> 00:34:35
a mistake or forgot something.
Look at it, mentally
491
00:34:34 --> 00:34:40
differentiated using the product
rule just to check that,
492
00:34:39 --> 00:34:45
in fact, it turns out to be the
same as the left-hand side.
493
00:34:43 --> 00:34:49
So, what do we get?
One over X times Y prime plus Y
494
00:34:47 --> 00:34:53
times the derivative of one over
X, which indeed is negative one
495
00:34:53 --> 00:34:59
over X2.
And now, finally,
496
00:34:55 --> 00:35:01
that's 3A, continue,
do the integration.
497
00:34:58 --> 00:35:04
So, you're going to get,
let's see if we can do it all
498
00:35:02 --> 00:35:08
on one board,
one over X times Y is equal to
499
00:35:06 --> 00:35:12
X plus a constant,
X, sorry, X2 over two plus a
500
00:35:09 --> 00:35:15
constant.
And, the final step will be,
501
00:35:15 --> 00:35:21
therefore, now I want to
isolate Y by itself.
502
00:35:21 --> 00:35:27
So, Y will be equal to multiply
through by X.
503
00:35:26 --> 00:35:32
X3 over two plus C times X.
And, that's the solution.
504
00:35:34 --> 00:35:40
OK, let's do one a little
slightly more complicated.
505
00:35:40 --> 00:35:46
Let's try this one.
Now, my equation is going to be
506
00:35:45 --> 00:35:51
one, I'll still keep two,
Y and X, as the variables.
507
00:35:51 --> 00:35:57
I'll use T and F for a minute
or two.
508
00:35:57 --> 00:36:03
One plus cosine X,
so, I'm not going to give you
509
00:36:03 --> 00:36:09
this one in standard form
either.
510
00:36:07 --> 00:36:13
It's a trick question.
Y prime minus sine X times Y is
511
00:36:14 --> 00:36:20
equal to anything reasonable,
I guess.
512
00:36:19 --> 00:36:25
I think X, 2X,
make it more exciting.
513
00:36:24 --> 00:36:30
OK, now, I think I should warn
you where the mistakes are just
514
00:36:32 --> 00:36:38
so that you can make all of
them.
515
00:36:38 --> 00:36:44
So, this is mistake number one.
You don't put it in standard
516
00:36:44 --> 00:36:50
form.
Mistake number two:
517
00:36:47 --> 00:36:53
generally people can do step
one fine.
518
00:36:51 --> 00:36:57
Mistake number two is,
this is my most common mistake,
519
00:36:57 --> 00:37:03
so I'm very sensitive to it.
But that doesn't mean if you
520
00:37:03 --> 00:37:09
make it, you'll get any sympathy
from me.
521
00:37:06 --> 00:37:12
I don't give sympathy to
myself.
522
00:37:08 --> 00:37:14
You are so intense,
so happy at having found the
523
00:37:11 --> 00:37:17
integrating factor,
you forget to multiply Q by the
524
00:37:15 --> 00:37:21
integrating factor also.
You just handle the left-hand
525
00:37:19 --> 00:37:25
side of the equation,
if you forget about the
526
00:37:22 --> 00:37:28
right-hand side.
So, the emphasis on the both
527
00:37:25 --> 00:37:31
here is the right-hand,
please include the Q.
528
00:37:29 --> 00:37:35
Please include the right-hand
side.
529
00:37:33 --> 00:37:39
Any other mistakes?
Well, nothing that I can think
530
00:37:37 --> 00:37:43
of.
Well, maybe only,
531
00:37:38 --> 00:37:44
anyway, we are not going to
make any mistakes the rest of
532
00:37:43 --> 00:37:49
this lecture.
So, what do we do?
533
00:37:45 --> 00:37:51
We write this in standard form.
So, it's going to look like Y
534
00:37:50 --> 00:37:56
prime minus sine X,
sine X divided by one plus
535
00:37:54 --> 00:38:00
cosine X times Y equals,
my heart sinks because I know
536
00:37:59 --> 00:38:05
I'm supposed to integrate
something like this.
537
00:38:04 --> 00:38:10
And, boy, that's going to give
me problems.
538
00:38:07 --> 00:38:13
Well, not yet.
With the integrating factor?
539
00:38:11 --> 00:38:17
The integrating factor is,
well, we want to calculate the
540
00:38:16 --> 00:38:22
integral of negative sine X over
one plus cosine.
541
00:38:20 --> 00:38:26
That's the integral of PDX.
And, after that,
542
00:38:23 --> 00:38:29
we have to exponentiate it.
Well, can you do this?
543
00:38:28 --> 00:38:34
Yeah, but if you stare at it a
little while,
544
00:38:31 --> 00:38:37
you can see that the top is the
derivative of the bottom.
545
00:38:38 --> 00:38:44
That is great.
That means it integrates to be
546
00:38:42 --> 00:38:48
the log of one plus cosine X.
Is that right,
547
00:38:46 --> 00:38:52
one over one plus cosine X
times the derivative of this,
548
00:38:51 --> 00:38:57
which is negative cosine X.
Therefore, the integrating
549
00:38:56 --> 00:39:02
factor is E to that.
In other words,
550
00:38:59 --> 00:39:05
it is one plus cosine X.
Therefore, so this was step
551
00:39:05 --> 00:39:11
zero.
Step one, we found the
552
00:39:08 --> 00:39:14
integrating factor.
And now, step two,
553
00:39:12 --> 00:39:18
we multiply through the
integrating factor.
554
00:39:17 --> 00:39:23
And what do we get?
We multiply through the
555
00:39:21 --> 00:39:27
standard for equation by the
integrating factor,
556
00:39:26 --> 00:39:32
if you do that,
what you get is,
557
00:39:29 --> 00:39:35
well, Y prime gets the
coefficient one plus cosine X,
558
00:39:35 --> 00:39:41
Y prime minus sign X equals 2X.
Oh, dear.
559
00:39:40 --> 00:39:46
Well, I hope somebody would
giggle at this point.
560
00:39:45 --> 00:39:51
What's giggle-able about it?
Well, that all this was totally
561
00:39:50 --> 00:39:56
wasted work.
It's called spinning your
562
00:39:53 --> 00:39:59
wheels.
No, it's not spinning your
563
00:39:56 --> 00:40:02
wheels.
It's doing what you're supposed
564
00:39:59 --> 00:40:05
to do, and finding out that you
wasted the entire time doing
565
00:40:05 --> 00:40:11
what you were supposed to do.
Well, in other words,
566
00:40:10 --> 00:40:16
that net effect of this is to
end up with the same equation we
567
00:40:16 --> 00:40:22
started with.
But, what is the point?
568
00:40:19 --> 00:40:25
The point of having done all
this was because now the
569
00:40:24 --> 00:40:30
left-hand side is exactly the
derivative of something,
570
00:40:29 --> 00:40:35
and the left-hand side should
be the derivative of what?
571
00:40:35 --> 00:40:41
Well, it should be the
derivative of one plus cosine X
572
00:40:39 --> 00:40:45
times Y, all prime.
Now, you can check that that's
573
00:40:43 --> 00:40:49
in fact the case.
It's one plus cosine X,
574
00:40:47 --> 00:40:53
Y prime, plus minus sine X,
the derivative of this side
575
00:40:51 --> 00:40:57
times Y.
So, if you had thought,
576
00:40:54 --> 00:41:00
in looking at the equation,
to say to yourself,
577
00:40:58 --> 00:41:04
this is a derivative of that,
maybe I'll just check right
578
00:41:03 --> 00:41:09
away to see if it's the
derivative of one plus cosine X
579
00:41:08 --> 00:41:14
sine.
You would have saved that work.
580
00:41:12 --> 00:41:18
Well, you don't have to be
brilliant or clever,
581
00:41:16 --> 00:41:22
or anything like that.
You can follow your nose,
582
00:41:19 --> 00:41:25
and it's just,
I want to give you a positive
583
00:41:23 --> 00:41:29
experience in solving linear
equations, not too negative.
584
00:41:28 --> 00:41:34
Anyway, so we got to this
point.
585
00:41:31 --> 00:41:37
So, now this is 2X,
and now we are ready to solve
586
00:41:37 --> 00:41:43
the equation,
which is the solution now will
587
00:41:42 --> 00:41:48
be one plus cosine X times Y is
equal to X2 plus a constant,
588
00:41:49 --> 00:41:55
and so Y is equal to X2 divided
by X2 plus a constant divided by
589
00:41:56 --> 00:42:02
one plus cosine X.
Suppose I have given you an
590
00:42:01 --> 00:42:07
initial condition,
which I didn't.
591
00:42:03 --> 00:42:09
But, suppose the initial
condition said that Y of zero
592
00:42:07 --> 00:42:13
were one, for instance.
Then, the solution would be,
593
00:42:10 --> 00:42:16
so, this is an if,
I'm throwing in at the end just
594
00:42:14 --> 00:42:20
to make it a little bit more of
a problem, how would I put,
595
00:42:17 --> 00:42:23
then I could evaluate the
constant by using the initial
596
00:42:21 --> 00:42:27
condition.
What would it be?
597
00:42:23 --> 00:42:29
This would be,
on the left-hand side,
598
00:42:25 --> 00:42:31
one, on the right-hand side
would be C over two.
599
00:42:30 --> 00:42:36
So, I would get one equals C
over two.
600
00:42:34 --> 00:42:40
Is that correct?
Cosine of zero is one,
601
00:42:39 --> 00:42:45
so that's two down below.
Therefore, C is equal to two,
602
00:42:46 --> 00:42:52
and that would then complete
the solution.
603
00:42:51 --> 00:42:57
We would be X2 plus two over
one plus cosine X.
604
00:42:57 --> 00:43:03
Now, you can do this in
general, of course,
605
00:43:03 --> 00:43:09
and get a general formula.
And, we will have occasion to
606
00:43:10 --> 00:43:16
use that next week.
But for now,
607
00:43:13 --> 00:43:19
why don't we concentrate on the
most interesting case,
608
00:43:18 --> 00:43:24
namely that of the most linear
equation, with constant
609
00:43:23 --> 00:43:29
coefficient, that is,
so let's look at the linear
610
00:43:27 --> 00:43:33
equation with constant
coefficient, because that's the
611
00:43:32 --> 00:43:38
one that most closely models the
conduction and diffusion
612
00:43:37 --> 00:43:43
equations.
So, what I'm interested in,
613
00:43:41 --> 00:43:47
is since this is the,
of them all,
614
00:43:43 --> 00:43:49
probably it's the most
important case is the one where
615
00:43:47 --> 00:43:53
P is a constant because of its
application to that.
616
00:43:50 --> 00:43:56
And, many of the other,
the bank account,
617
00:43:53 --> 00:43:59
for example,
all of those will use a
618
00:43:55 --> 00:44:01
constant coefficient.
So, how is the thing going to
619
00:43:58 --> 00:44:04
look?
Well, I will use the cooling.
620
00:44:01 --> 00:44:07
Let's use the temperature
model, for example.
621
00:44:05 --> 00:44:11
The temperature model,
the equation will be DTDT plus
622
00:44:09 --> 00:44:15
KT is equal to.
Now, notice on the right-hand
623
00:44:13 --> 00:44:19
side, this is a common error.
You don't put TE.
624
00:44:16 --> 00:44:22
You have to put KTE because
that's what the equation says.
625
00:44:21 --> 00:44:27
If you think units,
you won't have any trouble.
626
00:44:25 --> 00:44:31
Units have to be compatible on
both sides of a differential
627
00:44:30 --> 00:44:36
equation.
And therefore,
628
00:44:32 --> 00:44:38
whatever the units were for
capital KT, I'd have to have the
629
00:44:36 --> 00:44:42
same units on the right-hand
side, which indicates I cannot
630
00:44:40 --> 00:44:46
have KT on the left of the
differential equation,
631
00:44:43 --> 00:44:49
and just T on the right,
and expect the units to be
632
00:44:47 --> 00:44:53
compatible.
That's not possible.
633
00:44:49 --> 00:44:55
So, that's a good way of
remembering that if you're
634
00:44:52 --> 00:44:58
modeling temperature or
concentration,
635
00:44:54 --> 00:45:00
you have to have the K on both
sides.
636
00:44:57 --> 00:45:03
OK, let's do,
now, a lot of this we are going
637
00:45:00 --> 00:45:06
to do in our head now because
this is really too easy.
638
00:45:05 --> 00:45:11
What's the integrating factor?
Well, the integrating factor is
639
00:45:10 --> 00:45:16
going to be the integral of K,
the coefficient now is just K.
640
00:45:16 --> 00:45:22
P is a constant,
K, and if I integrate KDT,
641
00:45:20 --> 00:45:26
I get KT, and I exponentiate
that.
642
00:45:23 --> 00:45:29
So, the integrating factor is E
to the KT.
643
00:45:28 --> 00:45:34
I multiply through both sides,
multiply by E to the KT,
644
00:45:34 --> 00:45:40
and what's the resulting
equation?
645
00:45:38 --> 00:45:44
Well, it's going to be ,
I'll write it in the compact
646
00:45:44 --> 00:45:50
form.
It's going to be E to the KT
647
00:45:47 --> 00:45:53
times T, all prime.
The differentiation is now,
648
00:45:53 --> 00:45:59
of course, with respect to the
time.
649
00:45:57 --> 00:46:03
And, that's equal to KTE,
whatever that is,
650
00:46:02 --> 00:46:08
times E to the KT.
This is a function of T,
651
00:46:08 --> 00:46:14
of course, the function of
little time, sorry,
652
00:46:12 --> 00:46:18
little T time.
OK, and now,
653
00:46:15 --> 00:46:21
finally, we are going to
integrate.
654
00:46:18 --> 00:46:24
What's the answer?
Well, it is E to the,
655
00:46:22 --> 00:46:28
so, are we going to get E to
the KT times T is,
656
00:46:27 --> 00:46:33
sorry, K little t,
K times time times the
657
00:46:31 --> 00:46:37
temperature is equal to the
integral of KTE.
658
00:46:37 --> 00:46:43
I'll put the fact that it's a
function of T inside just to
659
00:46:42 --> 00:46:48
remind you, E to the KT,
and now I'll put the arbitrary
660
00:46:46 --> 00:46:52
constant.
Let's put in the arbitrary
661
00:46:49 --> 00:46:55
constant explicitly.
So, what will T be?
662
00:46:53 --> 00:46:59
OK, T will look like this,
finally.
663
00:46:56 --> 00:47:02
It will be E to the negative
KT.
664
00:46:59 --> 00:47:05
That's on the outside.
Then, you will integrate.
665
00:47:04 --> 00:47:10
Of course, the difficulty of
doing this integral depends
666
00:47:08 --> 00:47:14
entirely upon how this external
temperature varies.
667
00:47:12 --> 00:47:18
But anyways,
it's going to be K times that
668
00:47:16 --> 00:47:22
function, which I haven't
specified, E to the KT plus C
669
00:47:20 --> 00:47:26
times E to the negative KT.
Now, some people,
670
00:47:24 --> 00:47:30
many, in fact,
that almost always,
671
00:47:27 --> 00:47:33
in the engineering literature,
almost never write indefinite
672
00:47:32 --> 00:47:38
integrals because an indefinite
integral is indefinite.
673
00:47:38 --> 00:47:44
In other words,
this covers not just one
674
00:47:40 --> 00:47:46
function, but a whole multitude
of functions which differ from
675
00:47:44 --> 00:47:50
each other by an arbitrary
constant.
676
00:47:46 --> 00:47:52
So, in a formula like this,
there's a certain vagueness,
677
00:47:49 --> 00:47:55
and it's further compounded by
the fact that I don't know
678
00:47:53 --> 00:47:59
whether the arbitrary constant
is here.
679
00:47:55 --> 00:48:01
I seem to have put it
explicitly on the outside the
680
00:47:58 --> 00:48:04
way you're used to doing from
calculus.
681
00:48:02 --> 00:48:08
Many people,
therefore, prefer,
682
00:48:04 --> 00:48:10
and I think you should learn
this, to do what is done in the
683
00:48:08 --> 00:48:14
very first section of the notes
called definite integral
684
00:48:13 --> 00:48:19
solutions.
If there's an initial condition
685
00:48:16 --> 00:48:22
saying that the internal
temperature at time zero is some
686
00:48:20 --> 00:48:26
given value, what they like to
do is make this thing definite
687
00:48:25 --> 00:48:31
by integrating here from zero to
T, and making this a dummy
688
00:48:30 --> 00:48:36
variable.
You see, what that does is it
689
00:48:35 --> 00:48:41
gives you a particular function,
whereas, I'm sorry I didn't put
690
00:48:42 --> 00:48:48
in the DT one minus two.
What it does is that when time
691
00:48:48 --> 00:48:54
is zero, all this automatically
disappears, and the arbitrary
692
00:48:55 --> 00:49:01
constant will then be,
it's T.
693
00:49:00 --> 00:49:06
So, in other words,
C times this,
694
00:49:02 --> 00:49:08
which is one,
is that equal to [T?].
695
00:49:05 --> 00:49:11
In other words,
if I make this zero,
696
00:49:07 --> 00:49:13
that I can write C as equal to
this arbitrary starting value.
697
00:49:12 --> 00:49:18
Now, when you do this,
the essential thing,
698
00:49:15 --> 00:49:21
and we're going to come back to
this next week,
699
00:49:18 --> 00:49:24
but right away,
because K is positive,
700
00:49:21 --> 00:49:27
I want to emphasize that so
much at the beginning of the
701
00:49:25 --> 00:49:31
period, I want to conclude by
showing you what its
702
00:49:29 --> 00:49:35
significance is.
This part disappears because K
703
00:49:34 --> 00:49:40
is positive.
The conductivity is positive.
704
00:49:38 --> 00:49:44
This part disappears as T goes
to zero.
705
00:49:41 --> 00:49:47
This goes to zero as T goes to
infinity.
706
00:49:45 --> 00:49:51
So, this is a solution that
remains.
707
00:49:48 --> 00:49:54
This, therefore,
is called the steady state
708
00:49:52 --> 00:49:58
solution, the thing which the
temperature behaves like,
709
00:49:57 --> 00:50:03
as T goes to infinity.
This is called the trangent.
710
00:50:01 --> 00:50:07
because it disappears as T goes
to infinity.
711
00:50:07 --> 00:50:13
It depends on the initial
condition, but it disappears,
712
00:50:11 --> 00:50:17
which shows you,
then, in the long run for this
713
00:50:15 --> 00:50:21
type of problem the initial
condition makes no difference.
714
00:50:20 --> 00:50:26
The function behaves always the
same way as T goes to infinity.