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OK, let's get started.
I'm assuming that,
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A, you went recitation
yesterday, B,
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that even if you didn't,
you know how to separate
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variables, and you know how to
construct simple models,
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solve physical problems with
differential equations,
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and possibly even solve them.
So, you should have learned
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that either in high school,
or 18.01 here,
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or, yeah.
So, I'm going to start from
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that point, assume you know
that.
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I'm not going to tell you what
differential equations are,
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or what modeling is.
If you still are uncertain
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about those things,
the book has a very long and
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good explanation of it.
Just read that stuff.
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So, we are talking about first
order ODEs.
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ODE: I'll only use two
acronyms.
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ODE is ordinary differential
equations.
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I think all of MIT knows that,
whether they've been taking the
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course or not.
So, we are talking about
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first-order ODEs,
which in standard form,
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are written,
you isolate the derivative of y
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with respect to,
x, let's say,
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on the left-hand side,
and on the right-hand side you
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write everything else.
You can't always do this very
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well, but for today,
I'm going to assume that it has
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been done and it's doable.
So, for example,
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some of the ones that will be
considered either today or in
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the problem set are things like
y prime equals x over y.
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That's pretty simple.
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The problem set has y prime
equals, let's see,
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x minus y squared.
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And, it also has y prime equals
y minus x squared.
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There are others,
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too.
Now, when you look at this,
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this, of course,
you can solve by separating
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variables.
So, this is solvable.
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This one is-- and neither of
these can you separate
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variables.
And they look extremely
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similar.
But they are extremely
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dissimilar.
The most dissimilar about them
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is that this one is easily
solvable.
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And you will learn,
if you don't know already,
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next time next Friday how to
solve this one.
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This one, which looks almost
the same, is unsolvable in a
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certain sense.
Namely, there are no elementary
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functions which you can write
down, which will give a solution
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of that differential equation.
So, right away,
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one confronts the most
significant fact that even for
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the simplest possible
differential equations,
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those which only involve the
first derivative,
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it's possible to write down
extremely looking simple guys.
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I'll put this one up in blue to
indicate that it's bad.
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Whoops, sorry,
I mean, not really bad,
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but recalcitrant.
It's not solvable in the
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ordinary sense in which you
think of an equation is
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solvable.
And, since those equations are
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the rule rather than the
exception, I'm going about this
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first day to not solving a
single differential equation,
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but indicating to you what you
do when you meet a blue equation
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like that.
What do you do with it?
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So, this first day is going to
be devoted to geometric ways of
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looking at differential
equations and numerical.
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At the very end,
I'll talk a little bit about
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numerical ways.
And you'll work on both of
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those for the first problem set.
So, what's our geometric view
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of differential equations?
Well, it's something that's
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contrasted with the usual
procedures, by which you solve
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things and find elementary
functions which solve them.
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I'll call that the analytic
method.
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So, on the one hand,
we have the analytic ideas,
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in which you write down
explicitly the equation,
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y prime equals f of x,y.
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And, you look for certain
functions, which are called its
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solutions.
Now, so there's the ODE.
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And, y1 of x,
notice I don't use a separate
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letter.
I don't use g or h or something
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like that for the solution
because the letters multiply so
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quickly, that is,
multiply in the sense of
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rabbits, that after a while,
if you keep using different
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letters for each new idea,
you can't figure out what
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you're talking about.
So, I'll use y1 means,
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it's a solution of this
differential equation.
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Of course, the differential
equation has many solutions
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containing an arbitrary
constant.
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So, we'll call this the
solution.
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Now, the geometric view,
the geometric guy that
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corresponds to this version of
writing the equation,
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is something called a direction
field.
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And, the solution is,
from the geometric point of
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view, something called an
integral curve.
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So, let me explain if you don't
know what the direction field
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is.
I know for some of you,
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00:06:18,000 --> 00:06:24,000
I'm reviewing what you learned
in high school.
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Those of you who had the BC
syllabus in high school should
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know these things.
But, it never hurts to get a
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little more practice.
And, in any event,
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I think the computer stuff that
you will be doing on the problem
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set, a certain amount of it
should be novel to you.
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It was novel to me,
so why not to you?
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So, what's a direction field?
Well, the direction field is,
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you take the plane,
and in each point of the
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plane-- of course,
that's an impossibility.
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But, you pick some points of
the plane.
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You draw what's called a little
line element.
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So, there is a point.
It's a little line,
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and the only thing which
distinguishes it outside of its
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position in the plane,
so here's the point,
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(x,y), at which we are drawing
this line element,
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is its slope.
And, what is its slope?
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Its slope is to be f of x,y.
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And now, You fill up the plane
with these things until you're
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tired of putting then in.
So, I'm going to get tired
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pretty quickly.
So, I don't know,
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let's not make them all go the
same way.
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That sort of seems cheating.
How about here?
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Here's a few randomly chosen
line elements that I put in,
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and I putted the slopes at
random since I didn't have any
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particular differential equation
in mind.
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Now, the integral curve,
so those are the line elements.
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The integral curve is a curve,
which goes through the plane,
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and at every point is tangent
to the line element there.
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So, this is the integral curve.
Hey, wait a minute,
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I thought tangents were the
line element there didn't even
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touch it.
Well, I can't fill up the plane
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with line elements.
Here, at this point,
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there was a line element,
which I didn't bother drawing
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in.
And, it was tangent to that.
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Same thing over here:
if I drew the line element
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here, I would find that the
curve had exactly the right
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slope there.
So, the point is the integral,
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what distinguishes the integral
curve is that everywhere it has
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the direction,
that's the way I'll indicate
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that it's tangent,
has the direction of the field
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everywhere at all points on the
curve, of course,
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where it doesn't go.
It doesn't have any mission to
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fulfill.
Now, I say that this integral
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curve is the graph of the
solution to the differential
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equation.
In other words,
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writing down analytically the
differential equation is the
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same geometrically as drawing
this direction field,
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and solving analytically for a
solution of the differential
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equation is the same thing as
geometrically drawing an
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integral curve.
So, what am I saying?
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I say that an integral curve,
all right, let me write it this
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way.
I'll make a little theorem out
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of it, that y1 of x is
a solution to the differential
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equation if, and only if,
the graph, the curve associated
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with this, the graph of y1 of x
is an integral curve.
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Integral curve of what?
Well, of the direction field
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associated with that equation.
But there isn't quite enough
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room to write that on the board.
But, you could put it in your
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notes, if you take notes.
So, this is the relation
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between the two,
the integral curves of the
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graphs or solutions.
Now, why is that so?
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Well, in fact,
all I have to do to prove this,
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if you can call it a proof at
all, is simply to translate what
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each side really means.
What does it really mean to say
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that a given function is a
solution to the differential
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equation?
Well, it means that if you plug
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it into the differential
equation, it satisfies it.
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Okay, what is that?
So, how do I plug it into the
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differential equation and check
that it satisfies it?
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Well, doing it in the abstract,
I first calculate its
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derivative.
And then, how will it look
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after I plugged it into the
differential equation?
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Well, I don't do anything to
the x, but wherever I see y,
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I plug in this particular
function.
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So, in notation,
that would be written this way.
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So, for this to be a solution
means this, that that equation
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is satisfied.
Okay, what does it mean for the
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graph to be an integral curve?
Well, it means that at each
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point, the slope of this curve,
it means that the slope of y1
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of x should be,
at each point, x1 y1.
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It should be equal to the slope
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of the direction field at that
point.
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And then, what is the slope of
the direction field at that
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point?
Well, it is f of that
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particular, well,
at the point,
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x, y1 of x.
If you like,
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you can put a subscript,
one, on there,
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send a one here or a zero
there, to indicate that you mean
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a particular point.
But, it looks better if you
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don't.
But, there's some possibility
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of confusion.
I admit to that.
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So, the slope of the direction
field, what is that slope?
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Well, by the way,
I calculated the direction
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field.
Its slope at the point was to
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be x, whatever the value of x
was, and whatever the value of
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y1 of x was,
substituted into the right-hand
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side of the equation.
So, what the slope of this
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function of that curve of the
graph should be equal to the
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slope of the direction field.
Now, what does this say?
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00:13:01,000 --> 00:13:07,000
Well, what's the slope of y1 of
x?
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That's y1 prime of x.
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That's from the first day of
18.01, calculus.
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What's the slope of the
direction field?
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This?
Well, it's this.
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And, that's with the right hand
side.
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So, saying these two guys are
the same or equal,
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is exactly, analytically,
the same as saying these two
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00:13:21,000 --> 00:13:27,000
guys are equal.
So, in other words,
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the proof consists of,
what does this really mean?
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00:13:26,000 --> 00:13:32,000
What does this really mean?
And after you see what both
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really mean, you say,
yeah, they're the same.
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So, I don't how to write that.
It's okay: same,
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same, how's that?
This is the same as that.
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Okay, well, this leaves us the
interesting question of how do
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00:13:52,000 --> 00:13:58,000
you draw a direction from the,
well, this being 2003,
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00:13:58,000 --> 00:14:04,000
mostly computers draw them for
you.
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00:14:04,000 --> 00:14:10,000
Nonetheless,
you do have to know a certain
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00:14:07,000 --> 00:14:13,000
amount.
I've given you a couple of
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00:14:09,000 --> 00:14:15,000
exercises where you have to draw
the direction field yourself.
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00:14:14,000 --> 00:14:20,000
This is so you get a feeling
for it, and also because humans
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don't draw direction fields the
same way computers do.
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So, let's first of all,
how did computers do it?
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00:14:27,000 --> 00:14:33,000
They are very stupid.
There's no problem.
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Since they go very fast and
have unlimited amounts of energy
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00:14:37,000 --> 00:14:43,000
to waste, the computer method is
the naive one.
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00:14:42,000 --> 00:14:48,000
You pick the point.
You pick a point,
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and generally,
they are usually equally
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spaced.
You determine some spacing,
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00:14:52,000 --> 00:14:58,000
that one: blah,
blah, blah, blah,
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00:14:55,000 --> 00:15:01,000
blah, blah, blah,
equally spaced.
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And, at each point,
it computes f of x,
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00:15:04,000 --> 00:15:10,000
y at the point,
finds, meets,
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and computes the value of f of
(x, y), that function,
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and the next thing is,
on the screen,
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it draws, at (x,
y), the little line element
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having slope f of x,y.
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In other words,
it does what the differential
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00:15:30,000 --> 00:15:36,000
equation tells it to do.
And the only thing that it does
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is you can, if you are telling
the thing to draw the direction
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00:15:40,000 --> 00:15:46,000
field, about the only option you
have is telling what the spacing
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should be, and sometimes people
don't like to see a whole line.
234
00:15:46,000 --> 00:15:52,000
They only like to see a little
bit of a half line.
235
00:15:49,000 --> 00:15:55,000
And, you can sometimes tell,
according to the program,
236
00:15:52,000 --> 00:15:58,000
tell the computer how long you
want that line to be,
237
00:15:55,000 --> 00:16:01,000
if you want it teeny or a
little bigger.
238
00:15:57,000 --> 00:16:03,000
Once in awhile you want you
want it narrower on it,
239
00:16:00,000 --> 00:16:06,000
but not right now.
Okay, that's what a computer
240
00:16:04,000 --> 00:16:10,000
does.
What does a human do?
241
00:16:05,000 --> 00:16:11,000
This is what it means to be
human.
242
00:16:08,000 --> 00:16:14,000
You use your intelligence.
From a human point of view,
243
00:16:12,000 --> 00:16:18,000
this stuff has been done in the
wrong order.
244
00:16:15,000 --> 00:16:21,000
And the reason it's been done
in the wrong order:
245
00:16:18,000 --> 00:16:24,000
because for each new point,
it requires a recalculation of
246
00:16:22,000 --> 00:16:28,000
f of (x, y).
247
00:16:24,000 --> 00:16:30,000
That is horrible.
The computer doesn't mind,
248
00:16:27,000 --> 00:16:33,000
but a human does.
So, for a human,
249
00:16:31,000 --> 00:16:37,000
the way to do it is not to
begin by picking the point,
250
00:16:35,000 --> 00:16:41,000
but to begin by picking the
slope that you would like to
251
00:16:40,000 --> 00:16:46,000
see.
So, you begin by taking the
252
00:16:42,000 --> 00:16:48,000
slope.
Let's call it the value of the
253
00:16:45,000 --> 00:16:51,000
slope, C.
So, you pick a number.
254
00:16:48,000 --> 00:16:54,000
C is two.
I want to see where are all the
255
00:16:51,000 --> 00:16:57,000
points in the plane where the
slope of that line element would
256
00:16:56,000 --> 00:17:02,000
be two?
Well, they will satisfy an
257
00:16:58,000 --> 00:17:04,000
equation.
The equation is f of (x,
258
00:17:02,000 --> 00:17:08,000
y) equals, in general,
it will be C.
259
00:17:07,000 --> 00:17:13,000
So, what you do is plot this,
plot the equation,
260
00:17:10,000 --> 00:17:16,000
plot this equation.
Notice, it's not the
261
00:17:14,000 --> 00:17:20,000
differential equation.
You can't exactly plot a
262
00:17:17,000 --> 00:17:23,000
differential equation.
It's a curve,
263
00:17:20,000 --> 00:17:26,000
an ordinary curve.
But which curve will depend;
264
00:17:24,000 --> 00:17:30,000
it's, in fact,
from the 18.02 point of view,
265
00:17:28,000 --> 00:17:34,000
the level curve of C,
sorry, it's a level curve of f
266
00:17:32,000 --> 00:17:38,000
of (x, y), the function f of x
and y corresponding to the level
267
00:17:37,000 --> 00:17:43,000
of value C.
But we are not going to call it
268
00:17:42,000 --> 00:17:48,000
that because this is not 18.02.
Instead, we're going to call it
269
00:17:48,000 --> 00:17:54,000
an isocline.
And then, you plot,
270
00:17:51,000 --> 00:17:57,000
well, you've done it.
So, you've got this isocline,
271
00:17:56,000 --> 00:18:02,000
except I'm going to use a
solution curve,
272
00:18:00,000 --> 00:18:06,000
solid lines,
only for integral curves.
273
00:18:03,000 --> 00:18:09,000
When we do plot isoclines,
to indicate that they are not
274
00:18:09,000 --> 00:18:15,000
solutions, we'll use dashed
lines for doing them.
275
00:18:15,000 --> 00:18:21,000
One of the computer things does
and the other one doesn't.
276
00:18:18,000 --> 00:18:24,000
But they use different colors,
also.
277
00:18:20,000 --> 00:18:26,000
There are different ways of
telling you what's an isocline
278
00:18:23,000 --> 00:18:29,000
and what's the solution curve.
So, and what do you do?
279
00:18:26,000 --> 00:18:32,000
So, these are all the points
where the slope is going to be
280
00:18:29,000 --> 00:18:35,000
C.
And now, what you do is draw in
281
00:18:32,000 --> 00:18:38,000
as many as you want of line
elements having slope C.
282
00:18:35,000 --> 00:18:41,000
Notice how efficient that is.
If you want 50 million of them
283
00:18:39,000 --> 00:18:45,000
and have the time,
draw in 50 million.
284
00:18:41,000 --> 00:18:47,000
If two or three are enough,
draw in two or three.
285
00:18:45,000 --> 00:18:51,000
You will be looking at the
picture.
286
00:18:47,000 --> 00:18:53,000
You will see what the curve
looks like, and that will give
287
00:18:51,000 --> 00:18:57,000
you your judgment as to how you
are to do that.
288
00:18:54,000 --> 00:19:00,000
So, in general,
a picture drawn that way,
289
00:18:57,000 --> 00:19:03,000
so let's say,
an isocline corresponding to C
290
00:18:59,000 --> 00:19:05,000
equals zero.
The line elements,
291
00:19:03,000 --> 00:19:09,000
and I think for an isocline,
for the purposes of this
292
00:19:07,000 --> 00:19:13,000
lecture, it would be a good idea
to put isoclines.
293
00:19:10,000 --> 00:19:16,000
Okay, so I'm going to put
solution curves in pink,
294
00:19:14,000 --> 00:19:20,000
or whatever this color is,
and isoclines are going to be
295
00:19:18,000 --> 00:19:24,000
in orange, I guess.
So, isocline,
296
00:19:21,000 --> 00:19:27,000
represented by a dashed line,
and now you will put in the
297
00:19:25,000 --> 00:19:31,000
line elements of,
we'll need lots of chalk for
298
00:19:28,000 --> 00:19:34,000
that.
So, I'll use white chalk.
299
00:19:32,000 --> 00:19:38,000
Y horizontal?
Because according to this the
300
00:19:34,000 --> 00:19:40,000
slope is supposed to be zero
there.
301
00:19:37,000 --> 00:19:43,000
And at the same way,
how about an isocline where the
302
00:19:40,000 --> 00:19:46,000
slope is negative one?
Let's suppose here C is equal
303
00:19:44,000 --> 00:19:50,000
to negative one.
Okay, then it will look like
304
00:19:47,000 --> 00:19:53,000
this.
These are supposed to be lines
305
00:19:49,000 --> 00:19:55,000
of slope negative one.
Don't shoot me if they are not.
306
00:19:53,000 --> 00:19:59,000
So, that's the principle.
So, this is how you will fill
307
00:19:56,000 --> 00:20:02,000
up the plane to draw a direction
field: by plotting the isoclines
308
00:20:01,000 --> 00:20:07,000
first.
And then, once you have the
309
00:20:04,000 --> 00:20:10,000
isoclines there,
you will have line elements.
310
00:20:07,000 --> 00:20:13,000
And you can draw a direction
field.
311
00:20:09,000 --> 00:20:15,000
Okay, so, for the next few
minutes, I'd like to work a
312
00:20:12,000 --> 00:20:18,000
couple of examples for you to
show how this works out in
313
00:20:15,000 --> 00:20:21,000
practice.
314
00:20:34,000 --> 00:20:40,000
So, the first equation is going
to be y prime equals minus x
315
00:20:45,000 --> 00:20:51,000
over y.
Okay, first thing,
316
00:20:53,000 --> 00:20:59,000
what are the isoclines?
Well, the isoclines are going
317
00:21:03,000 --> 00:21:09,000
to be y.
Well, negative x over y is
318
00:21:08,000 --> 00:21:14,000
equal to C.
Maybe I better make two steps
319
00:21:12,000 --> 00:21:18,000
out of this.
Minus x over y is equal to C.
320
00:21:16,000 --> 00:21:22,000
But, of course,
nobody draws a curve in that
321
00:21:19,000 --> 00:21:25,000
form.
You'll want it in the form y
322
00:21:22,000 --> 00:21:28,000
equals minus one over
C times x.
323
00:21:26,000 --> 00:21:32,000
So, there's our isocline.
Why don't I put that up in
324
00:21:32,000 --> 00:21:38,000
orange since it's going to be,
that's the color I'll draw it
325
00:21:36,000 --> 00:21:42,000
in.
In other words,
326
00:21:38,000 --> 00:21:44,000
for different values of C,
now this thing is aligned.
327
00:21:42,000 --> 00:21:48,000
It's aligned,
in fact, through the origin.
328
00:21:45,000 --> 00:21:51,000
This looks pretty simple.
Okay, so here's our plane.
329
00:21:50,000 --> 00:21:56,000
The isoclines are going to be
lines through the origin.
330
00:21:54,000 --> 00:22:00,000
And now, let's put them in,
suppose, for example,
331
00:21:58,000 --> 00:22:04,000
C is equal to one.
Well, if C is equal to one,
332
00:22:06,000 --> 00:22:12,000
then it's the line,
y equals minus x.
333
00:22:14,000 --> 00:22:20,000
So, this is the isocline.
I'll put, down here,
334
00:22:23,000 --> 00:22:29,000
C equals minus one.
And, along it,
335
00:22:30,000 --> 00:22:36,000
no, something's wrong.
I'm sorry?
336
00:22:38,000 --> 00:22:44,000
C is one, not negative one,
right, thanks.
337
00:22:42,000 --> 00:22:48,000
Thanks.
So, C equals one.
338
00:22:44,000 --> 00:22:50,000
So, it should be little line
segments of slope one will be
339
00:22:50,000 --> 00:22:56,000
the line elements,
things of slope one.
340
00:22:54,000 --> 00:23:00,000
OK, now how about C equals
negative one?
341
00:23:00,000 --> 00:23:06,000
If C equals negative one,
then it's the line,
342
00:23:03,000 --> 00:23:09,000
y equals x.
And so, that's the isocline.
343
00:23:07,000 --> 00:23:13,000
Notice, still dash because
these are isoclines.
344
00:23:11,000 --> 00:23:17,000
Here, C is negative one.
And so, the slope elements look
345
00:23:15,000 --> 00:23:21,000
like this.
Notice, they are perpendicular.
346
00:23:19,000 --> 00:23:25,000
Now, notice that they are
always going to be perpendicular
347
00:23:23,000 --> 00:23:29,000
to the line because the slope of
this line is minus one over C.
348
00:23:30,000 --> 00:23:36,000
But, the slope of the line
element is going to be C.
349
00:23:33,000 --> 00:23:39,000
Those numbers,
minus one over C and C,
350
00:23:36,000 --> 00:23:42,000
are negative reciprocals.
And, you know that two lines
351
00:23:40,000 --> 00:23:46,000
whose slopes are negative
reciprocals are perpendicular.
352
00:23:44,000 --> 00:23:50,000
So, the line elements are going
to be perpendicular to these.
353
00:23:49,000 --> 00:23:55,000
And therefore,
I hardly even have to bother
354
00:23:52,000 --> 00:23:58,000
calculating, doing any more
calculation.
355
00:23:55,000 --> 00:24:01,000
Here's going to be a,
well, how about this one?
356
00:24:00,000 --> 00:24:06,000
Here's a controversial
isocline.
357
00:24:02,000 --> 00:24:08,000
Is that an isocline?
Well, wait a minute.
358
00:24:05,000 --> 00:24:11,000
That doesn't correspond to
anything looking like this.
359
00:24:10,000 --> 00:24:16,000
Ah-ha, but it would if I put C
multiplied through by C.
360
00:24:14,000 --> 00:24:20,000
And then, it would correspond
to C being zero.
361
00:24:18,000 --> 00:24:24,000
In other words,
don't write it like this.
362
00:24:21,000 --> 00:24:27,000
Multiply through by C.
It will read C y equals
363
00:24:25,000 --> 00:24:31,000
negative x.
And then, when C is zero,
364
00:24:29,000 --> 00:24:35,000
I have x equals zero,
which is exactly the y-axis.
365
00:24:35,000 --> 00:24:41,000
So, that really is included.
How about the x-axis?
366
00:24:38,000 --> 00:24:44,000
Well, the x-axis is not
included.
367
00:24:40,000 --> 00:24:46,000
However, most people include it
anyway.
368
00:24:43,000 --> 00:24:49,000
This is very common to be a
sort of sloppy and bending the
369
00:24:47,000 --> 00:24:53,000
edges of corners a little bit,
and hoping nobody will notice.
370
00:24:51,000 --> 00:24:57,000
We'll say that corresponds to C
equals infinity.
371
00:24:55,000 --> 00:25:01,000
I hope nobody wants to fight
about that.
372
00:24:58,000 --> 00:25:04,000
If you do, go fight with
somebody else.
373
00:25:02,000 --> 00:25:08,000
So, if C is infinity,
that means the little line
374
00:25:05,000 --> 00:25:11,000
segment should have infinite
slope, and by common consent,
375
00:25:10,000 --> 00:25:16,000
that means it should be
vertical.
376
00:25:12,000 --> 00:25:18,000
And so, we can even count this
as sort of an isocline.
377
00:25:17,000 --> 00:25:23,000
And, I'll make the dashes
smaller, indicate it has a lower
378
00:25:21,000 --> 00:25:27,000
status than the others.
And, I'll put this in,
379
00:25:25,000 --> 00:25:31,000
do this weaselly thing of
putting it in quotation marks to
380
00:25:29,000 --> 00:25:35,000
indicate that I'm not
responsible for it.
381
00:25:34,000 --> 00:25:40,000
Okay, now, we now have to put
it the integral curves.
382
00:25:39,000 --> 00:25:45,000
Well, nothing could be easier.
I'm looking for curves which
383
00:25:45,000 --> 00:25:51,000
are everywhere perpendicular to
these rays.
384
00:25:50,000 --> 00:25:56,000
Well, you know from geometry
that those are circles.
385
00:25:55,000 --> 00:26:01,000
So, the integral curves are
circles.
386
00:26:00,000 --> 00:26:06,000
And, it's an elementary
exercise, which I would not
387
00:26:04,000 --> 00:26:10,000
deprive you of the pleasure of.
Solve the ODE by separation of
388
00:26:08,000 --> 00:26:14,000
variables.
In other words,
389
00:26:10,000 --> 00:26:16,000
we've gotten the,
so the circles are ones with a
390
00:26:14,000 --> 00:26:20,000
center at the origin,
of course, equal some constant.
391
00:26:18,000 --> 00:26:24,000
I'll call it C1,
so it's not confused with this
392
00:26:22,000 --> 00:26:28,000
C.
They look like that,
393
00:26:24,000 --> 00:26:30,000
and now you should solve this
by separating variables,
394
00:26:28,000 --> 00:26:34,000
and just confirm that the
solutions are,
395
00:26:31,000 --> 00:26:37,000
in fact, those circles.
One interesting thing,
396
00:26:36,000 --> 00:26:42,000
and so I confirm this,
I won't do it because I want to
397
00:26:40,000 --> 00:26:46,000
do geometric and numerical
things today.
398
00:26:42,000 --> 00:26:48,000
So, if you solve it by
separating variables,
399
00:26:45,000 --> 00:26:51,000
one interesting thing to note
is that if I write the solution
400
00:26:49,000 --> 00:26:55,000
as y equals y1 of x, well,
401
00:26:52,000 --> 00:26:58,000
it'll look something like the
square root of C1 minus,
402
00:26:56,000 --> 00:27:02,000
let's make this squared because
that's the way people usually
403
00:27:00,000 --> 00:27:06,000
put the radius,
minus x squared.
404
00:27:03,000 --> 00:27:09,000
And so, a solution,
405
00:27:06,000 --> 00:27:12,000
a typical solution looks like
this.
406
00:27:09,000 --> 00:27:15,000
Well, what's the solution over
here?
407
00:27:11,000 --> 00:27:17,000
Well, that one solution will be
goes from here to here.
408
00:27:15,000 --> 00:27:21,000
If you like,
it has a negative side to it.
409
00:27:18,000 --> 00:27:24,000
So, I'll make,
let's say, plus.
410
00:27:21,000 --> 00:27:27,000
There's another solution,
which has a negative value.
411
00:27:25,000 --> 00:27:31,000
But let's use the one with the
positive value of the square
412
00:27:29,000 --> 00:27:35,000
root.
My point is this,
413
00:27:32,000 --> 00:27:38,000
that that solution,
the domain of that solution,
414
00:27:35,000 --> 00:27:41,000
really only goes from here to
here.
415
00:27:38,000 --> 00:27:44,000
It's not the whole x-axis.
It's just a limited piece of
416
00:27:42,000 --> 00:27:48,000
the x-axis where that solution
is defined.
417
00:27:45,000 --> 00:27:51,000
There's no way of extending it
further.
418
00:27:48,000 --> 00:27:54,000
And, there's no way of
predicting, by looking at the
419
00:27:52,000 --> 00:27:58,000
differential equation,
that a typical solution was
420
00:27:56,000 --> 00:28:02,000
going to have a limited domain
like that.
421
00:28:01,000 --> 00:28:07,000
In other words,
you could find a solution,
422
00:28:04,000 --> 00:28:10,000
but how far out is it going to
go?
423
00:28:07,000 --> 00:28:13,000
Sometimes, it's impossible to
tell, except by either finding
424
00:28:12,000 --> 00:28:18,000
it explicitly,
or by asking a computer to draw
425
00:28:16,000 --> 00:28:22,000
a picture of it,
and seeing if that gives you
426
00:28:19,000 --> 00:28:25,000
some insight.
It's one of the many
427
00:28:22,000 --> 00:28:28,000
difficulties in handling
differential equations.
428
00:28:26,000 --> 00:28:32,000
You don't know what the domain
of a solution is going to be
429
00:28:31,000 --> 00:28:37,000
until you've actually calculated
it.
430
00:28:36,000 --> 00:28:42,000
Now, a slightly more
complicated example is going to
431
00:28:40,000 --> 00:28:46,000
be, let's see, y prime
equals one plus x minus y.
432
00:28:43,000 --> 00:28:49,000
It's not a lot more
433
00:28:46,000 --> 00:28:52,000
complicated, and as a computer
exercise, you will work with,
434
00:28:51,000 --> 00:28:57,000
still, more complicated ones.
But here, the isoclines would
435
00:28:56,000 --> 00:29:02,000
be what?
Well, I set that equal to C.
436
00:29:00,000 --> 00:29:06,000
Can you do the algebra in your
head?
437
00:29:02,000 --> 00:29:08,000
An isocline will have the
equation: this equals C.
438
00:29:07,000 --> 00:29:13,000
So, I'm going to put the y on
the right hand side,
439
00:29:11,000 --> 00:29:17,000
and that C on the left hand
side.
440
00:29:13,000 --> 00:29:19,000
So, it will have the equation y
equals one plus x minus C,
441
00:29:19,000 --> 00:29:25,000
or a nicer way to
write it would be x plus one
442
00:29:23,000 --> 00:29:29,000
minus C.
I guess it really doesn't
443
00:29:28,000 --> 00:29:34,000
matter.
So there's the equation of the
444
00:29:31,000 --> 00:29:37,000
isocline.
Let's quickly draw the
445
00:29:34,000 --> 00:29:40,000
direction field.
And notice, by the way,
446
00:29:36,000 --> 00:29:42,000
it's a simple equation,
but you cannot separate
447
00:29:39,000 --> 00:29:45,000
variables.
So, I will not,
448
00:29:41,000 --> 00:29:47,000
today at any rate,
be able to check the answer.
449
00:29:44,000 --> 00:29:50,000
I will not be able to get an
analytic answer.
450
00:29:47,000 --> 00:29:53,000
All we'll be able to do now is
get a geometric answer.
451
00:29:50,000 --> 00:29:56,000
But notice how quickly,
relatively quickly,
452
00:29:53,000 --> 00:29:59,000
one can get it.
So, I'm feeling for how the
453
00:29:56,000 --> 00:30:02,000
solutions behave to this
equation.
454
00:30:00,000 --> 00:30:06,000
All right, let's see,
what should we plot first?
455
00:30:05,000 --> 00:30:11,000
I like C equals one,
no, don't do C equals one.
456
00:30:10,000 --> 00:30:16,000
Let's do C equals zero,
first.
457
00:30:13,000 --> 00:30:19,000
C equals zero.
That's the line.
458
00:30:16,000 --> 00:30:22,000
y equals x plus 1.
459
00:30:19,000 --> 00:30:25,000
Okay, let me run and get that
chalk.
460
00:30:23,000 --> 00:30:29,000
So, I'll isoclines are in
orange.
461
00:30:27,000 --> 00:30:33,000
If so, when C equals zero,
y equals x plus one.
462
00:30:32,000 --> 00:30:38,000
So, let's say it's this curve.
C equals zero.
463
00:30:38,000 --> 00:30:44,000
How about C equals negative
one?
464
00:30:42,000 --> 00:30:48,000
Then it's y equals x plus two.
465
00:30:47,000 --> 00:30:53,000
It's this curve.
Well, let's label it down here.
466
00:30:53,000 --> 00:30:59,000
So, this is C equals negative
one.
467
00:30:57,000 --> 00:31:03,000
C equals negative two would be
y equals x, no,
468
00:31:02,000 --> 00:31:08,000
what am I doing?
C equals negative one is y
469
00:31:08,000 --> 00:31:14,000
equals x plus two.
That's right.
470
00:31:12,000 --> 00:31:18,000
Well, how about the other side?
If C equals plus one,
471
00:31:16,000 --> 00:31:22,000
well, then it's going to go
through the origin.
472
00:31:20,000 --> 00:31:26,000
It looks like a little more
room down here.
473
00:31:24,000 --> 00:31:30,000
How about, so if this is going
to be C equals one,
474
00:31:28,000 --> 00:31:34,000
then I sort of get the idea.
C equals two will look like
475
00:31:34,000 --> 00:31:40,000
this.
They're all going to be
476
00:31:37,000 --> 00:31:43,000
parallel lines because all
that's changing is the
477
00:31:42,000 --> 00:31:48,000
y-intercept, as I do this thing.
So, here, it's C equals two.
478
00:31:47,000 --> 00:31:53,000
That's probably enough.
All right, let's put it in the
479
00:31:53,000 --> 00:31:59,000
line elements.
All right, C equals negative
480
00:31:57,000 --> 00:32:03,000
one.
These will be perpendicular.
481
00:32:00,000 --> 00:32:06,000
C equals zero,
like this.
482
00:32:04,000 --> 00:32:10,000
C equals one.
Oh, this is interesting.
483
00:32:06,000 --> 00:32:12,000
I can't even draw in the line
elements because they seem to
484
00:32:10,000 --> 00:32:16,000
coincide with the curve itself,
with the line itself.
485
00:32:14,000 --> 00:32:20,000
They write y along the line,
and that makes it hard to draw
486
00:32:18,000 --> 00:32:24,000
them in.
How about C equals two?
487
00:32:20,000 --> 00:32:26,000
Well, here, the line elements
will be slanty.
488
00:32:23,000 --> 00:32:29,000
They'll have slope two,
so a pretty slanty up.
489
00:32:26,000 --> 00:32:32,000
And, I can see if a C equals
three in the same way.
490
00:32:31,000 --> 00:32:37,000
There are going to be even more
slantier up.
491
00:32:34,000 --> 00:32:40,000
And here, they're going to be
even more slanty down.
492
00:32:37,000 --> 00:32:43,000
This is not very scientific
terminology or mathematical,
493
00:32:41,000 --> 00:32:47,000
but you get the idea.
Okay, so there's our quick
494
00:32:45,000 --> 00:32:51,000
version of the direction field.
All we have to do is put in
495
00:32:49,000 --> 00:32:55,000
some integral curves now.
Well, it looks like it's doing
496
00:32:53,000 --> 00:32:59,000
this.
It gets less slanty here.
497
00:32:55,000 --> 00:33:01,000
It levels out,
has slope zero.
498
00:32:59,000 --> 00:33:05,000
And now, in this part of the
plain, the slope seems to be
499
00:33:03,000 --> 00:33:09,000
rising.
So, it must do something like
500
00:33:06,000 --> 00:33:12,000
that.
This guy must do something like
501
00:33:08,000 --> 00:33:14,000
this.
I'm a little doubtful of what I
502
00:33:11,000 --> 00:33:17,000
should be doing here.
Or, how about going from the
503
00:33:15,000 --> 00:33:21,000
other side?
Well, it rises,
504
00:33:17,000 --> 00:33:23,000
gets a little,
should it cross this?
505
00:33:20,000 --> 00:33:26,000
What should I do?
Well, there's one integral
506
00:33:23,000 --> 00:33:29,000
curve, which is easy to see.
It's this one.
507
00:33:26,000 --> 00:33:32,000
This line is both an isocline
and an integral curve.
508
00:33:32,000 --> 00:33:38,000
It's everything,
except drawable,
509
00:33:35,000 --> 00:33:41,000
[LAUGHTER] so,
you understand this is the same
510
00:33:41,000 --> 00:33:47,000
line.
It's both orange and pink at
511
00:33:45,000 --> 00:33:51,000
the same time.
But I don't know what
512
00:33:49,000 --> 00:33:55,000
combination color that would
make.
513
00:33:53,000 --> 00:33:59,000
It doesn't look like a line,
but be sympathetic.
514
00:34:00,000 --> 00:34:06,000
Now, the question is,
what's happening in this
515
00:34:04,000 --> 00:34:10,000
corridor?
In the corridor,
516
00:34:06,000 --> 00:34:12,000
that's not a mathematical word
either, between the isoclines
517
00:34:12,000 --> 00:34:18,000
for, well, what are they?
They are the isoclines for C
518
00:34:18,000 --> 00:34:24,000
equals two, and C equals zero.
How does that corridor look?
519
00:34:23,000 --> 00:34:29,000
Well: something like this.
Over here, the lines all look
520
00:34:29,000 --> 00:34:35,000
like that.
And here, they all look like
521
00:34:33,000 --> 00:34:39,000
this.
The slope is two.
522
00:34:36,000 --> 00:34:42,000
And, a hapless solution gets in
there.
523
00:34:39,000 --> 00:34:45,000
What's it to do?
Well, do you see that if a
524
00:34:43,000 --> 00:34:49,000
solution gets in that corridor,
an integral curve gets in that
525
00:34:49,000 --> 00:34:55,000
corridor, no escape is possible.
It's like a lobster trap.
526
00:34:54,000 --> 00:35:00,000
The lobster can walk in.
But it cannot walk out because
527
00:34:58,000 --> 00:35:04,000
things are always going in.
How could it escape?
528
00:35:03,000 --> 00:35:09,000
Well, it would have to double
back, somehow,
529
00:35:06,000 --> 00:35:12,000
and remember,
to escape, it has to be,
530
00:35:10,000 --> 00:35:16,000
to escape on the left side,
it must be going horizontally.
531
00:35:17,000 --> 00:35:23,000
But, how could it do that
without doubling back first and
532
00:35:20,000 --> 00:35:26,000
having the wrong slope?
The slope of everything in this
533
00:35:24,000 --> 00:35:30,000
corridor is positive,
and to double back and escape,
534
00:35:28,000 --> 00:35:34,000
it would at some point have to
have negative slope.
535
00:35:32,000 --> 00:35:38,000
It can't do that.
Well, could it escape on the
536
00:35:35,000 --> 00:35:41,000
right-hand side?
No, because at the moment when
537
00:35:39,000 --> 00:35:45,000
it wants to cross,
it will have to have a slope
538
00:35:42,000 --> 00:35:48,000
less than this line.
But all these spiky guys are
539
00:35:46,000 --> 00:35:52,000
pointing; it can't escape that
way either.
540
00:35:50,000 --> 00:35:56,000
So, no escape is possible.
It has to continue on,
541
00:35:53,000 --> 00:35:59,000
there.
But, more than that is true.
542
00:35:56,000 --> 00:36:02,000
So, a solution can't escape.
Once it's in there,
543
00:36:01,000 --> 00:36:07,000
it can't escape.
It's like, what do they call
544
00:36:04,000 --> 00:36:10,000
those plants,
I forget, pitcher plants.
545
00:36:07,000 --> 00:36:13,000
All they hear is they are going
down.
546
00:36:10,000 --> 00:36:16,000
So, it looks like that.
And so, the poor little insect
547
00:36:14,000 --> 00:36:20,000
falls in.
They could climb up the walls
548
00:36:17,000 --> 00:36:23,000
except that all the hairs are
going the wrong direction,
549
00:36:22,000 --> 00:36:28,000
and it can't get over them.
Well, let's think of it that
550
00:36:26,000 --> 00:36:32,000
way: this poor trap solution.
So, it does what it has to do.
551
00:36:32,000 --> 00:36:38,000
Now, there's more to it than
that.
552
00:36:35,000 --> 00:36:41,000
Because there are two
principles involved here that
553
00:36:39,000 --> 00:36:45,000
you should know,
that help a lot in drawing
554
00:36:43,000 --> 00:36:49,000
these pictures.
Principle number one is that
555
00:36:46,000 --> 00:36:52,000
two integral curves cannot cross
at an angle.
556
00:36:50,000 --> 00:36:56,000
Two integral curves can't
cross, I mean,
557
00:36:53,000 --> 00:36:59,000
by crossing at an angle like
that.
558
00:36:56,000 --> 00:37:02,000
I'll indicate what I mean by a
picture like that.
559
00:37:02,000 --> 00:37:08,000
Now, why not?
This is an important principle.
560
00:37:05,000 --> 00:37:11,000
Let's put that up in the white
box.
561
00:37:08,000 --> 00:37:14,000
They can't cross because if two
integral curves,
562
00:37:12,000 --> 00:37:18,000
are trying to cross,
well, one will look like this.
563
00:37:16,000 --> 00:37:22,000
It's an integral curve because
it has this slope.
564
00:37:20,000 --> 00:37:26,000
And, the other integral curve
has this slope.
565
00:37:24,000 --> 00:37:30,000
And now, they fight with each
other.
566
00:37:27,000 --> 00:37:33,000
What is the true slope at that
point?
567
00:37:32,000 --> 00:37:38,000
Well, the direction field only
allows you to have one slope.
568
00:37:36,000 --> 00:37:42,000
If there's a line element at
that point, it has a definite
569
00:37:40,000 --> 00:37:46,000
slope.
And therefore,
570
00:37:41,000 --> 00:37:47,000
it cannot have both the slope
and that one.
571
00:37:44,000 --> 00:37:50,000
It's as simple as that.
So, the reason is you can't
572
00:37:48,000 --> 00:37:54,000
have two slopes.
The direction field doesn't
573
00:37:51,000 --> 00:37:57,000
allow it.
Well, that's a big,
574
00:37:53,000 --> 00:37:59,000
big help because if I know,
here's an integral curve,
575
00:37:57,000 --> 00:38:03,000
and if I know that none of
these other pink integral curves
576
00:38:01,000 --> 00:38:07,000
are allowed to cross it,
how else can I do it?
577
00:38:06,000 --> 00:38:12,000
Well, they can't escape.
They can't cross.
578
00:38:09,000 --> 00:38:15,000
It's sort of clear that they
must get closer and closer to
579
00:38:13,000 --> 00:38:19,000
it.
You know, I'd have to work a
580
00:38:16,000 --> 00:38:22,000
little to justify that.
But I think that nobody would
581
00:38:20,000 --> 00:38:26,000
have any doubt of it who did a
little experimentation.
582
00:38:24,000 --> 00:38:30,000
In other words,
all these curves joined that
583
00:38:28,000 --> 00:38:34,000
little tube and get closer and
closer to this line,
584
00:38:32,000 --> 00:38:38,000
y equals x.
And there, without solving the
585
00:38:37,000 --> 00:38:43,000
differential equation,
it's clear that all of these
586
00:38:42,000 --> 00:38:48,000
solutions, how do they behave?
As x goes to infinity,
587
00:38:47,000 --> 00:38:53,000
they become asymptotic to,
they become closer and closer
588
00:38:52,000 --> 00:38:58,000
to the solution,
x.
589
00:38:54,000 --> 00:39:00,000
Is x a solution?
Yeah, because y equals x is an
590
00:38:58,000 --> 00:39:04,000
integral curve.
Is x a solution?
591
00:39:02,000 --> 00:39:08,000
Yeah, because if I plug in y
equals x, I get what?
592
00:39:07,000 --> 00:39:13,000
On the right-hand side,
I get one.
593
00:39:10,000 --> 00:39:16,000
And on the left-hand side,
I get one.
594
00:39:14,000 --> 00:39:20,000
One equals one.
So, this is a solution.
595
00:39:18,000 --> 00:39:24,000
Let's indicate that it's a
solution.
596
00:39:21,000 --> 00:39:27,000
So, analytically,
we've discovered an analytic
597
00:39:26,000 --> 00:39:32,000
solution to the differential
equation, namely,
598
00:39:31,000 --> 00:39:37,000
Y equals X, just by this
geometric process.
599
00:39:37,000 --> 00:39:43,000
Now, there's one more principle
like that, which is less
600
00:39:41,000 --> 00:39:47,000
obvious.
But you do have to know it.
601
00:39:44,000 --> 00:39:50,000
So, you are not allowed to
cross.
602
00:39:46,000 --> 00:39:52,000
That's clear.
But it's much,
603
00:39:49,000 --> 00:39:55,000
much, much, much,
much less obvious that two
604
00:39:52,000 --> 00:39:58,000
integral curves cannot touch.
That is, they cannot even be
605
00:39:57,000 --> 00:40:03,000
tangent.
Two integral curves cannot be
606
00:40:00,000 --> 00:40:06,000
tangent.
607
00:40:10,000 --> 00:40:16,000
I'll indicate that by the word
touch, which is what a lot of
608
00:40:19,000 --> 00:40:25,000
people say.
In other words,
609
00:40:23,000 --> 00:40:29,000
if this is illegal,
so is this.
610
00:40:28,000 --> 00:40:34,000
It can't happen.
You know, without that,
611
00:40:33,000 --> 00:40:39,000
for example,
it might be,
612
00:40:35,000 --> 00:40:41,000
I might feel that there would
be nothing in this to prevent
613
00:40:39,000 --> 00:40:45,000
those curves from joining.
Why couldn't these pink curves
614
00:40:43,000 --> 00:40:49,000
join the line,
y equals x?
615
00:40:45,000 --> 00:40:51,000
You know, it's a solution.
They just pitch a ride,
616
00:40:49,000 --> 00:40:55,000
as it were.
The answer is they cannot do
617
00:40:52,000 --> 00:40:58,000
that because they have to just
get asymptotic to it,
618
00:40:55,000 --> 00:41:01,000
ever, ever closer.
They can't join y equals x
619
00:40:59,000 --> 00:41:05,000
because at the point where they
join, you have that situation.
620
00:41:05,000 --> 00:41:11,000
Now, why can't you to have
this?
621
00:41:09,000 --> 00:41:15,000
That's much more sophisticated
than this, and the reason is
622
00:41:17,000 --> 00:41:23,000
because of something called the
Existence and Uniqueness
623
00:41:24,000 --> 00:41:30,000
Theorem, which says that there
is through a point,
624
00:41:31,000 --> 00:41:37,000
x zero y zero,
that y prime equals f of
625
00:41:38,000 --> 00:41:44,000
(x, y) has only one,
626
00:41:43,000 --> 00:41:49,000
and only one solution.
One has one solution.
627
00:41:49,000 --> 00:41:55,000
In mathematics speak,
that means at least one
628
00:41:53,000 --> 00:41:59,000
solution.
It doesn't mean it has just one
629
00:41:56,000 --> 00:42:02,000
solution.
That's mathematical convention.
630
00:41:59,000 --> 00:42:05,000
It has one solution,
at least one solution.
631
00:42:02,000 --> 00:42:08,000
But, the killer is,
only one solution.
632
00:42:06,000 --> 00:42:12,000
That's what you have to say in
mathematics if you want just
633
00:42:10,000 --> 00:42:16,000
one, one, and only one solution
through the point
634
00:42:15,000 --> 00:42:21,000
x zero y zero.
So, the fact that it has one,
635
00:42:18,000 --> 00:42:24,000
that is the existence part.
The fact that it has only one
636
00:42:23,000 --> 00:42:29,000
is the uniqueness part of the
theorem.
637
00:42:26,000 --> 00:42:32,000
Now, like all good mathematical
theorems, this one does have
638
00:42:31,000 --> 00:42:37,000
hypotheses.
So, this is not going to be a
639
00:42:35,000 --> 00:42:41,000
course, I warn you,
those of you who are
640
00:42:39,000 --> 00:42:45,000
theoretically inclined,
very rich in hypotheses.
641
00:42:44,000 --> 00:42:50,000
But, hypotheses for those one
or that f of (x,
642
00:42:48,000 --> 00:42:54,000
y) should be a
continuous function.
643
00:42:52,000 --> 00:42:58,000
Now, like polynomial,
signs, should be continuous
644
00:42:57,000 --> 00:43:03,000
near, in the vicinity of that
point.
645
00:43:02,000 --> 00:43:08,000
That guarantees existence,
and what guarantees uniqueness
646
00:43:08,000 --> 00:43:14,000
is the hypothesis that you would
not guess by yourself.
647
00:43:14,000 --> 00:43:20,000
Neither would I.
What guarantees the uniqueness
648
00:43:19,000 --> 00:43:25,000
is that also,
it's partial derivative with
649
00:43:24,000 --> 00:43:30,000
respect to y should be
continuous, should be continuous
650
00:43:30,000 --> 00:43:36,000
near x zero y zero.
651
00:43:35,000 --> 00:43:41,000
Well, I have to make a
decision.
652
00:43:38,000 --> 00:43:44,000
I don't have time to talk about
Euler's method.
653
00:43:43,000 --> 00:43:49,000
I'll refer you to the,
there's one page of notes,
654
00:43:49,000 --> 00:43:55,000
and I couldn't do any more than
just repeat what's on those
655
00:43:55,000 --> 00:44:01,000
notes.
So, I'll trust you to read
656
00:43:59,000 --> 00:44:05,000
that.
And instead,
657
00:44:02,000 --> 00:44:08,000
let me give you an example
which will solidify these things
658
00:44:09,000 --> 00:44:15,000
in your mind a little bit.
I think that's a better course.
659
00:44:17,000 --> 00:44:23,000
The example is not in your
notes, and therefore,
660
00:44:22,000 --> 00:44:28,000
remember, you heard it here
first.
661
00:44:27,000 --> 00:44:33,000
Okay, so what's the example?
So, there is that differential
662
00:44:34,000 --> 00:44:40,000
equation.
Now, let's just solve it by
663
00:44:38,000 --> 00:44:44,000
separating variables.
Can you do it in your head?
664
00:44:42,000 --> 00:44:48,000
dy over dx, put all the y's on
the left.
665
00:44:44,000 --> 00:44:50,000
It will look like dy over one
minus y.
666
00:44:48,000 --> 00:44:54,000
Put all the dx's on the left.
So, the dx here goes on the
667
00:44:52,000 --> 00:44:58,000
right, rather.
That will be dx.
668
00:44:54,000 --> 00:45:00,000
And then, the x goes down into
the denominator.
669
00:44:57,000 --> 00:45:03,000
So now, it looks like that.
And, if I integrate both sides,
670
00:45:03,000 --> 00:45:09,000
I get the log of one minus y,
I guess, maybe with a,
671
00:45:08,000 --> 00:45:14,000
I never bothered with that,
but you can.
672
00:45:12,000 --> 00:45:18,000
It should be absolute values.
All right, put an absolute
673
00:45:17,000 --> 00:45:23,000
value, plus a constant.
And now, if I exponentiate both
674
00:45:23,000 --> 00:45:29,000
sides, the constant is positive.
So, this is going to look like
675
00:45:29,000 --> 00:45:35,000
y.
One minus y equals x
676
00:45:33,000 --> 00:45:39,000
And, the constant will be e to
677
00:45:36,000 --> 00:45:42,000
the C1.
And, I'll just make that a new
678
00:45:39,000 --> 00:45:45,000
constant, Cx.
And now, by letting C be
679
00:45:42,000 --> 00:45:48,000
negative, that's why you can get
rid of the absolute values,
680
00:45:45,000 --> 00:45:51,000
if you allow C to have negative
values as well as positive
681
00:45:49,000 --> 00:45:55,000
values.
Let's write this in a more
682
00:45:51,000 --> 00:45:57,000
human form.
So, y is equal to one minus Cx.
683
00:45:53,000 --> 00:45:59,000
Good, all right,
684
00:45:55,000 --> 00:46:01,000
let's just plot those.
So, these are the solutions.
685
00:46:00,000 --> 00:46:06,000
It's a pretty easy equation,
pretty easy solution method,
686
00:46:05,000 --> 00:46:11,000
just separation of variables.
What do they look like?
687
00:46:11,000 --> 00:46:17,000
Well, these are all lines whose
intercept is at one.
688
00:46:16,000 --> 00:46:22,000
And, they have any slope
whatsoever.
689
00:46:19,000 --> 00:46:25,000
So, these are the lines that
look like that.
690
00:46:24,000 --> 00:46:30,000
Okay, now let me ask,
existence and uniqueness.
691
00:46:29,000 --> 00:46:35,000
Existence: through which points
of the plane does the solution
692
00:46:35,000 --> 00:46:41,000
go?
Answer: through every point of
693
00:46:39,000 --> 00:46:45,000
the plane, through any point
here, I can find one and only
694
00:46:44,000 --> 00:46:50,000
one of those lines,
except for these stupid guys
695
00:46:48,000 --> 00:46:54,000
here on the stalk of the flower.
Here, for each of these points,
696
00:46:53,000 --> 00:46:59,000
there is no existence.
There is no solution to this
697
00:46:57,000 --> 00:47:03,000
differential equation,
which goes through any of these
698
00:47:02,000 --> 00:47:08,000
wiggly points on the y-axis,
with one exception.
699
00:47:07,000 --> 00:47:13,000
This point is oversupplied.
At this point,
700
00:47:10,000 --> 00:47:16,000
it's not existence that fails.
It's uniqueness that fails:
701
00:47:14,000 --> 00:47:20,000
no uniqueness.
There are lots of things which
702
00:47:18,000 --> 00:47:24,000
go through here.
Now, is that a violation of the
703
00:47:21,000 --> 00:47:27,000
existence and uniqueness
theorem?
704
00:47:24,000 --> 00:47:30,000
It cannot be a violation
because the theorem has no
705
00:47:28,000 --> 00:47:34,000
exceptions.
Otherwise, it wouldn't be a
706
00:47:31,000 --> 00:47:37,000
theorem.
So, let's take a look.
707
00:47:34,000 --> 00:47:40,000
What's wrong?
We thought we solved it modulo,
708
00:47:37,000 --> 00:47:43,000
putting the absolute value
signs on the log.
709
00:47:40,000 --> 00:47:46,000
What's wrong?
The answer: what's wrong is to
710
00:47:43,000 --> 00:47:49,000
use the theorem you must write
the differential equation in
711
00:47:48,000 --> 00:47:54,000
standard form,
in the green form I gave you.
712
00:47:51,000 --> 00:47:57,000
Let's write the differential
equation the way we were
713
00:47:54,000 --> 00:48:00,000
supposed to.
It says dy / dx equals one
714
00:47:57,000 --> 00:48:03,000
minus y divided by x.
715
00:48:02,000 --> 00:48:08,000
And now, I see,
the right-hand side is not
716
00:48:05,000 --> 00:48:11,000
continuous, in fact,
not even defined when x equals
717
00:48:09,000 --> 00:48:15,000
zero, when along the y-axis.
And therefore,
718
00:48:12,000 --> 00:48:18,000
the existence and uniqueness is
not guaranteed along the line,
719
00:48:16,000 --> 00:48:22,000
x equals zero of the y-axis.
And, in fact,
720
00:48:20,000 --> 00:48:26,000
we see that it failed.
Now, as a practical matter,
721
00:48:23,000 --> 00:48:29,000
it's the way existence and
uniqueness fails in all ordinary
722
00:48:28,000 --> 00:48:34,000
life work with differential
equations is not through
723
00:48:32,000 --> 00:48:38,000
sophisticated examples that
mathematicians can construct.
724
00:48:38,000 --> 00:48:44,000
But normally,
because f of (x,
725
00:48:40,000 --> 00:48:46,000
y) will fail to be
defined somewhere,
726
00:48:43,000 --> 00:48:49,000
and those will be the bad
points.
727
00:48:46,000 --> 00:48:52,000
Thanks.