WEBVTT
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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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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,
00:12:15.000 --> 00:12:21.000
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
00:12:22.000 --> 00:12:28.000
a particular point.
But, it looks better if you
00:12:26.000 --> 00:12:32.000
don't.
But, there's some possibility
00:12:28.000 --> 00:12:34.000
of confusion.
I admit to that.
00:12:32.000 --> 00:12:38.000
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
00:12:52.000 --> 00:12:58.000
function of that curve of the
graph should be equal to the
00:12:56.000 --> 00:13:02.000
slope of the direction field.
Now, what does this say?
00:13:01.000 --> 00:13:07.000
Well, what's the slope of y1 of
x?
00:13:03.000 --> 00:13:09.000
That's y1 prime of x.
00:13:05.000 --> 00:13:11.000
That's from the first day of
18.01, calculus.
00:13:08.000 --> 00:13:14.000
What's the slope of the
direction field?
00:13:11.000 --> 00:13:17.000
This?
Well, it's this.
00:13:12.000 --> 00:13:18.000
And, that's with the right hand
side.
00:13:14.000 --> 00:13:20.000
So, saying these two guys are
the same or equal,
00:13:17.000 --> 00:13:23.000
is exactly, analytically,
the same as saying these two
00:13:21.000 --> 00:13:27.000
guys are equal.
So, in other words,
00:13:23.000 --> 00:13:29.000
the proof consists of,
what does this really mean?
00:13:26.000 --> 00:13:32.000
What does this really mean?
And after you see what both
00:13:29.000 --> 00:13:35.000
really mean, you say,
yeah, they're the same.
00:13:34.000 --> 00:13:40.000
So, I don't how to write that.
It's okay: same,
00:13:39.000 --> 00:13:45.000
same, how's that?
This is the same as that.
00:13:44.000 --> 00:13:50.000
Okay, well, this leaves us the
interesting question of how do
00:13:52.000 --> 00:13:58.000
you draw a direction from the,
well, this being 2003,
00:13:58.000 --> 00:14:04.000
mostly computers draw them for
you.
00:14:04.000 --> 00:14:10.000
Nonetheless,
you do have to know a certain
00:14:07.000 --> 00:14:13.000
amount.
I've given you a couple of
00:14:09.000 --> 00:14:15.000
exercises where you have to draw
the direction field yourself.
00:14:14.000 --> 00:14:20.000
This is so you get a feeling
for it, and also because humans
00:14:19.000 --> 00:14:25.000
don't draw direction fields the
same way computers do.
00:14:23.000 --> 00:14:29.000
So, let's first of all,
how did computers do it?
00:14:27.000 --> 00:14:33.000
They are very stupid.
There's no problem.
00:14:32.000 --> 00:14:38.000
Since they go very fast and
have unlimited amounts of energy
00:14:37.000 --> 00:14:43.000
to waste, the computer method is
the naive one.
00:14:42.000 --> 00:14:48.000
You pick the point.
You pick a point,
00:14:45.000 --> 00:14:51.000
and generally,
they are usually equally
00:14:49.000 --> 00:14:55.000
spaced.
You determine some spacing,
00:14:52.000 --> 00:14:58.000
that one: blah,
blah, blah, blah,
00:14:55.000 --> 00:15:01.000
blah, blah, blah,
equally spaced.
00:15:00.000 --> 00:15:06.000
And, at each point,
it computes f of x,
00:15:04.000 --> 00:15:10.000
y at the point,
finds, meets,
00:15:08.000 --> 00:15:14.000
and computes the value of f of
(x, y), that function,
00:15:14.000 --> 00:15:20.000
and the next thing is,
on the screen,
00:15:17.000 --> 00:15:23.000
it draws, at (x,
y), the little line element
00:15:22.000 --> 00:15:28.000
having slope f of x,y.
00:15:26.000 --> 00:15:32.000
In other words,
it does what the differential
00:15:30.000 --> 00:15:36.000
equation tells it to do.
And the only thing that it does
00:15:36.000 --> 00:15:42.000
is you can, if you are telling
the thing to draw the direction
00:15:40.000 --> 00:15:46.000
field, about the only option you
have is telling what the spacing
00:15:43.000 --> 00:15:49.000
should be, and sometimes people
don't like to see a whole line.
00:15:46.000 --> 00:15:52.000
They only like to see a little
bit of a half line.
00:15:49.000 --> 00:15:55.000
And, you can sometimes tell,
according to the program,
00:15:52.000 --> 00:15:58.000
tell the computer how long you
want that line to be,
00:15:55.000 --> 00:16:01.000
if you want it teeny or a
little bigger.
00:15:57.000 --> 00:16:03.000
Once in awhile you want you
want it narrower on it,
00:16:00.000 --> 00:16:06.000
but not right now.
Okay, that's what a computer
00:16:04.000 --> 00:16:10.000
does.
What does a human do?
00:16:05.000 --> 00:16:11.000
This is what it means to be
human.
00:16:08.000 --> 00:16:14.000
You use your intelligence.
From a human point of view,
00:16:12.000 --> 00:16:18.000
this stuff has been done in the
wrong order.
00:16:15.000 --> 00:16:21.000
And the reason it's been done
in the wrong order:
00:16:18.000 --> 00:16:24.000
because for each new point,
it requires a recalculation of
00:16:22.000 --> 00:16:28.000
f of (x, y).
00:16:24.000 --> 00:16:30.000
That is horrible.
The computer doesn't mind,
00:16:27.000 --> 00:16:33.000
but a human does.
So, for a human,
00:16:31.000 --> 00:16:37.000
the way to do it is not to
begin by picking the point,
00:16:35.000 --> 00:16:41.000
but to begin by picking the
slope that you would like to
00:16:40.000 --> 00:16:46.000
see.
So, you begin by taking the
00:16:42.000 --> 00:16:48.000
slope.
Let's call it the value of the
00:16:45.000 --> 00:16:51.000
slope, C.
So, you pick a number.
00:16:48.000 --> 00:16:54.000
C is two.
I want to see where are all the
00:16:51.000 --> 00:16:57.000
points in the plane where the
slope of that line element would
00:16:56.000 --> 00:17:02.000
be two?
Well, they will satisfy an
00:16:58.000 --> 00:17:04.000
equation.
The equation is f of (x,
00:17:02.000 --> 00:17:08.000
y) equals, in general,
it will be C.
00:17:07.000 --> 00:17:13.000
So, what you do is plot this,
plot the equation,
00:17:10.000 --> 00:17:16.000
plot this equation.
Notice, it's not the
00:17:14.000 --> 00:17:20.000
differential equation.
You can't exactly plot a
00:17:17.000 --> 00:17:23.000
differential equation.
It's a curve,
00:17:20.000 --> 00:17:26.000
an ordinary curve.
But which curve will depend;
00:17:24.000 --> 00:17:30.000
it's, in fact,
from the 18.02 point of view,
00:17:28.000 --> 00:17:34.000
the level curve of C,
sorry, it's a level curve of f
00:17:32.000 --> 00:17:38.000
of (x, y), the function f of x
and y corresponding to the level
00:17:37.000 --> 00:17:43.000
of value C.
But we are not going to call it
00:17:42.000 --> 00:17:48.000
that because this is not 18.02.
Instead, we're going to call it
00:17:48.000 --> 00:17:54.000
an isocline.
And then, you plot,
00:17:51.000 --> 00:17:57.000
well, you've done it.
So, you've got this isocline,
00:17:56.000 --> 00:18:02.000
except I'm going to use a
solution curve,
00:18:00.000 --> 00:18:06.000
solid lines,
only for integral curves.
00:18:03.000 --> 00:18:09.000
When we do plot isoclines,
to indicate that they are not
00:18:09.000 --> 00:18:15.000
solutions, we'll use dashed
lines for doing them.
00:18:15.000 --> 00:18:21.000
One of the computer things does
and the other one doesn't.
00:18:18.000 --> 00:18:24.000
But they use different colors,
also.
00:18:20.000 --> 00:18:26.000
There are different ways of
telling you what's an isocline
00:18:23.000 --> 00:18:29.000
and what's the solution curve.
So, and what do you do?
00:18:26.000 --> 00:18:32.000
So, these are all the points
where the slope is going to be
00:18:29.000 --> 00:18:35.000
C.
And now, what you do is draw in
00:18:32.000 --> 00:18:38.000
as many as you want of line
elements having slope C.
00:18:35.000 --> 00:18:41.000
Notice how efficient that is.
If you want 50 million of them
00:18:39.000 --> 00:18:45.000
and have the time,
draw in 50 million.
00:18:41.000 --> 00:18:47.000
If two or three are enough,
draw in two or three.
00:18:45.000 --> 00:18:51.000
You will be looking at the
picture.
00:18:47.000 --> 00:18:53.000
You will see what the curve
looks like, and that will give
00:18:51.000 --> 00:18:57.000
you your judgment as to how you
are to do that.
00:18:54.000 --> 00:19:00.000
So, in general,
a picture drawn that way,
00:18:57.000 --> 00:19:03.000
so let's say,
an isocline corresponding to C
00:18:59.000 --> 00:19:05.000
equals zero.
The line elements,
00:19:03.000 --> 00:19:09.000
and I think for an isocline,
for the purposes of this
00:19:07.000 --> 00:19:13.000
lecture, it would be a good idea
to put isoclines.
00:19:10.000 --> 00:19:16.000
Okay, so I'm going to put
solution curves in pink,
00:19:14.000 --> 00:19:20.000
or whatever this color is,
and isoclines are going to be
00:19:18.000 --> 00:19:24.000
in orange, I guess.
So, isocline,
00:19:21.000 --> 00:19:27.000
represented by a dashed line,
and now you will put in the
00:19:25.000 --> 00:19:31.000
line elements of,
we'll need lots of chalk for
00:19:28.000 --> 00:19:34.000
that.
So, I'll use white chalk.
00:19:32.000 --> 00:19:38.000
Y horizontal?
Because according to this the
00:19:34.000 --> 00:19:40.000
slope is supposed to be zero
there.
00:19:37.000 --> 00:19:43.000
And at the same way,
how about an isocline where the
00:19:40.000 --> 00:19:46.000
slope is negative one?
Let's suppose here C is equal
00:19:44.000 --> 00:19:50.000
to negative one.
Okay, then it will look like
00:19:47.000 --> 00:19:53.000
this.
These are supposed to be lines
00:19:49.000 --> 00:19:55.000
of slope negative one.
Don't shoot me if they are not.
00:19:53.000 --> 00:19:59.000
So, that's the principle.
So, this is how you will fill
00:19:56.000 --> 00:20:02.000
up the plane to draw a direction
field: by plotting the isoclines
00:20:01.000 --> 00:20:07.000
first.
And then, once you have the
00:20:04.000 --> 00:20:10.000
isoclines there,
you will have line elements.
00:20:07.000 --> 00:20:13.000
And you can draw a direction
field.
00:20:09.000 --> 00:20:15.000
Okay, so, for the next few
minutes, I'd like to work a
00:20:12.000 --> 00:20:18.000
couple of examples for you to
show how this works out in
00:20:15.000 --> 00:20:21.000
practice.
00:20:34.000 --> 00:20:40.000
So, the first equation is going
to be y prime equals minus x
00:20:45.000 --> 00:20:51.000
over y.
Okay, first thing,
00:20:53.000 --> 00:20:59.000
what are the isoclines?
Well, the isoclines are going
00:21:03.000 --> 00:21:09.000
to be y.
Well, negative x over y is
00:21:08.000 --> 00:21:14.000
equal to C.
Maybe I better make two steps
00:21:12.000 --> 00:21:18.000
out of this.
Minus x over y is equal to C.
00:21:16.000 --> 00:21:22.000
But, of course,
nobody draws a curve in that
00:21:19.000 --> 00:21:25.000
form.
You'll want it in the form y
00:21:22.000 --> 00:21:28.000
equals minus one over
C times x.
00:21:26.000 --> 00:21:32.000
So, there's our isocline.
Why don't I put that up in
00:21:32.000 --> 00:21:38.000
orange since it's going to be,
that's the color I'll draw it
00:21:36.000 --> 00:21:42.000
in.
In other words,
00:21:38.000 --> 00:21:44.000
for different values of C,
now this thing is aligned.
00:21:42.000 --> 00:21:48.000
It's aligned,
in fact, through the origin.
00:21:45.000 --> 00:21:51.000
This looks pretty simple.
Okay, so here's our plane.
00:21:50.000 --> 00:21:56.000
The isoclines are going to be
lines through the origin.
00:21:54.000 --> 00:22:00.000
And now, let's put them in,
suppose, for example,
00:21:58.000 --> 00:22:04.000
C is equal to one.
Well, if C is equal to one,
00:22:06.000 --> 00:22:12.000
then it's the line,
y equals minus x.
00:22:14.000 --> 00:22:20.000
So, this is the isocline.
I'll put, down here,
00:22:23.000 --> 00:22:29.000
C equals minus one.
And, along it,
00:22:30.000 --> 00:22:36.000
no, something's wrong.
I'm sorry?
00:22:38.000 --> 00:22:44.000
C is one, not negative one,
right, thanks.
00:22:42.000 --> 00:22:48.000
Thanks.
So, C equals one.
00:22:44.000 --> 00:22:50.000
So, it should be little line
segments of slope one will be
00:22:50.000 --> 00:22:56.000
the line elements,
things of slope one.
00:22:54.000 --> 00:23:00.000
OK, now how about C equals
negative one?
00:23:00.000 --> 00:23:06.000
If C equals negative one,
then it's the line,
00:23:03.000 --> 00:23:09.000
y equals x.
And so, that's the isocline.
00:23:07.000 --> 00:23:13.000
Notice, still dash because
these are isoclines.
00:23:11.000 --> 00:23:17.000
Here, C is negative one.
And so, the slope elements look
00:23:15.000 --> 00:23:21.000
like this.
Notice, they are perpendicular.
00:23:19.000 --> 00:23:25.000
Now, notice that they are
always going to be perpendicular
00:23:23.000 --> 00:23:29.000
to the line because the slope of
this line is minus one over C.
00:23:30.000 --> 00:23:36.000
But, the slope of the line
element is going to be C.
00:23:33.000 --> 00:23:39.000
Those numbers,
minus one over C and C,
00:23:36.000 --> 00:23:42.000
are negative reciprocals.
And, you know that two lines
00:23:40.000 --> 00:23:46.000
whose slopes are negative
reciprocals are perpendicular.
00:23:44.000 --> 00:23:50.000
So, the line elements are going
to be perpendicular to these.
00:23:49.000 --> 00:23:55.000
And therefore,
I hardly even have to bother
00:23:52.000 --> 00:23:58.000
calculating, doing any more
calculation.
00:23:55.000 --> 00:24:01.000
Here's going to be a,
well, how about this one?
00:24:00.000 --> 00:24:06.000
Here's a controversial
isocline.
00:24:02.000 --> 00:24:08.000
Is that an isocline?
Well, wait a minute.
00:24:05.000 --> 00:24:11.000
That doesn't correspond to
anything looking like this.
00:24:10.000 --> 00:24:16.000
Ah-ha, but it would if I put C
multiplied through by C.
00:24:14.000 --> 00:24:20.000
And then, it would correspond
to C being zero.
00:24:18.000 --> 00:24:24.000
In other words,
don't write it like this.
00:24:21.000 --> 00:24:27.000
Multiply through by C.
It will read C y equals
00:24:25.000 --> 00:24:31.000
negative x.
And then, when C is zero,
00:24:29.000 --> 00:24:35.000
I have x equals zero,
which is exactly the y-axis.
00:24:35.000 --> 00:24:41.000
So, that really is included.
How about the x-axis?
00:24:38.000 --> 00:24:44.000
Well, the x-axis is not
included.
00:24:40.000 --> 00:24:46.000
However, most people include it
anyway.
00:24:43.000 --> 00:24:49.000
This is very common to be a
sort of sloppy and bending the
00:24:47.000 --> 00:24:53.000
edges of corners a little bit,
and hoping nobody will notice.
00:24:51.000 --> 00:24:57.000
We'll say that corresponds to C
equals infinity.
00:24:55.000 --> 00:25:01.000
I hope nobody wants to fight
about that.
00:24:58.000 --> 00:25:04.000
If you do, go fight with
somebody else.
00:25:02.000 --> 00:25:08.000
So, if C is infinity,
that means the little line
00:25:05.000 --> 00:25:11.000
segment should have infinite
slope, and by common consent,
00:25:10.000 --> 00:25:16.000
that means it should be
vertical.
00:25:12.000 --> 00:25:18.000
And so, we can even count this
as sort of an isocline.
00:25:17.000 --> 00:25:23.000
And, I'll make the dashes
smaller, indicate it has a lower
00:25:21.000 --> 00:25:27.000
status than the others.
And, I'll put this in,
00:25:25.000 --> 00:25:31.000
do this weaselly thing of
putting it in quotation marks to
00:25:29.000 --> 00:25:35.000
indicate that I'm not
responsible for it.
00:25:34.000 --> 00:25:40.000
Okay, now, we now have to put
it the integral curves.
00:25:39.000 --> 00:25:45.000
Well, nothing could be easier.
I'm looking for curves which
00:25:45.000 --> 00:25:51.000
are everywhere perpendicular to
these rays.
00:25:50.000 --> 00:25:56.000
Well, you know from geometry
that those are circles.
00:25:55.000 --> 00:26:01.000
So, the integral curves are
circles.
00:26:00.000 --> 00:26:06.000
And, it's an elementary
exercise, which I would not
00:26:04.000 --> 00:26:10.000
deprive you of the pleasure of.
Solve the ODE by separation of
00:26:08.000 --> 00:26:14.000
variables.
In other words,
00:26:10.000 --> 00:26:16.000
we've gotten the,
so the circles are ones with a
00:26:14.000 --> 00:26:20.000
center at the origin,
of course, equal some constant.
00:26:18.000 --> 00:26:24.000
I'll call it C1,
so it's not confused with this
00:26:22.000 --> 00:26:28.000
C.
They look like that,
00:26:24.000 --> 00:26:30.000
and now you should solve this
by separating variables,
00:26:28.000 --> 00:26:34.000
and just confirm that the
solutions are,
00:26:31.000 --> 00:26:37.000
in fact, those circles.
One interesting thing,
00:26:36.000 --> 00:26:42.000
and so I confirm this,
I won't do it because I want to
00:26:40.000 --> 00:26:46.000
do geometric and numerical
things today.
00:26:42.000 --> 00:26:48.000
So, if you solve it by
separating variables,
00:26:45.000 --> 00:26:51.000
one interesting thing to note
is that if I write the solution
00:26:49.000 --> 00:26:55.000
as y equals y1 of x, well,
00:26:52.000 --> 00:26:58.000
it'll look something like the
square root of C1 minus,
00:26:56.000 --> 00:27:02.000
let's make this squared because
that's the way people usually
00:27:00.000 --> 00:27:06.000
put the radius,
minus x squared.
00:27:03.000 --> 00:27:09.000
And so, a solution,
00:27:06.000 --> 00:27:12.000
a typical solution looks like
this.
00:27:09.000 --> 00:27:15.000
Well, what's the solution over
here?
00:27:11.000 --> 00:27:17.000
Well, that one solution will be
goes from here to here.
00:27:15.000 --> 00:27:21.000
If you like,
it has a negative side to it.
00:27:18.000 --> 00:27:24.000
So, I'll make,
let's say, plus.
00:27:21.000 --> 00:27:27.000
There's another solution,
which has a negative value.
00:27:25.000 --> 00:27:31.000
But let's use the one with the
positive value of the square
00:27:29.000 --> 00:27:35.000
root.
My point is this,
00:27:32.000 --> 00:27:38.000
that that solution,
the domain of that solution,
00:27:35.000 --> 00:27:41.000
really only goes from here to
here.
00:27:38.000 --> 00:27:44.000
It's not the whole x-axis.
It's just a limited piece of
00:27:42.000 --> 00:27:48.000
the x-axis where that solution
is defined.
00:27:45.000 --> 00:27:51.000
There's no way of extending it
further.
00:27:48.000 --> 00:27:54.000
And, there's no way of
predicting, by looking at the
00:27:52.000 --> 00:27:58.000
differential equation,
that a typical solution was
00:27:56.000 --> 00:28:02.000
going to have a limited domain
like that.
00:28:01.000 --> 00:28:07.000
In other words,
you could find a solution,
00:28:04.000 --> 00:28:10.000
but how far out is it going to
go?
00:28:07.000 --> 00:28:13.000
Sometimes, it's impossible to
tell, except by either finding
00:28:12.000 --> 00:28:18.000
it explicitly,
or by asking a computer to draw
00:28:16.000 --> 00:28:22.000
a picture of it,
and seeing if that gives you
00:28:19.000 --> 00:28:25.000
some insight.
It's one of the many
00:28:22.000 --> 00:28:28.000
difficulties in handling
differential equations.
00:28:26.000 --> 00:28:32.000
You don't know what the domain
of a solution is going to be
00:28:31.000 --> 00:28:37.000
until you've actually calculated
it.
00:28:36.000 --> 00:28:42.000
Now, a slightly more
complicated example is going to
00:28:40.000 --> 00:28:46.000
be, let's see, y prime
equals one plus x minus y.
00:28:43.000 --> 00:28:49.000
It's not a lot more
00:28:46.000 --> 00:28:52.000
complicated, and as a computer
exercise, you will work with,
00:28:51.000 --> 00:28:57.000
still, more complicated ones.
But here, the isoclines would
00:28:56.000 --> 00:29:02.000
be what?
Well, I set that equal to C.
00:29:00.000 --> 00:29:06.000
Can you do the algebra in your
head?
00:29:02.000 --> 00:29:08.000
An isocline will have the
equation: this equals C.
00:29:07.000 --> 00:29:13.000
So, I'm going to put the y on
the right hand side,
00:29:11.000 --> 00:29:17.000
and that C on the left hand
side.
00:29:13.000 --> 00:29:19.000
So, it will have the equation y
equals one plus x minus C,
00:29:19.000 --> 00:29:25.000
or a nicer way to
write it would be x plus one
00:29:23.000 --> 00:29:29.000
minus C.
I guess it really doesn't
00:29:28.000 --> 00:29:34.000
matter.
So there's the equation of the
00:29:31.000 --> 00:29:37.000
isocline.
Let's quickly draw the
00:29:34.000 --> 00:29:40.000
direction field.
And notice, by the way,
00:29:36.000 --> 00:29:42.000
it's a simple equation,
but you cannot separate
00:29:39.000 --> 00:29:45.000
variables.
So, I will not,
00:29:41.000 --> 00:29:47.000
today at any rate,
be able to check the answer.
00:29:44.000 --> 00:29:50.000
I will not be able to get an
analytic answer.
00:29:47.000 --> 00:29:53.000
All we'll be able to do now is
get a geometric answer.
00:29:50.000 --> 00:29:56.000
But notice how quickly,
relatively quickly,
00:29:53.000 --> 00:29:59.000
one can get it.
So, I'm feeling for how the
00:29:56.000 --> 00:30:02.000
solutions behave to this
equation.
00:30:00.000 --> 00:30:06.000
All right, let's see,
what should we plot first?
00:30:05.000 --> 00:30:11.000
I like C equals one,
no, don't do C equals one.
00:30:10.000 --> 00:30:16.000
Let's do C equals zero,
first.
00:30:13.000 --> 00:30:19.000
C equals zero.
That's the line.
00:30:16.000 --> 00:30:22.000
y equals x plus 1.
00:30:19.000 --> 00:30:25.000
Okay, let me run and get that
chalk.
00:30:23.000 --> 00:30:29.000
So, I'll isoclines are in
orange.
00:30:27.000 --> 00:30:33.000
If so, when C equals zero,
y equals x plus one.
00:30:32.000 --> 00:30:38.000
So, let's say it's this curve.
C equals zero.
00:30:38.000 --> 00:30:44.000
How about C equals negative
one?
00:30:42.000 --> 00:30:48.000
Then it's y equals x plus two.
00:30:47.000 --> 00:30:53.000
It's this curve.
Well, let's label it down here.
00:30:53.000 --> 00:30:59.000
So, this is C equals negative
one.
00:30:57.000 --> 00:31:03.000
C equals negative two would be
y equals x, no,
00:31:02.000 --> 00:31:08.000
what am I doing?
C equals negative one is y
00:31:08.000 --> 00:31:14.000
equals x plus two.
That's right.
00:31:12.000 --> 00:31:18.000
Well, how about the other side?
If C equals plus one,
00:31:16.000 --> 00:31:22.000
well, then it's going to go
through the origin.
00:31:20.000 --> 00:31:26.000
It looks like a little more
room down here.
00:31:24.000 --> 00:31:30.000
How about, so if this is going
to be C equals one,
00:31:28.000 --> 00:31:34.000
then I sort of get the idea.
C equals two will look like
00:31:34.000 --> 00:31:40.000
this.
They're all going to be
00:31:37.000 --> 00:31:43.000
parallel lines because all
that's changing is the
00:31:42.000 --> 00:31:48.000
y-intercept, as I do this thing.
So, here, it's C equals two.
00:31:47.000 --> 00:31:53.000
That's probably enough.
All right, let's put it in the
00:31:53.000 --> 00:31:59.000
line elements.
All right, C equals negative
00:31:57.000 --> 00:32:03.000
one.
These will be perpendicular.
00:32:00.000 --> 00:32:06.000
C equals zero,
like this.
00:32:04.000 --> 00:32:10.000
C equals one.
Oh, this is interesting.
00:32:06.000 --> 00:32:12.000
I can't even draw in the line
elements because they seem to
00:32:10.000 --> 00:32:16.000
coincide with the curve itself,
with the line itself.
00:32:14.000 --> 00:32:20.000
They write y along the line,
and that makes it hard to draw
00:32:18.000 --> 00:32:24.000
them in.
How about C equals two?
00:32:20.000 --> 00:32:26.000
Well, here, the line elements
will be slanty.
00:32:23.000 --> 00:32:29.000
They'll have slope two,
so a pretty slanty up.
00:32:26.000 --> 00:32:32.000
And, I can see if a C equals
three in the same way.
00:32:31.000 --> 00:32:37.000
There are going to be even more
slantier up.
00:32:34.000 --> 00:32:40.000
And here, they're going to be
even more slanty down.
00:32:37.000 --> 00:32:43.000
This is not very scientific
terminology or mathematical,
00:32:41.000 --> 00:32:47.000
but you get the idea.
Okay, so there's our quick
00:32:45.000 --> 00:32:51.000
version of the direction field.
All we have to do is put in
00:32:49.000 --> 00:32:55.000
some integral curves now.
Well, it looks like it's doing
00:32:53.000 --> 00:32:59.000
this.
It gets less slanty here.
00:32:55.000 --> 00:33:01.000
It levels out,
has slope zero.
00:32:59.000 --> 00:33:05.000
And now, in this part of the
plain, the slope seems to be
00:33:03.000 --> 00:33:09.000
rising.
So, it must do something like
00:33:06.000 --> 00:33:12.000
that.
This guy must do something like
00:33:08.000 --> 00:33:14.000
this.
I'm a little doubtful of what I
00:33:11.000 --> 00:33:17.000
should be doing here.
Or, how about going from the
00:33:15.000 --> 00:33:21.000
other side?
Well, it rises,
00:33:17.000 --> 00:33:23.000
gets a little,
should it cross this?
00:33:20.000 --> 00:33:26.000
What should I do?
Well, there's one integral
00:33:23.000 --> 00:33:29.000
curve, which is easy to see.
It's this one.
00:33:26.000 --> 00:33:32.000
This line is both an isocline
and an integral curve.
00:33:32.000 --> 00:33:38.000
It's everything,
except drawable,
00:33:35.000 --> 00:33:41.000
[LAUGHTER] so,
you understand this is the same
00:33:41.000 --> 00:33:47.000
line.
It's both orange and pink at
00:33:45.000 --> 00:33:51.000
the same time.
But I don't know what
00:33:49.000 --> 00:33:55.000
combination color that would
make.
00:33:53.000 --> 00:33:59.000
It doesn't look like a line,
but be sympathetic.
00:34:00.000 --> 00:34:06.000
Now, the question is,
what's happening in this
00:34:04.000 --> 00:34:10.000
corridor?
In the corridor,
00:34:06.000 --> 00:34:12.000
that's not a mathematical word
either, between the isoclines
00:34:12.000 --> 00:34:18.000
for, well, what are they?
They are the isoclines for C
00:34:18.000 --> 00:34:24.000
equals two, and C equals zero.
How does that corridor look?
00:34:23.000 --> 00:34:29.000
Well: something like this.
Over here, the lines all look
00:34:29.000 --> 00:34:35.000
like that.
And here, they all look like
00:34:33.000 --> 00:34:39.000
this.
The slope is two.
00:34:36.000 --> 00:34:42.000
And, a hapless solution gets in
there.
00:34:39.000 --> 00:34:45.000
What's it to do?
Well, do you see that if a
00:34:43.000 --> 00:34:49.000
solution gets in that corridor,
an integral curve gets in that
00:34:49.000 --> 00:34:55.000
corridor, no escape is possible.
It's like a lobster trap.
00:34:54.000 --> 00:35:00.000
The lobster can walk in.
But it cannot walk out because
00:34:58.000 --> 00:35:04.000
things are always going in.
How could it escape?
00:35:03.000 --> 00:35:09.000
Well, it would have to double
back, somehow,
00:35:06.000 --> 00:35:12.000
and remember,
to escape, it has to be,
00:35:10.000 --> 00:35:16.000
to escape on the left side,
it must be going horizontally.
00:35:17.000 --> 00:35:23.000
But, how could it do that
without doubling back first and
00:35:20.000 --> 00:35:26.000
having the wrong slope?
The slope of everything in this
00:35:24.000 --> 00:35:30.000
corridor is positive,
and to double back and escape,
00:35:28.000 --> 00:35:34.000
it would at some point have to
have negative slope.
00:35:32.000 --> 00:35:38.000
It can't do that.
Well, could it escape on the
00:35:35.000 --> 00:35:41.000
right-hand side?
No, because at the moment when
00:35:39.000 --> 00:35:45.000
it wants to cross,
it will have to have a slope
00:35:42.000 --> 00:35:48.000
less than this line.
But all these spiky guys are
00:35:46.000 --> 00:35:52.000
pointing; it can't escape that
way either.
00:35:50.000 --> 00:35:56.000
So, no escape is possible.
It has to continue on,
00:35:53.000 --> 00:35:59.000
there.
But, more than that is true.
00:35:56.000 --> 00:36:02.000
So, a solution can't escape.
Once it's in there,
00:36:01.000 --> 00:36:07.000
it can't escape.
It's like, what do they call
00:36:04.000 --> 00:36:10.000
those plants,
I forget, pitcher plants.
00:36:07.000 --> 00:36:13.000
All they hear is they are going
down.
00:36:10.000 --> 00:36:16.000
So, it looks like that.
And so, the poor little insect
00:36:14.000 --> 00:36:20.000
falls in.
They could climb up the walls
00:36:17.000 --> 00:36:23.000
except that all the hairs are
going the wrong direction,
00:36:22.000 --> 00:36:28.000
and it can't get over them.
Well, let's think of it that
00:36:26.000 --> 00:36:32.000
way: this poor trap solution.
So, it does what it has to do.
00:36:32.000 --> 00:36:38.000
Now, there's more to it than
that.
00:36:35.000 --> 00:36:41.000
Because there are two
principles involved here that
00:36:39.000 --> 00:36:45.000
you should know,
that help a lot in drawing
00:36:43.000 --> 00:36:49.000
these pictures.
Principle number one is that
00:36:46.000 --> 00:36:52.000
two integral curves cannot cross
at an angle.
00:36:50.000 --> 00:36:56.000
Two integral curves can't
cross, I mean,
00:36:53.000 --> 00:36:59.000
by crossing at an angle like
that.
00:36:56.000 --> 00:37:02.000
I'll indicate what I mean by a
picture like that.
00:37:02.000 --> 00:37:08.000
Now, why not?
This is an important principle.
00:37:05.000 --> 00:37:11.000
Let's put that up in the white
box.
00:37:08.000 --> 00:37:14.000
They can't cross because if two
integral curves,
00:37:12.000 --> 00:37:18.000
are trying to cross,
well, one will look like this.
00:37:16.000 --> 00:37:22.000
It's an integral curve because
it has this slope.
00:37:20.000 --> 00:37:26.000
And, the other integral curve
has this slope.
00:37:24.000 --> 00:37:30.000
And now, they fight with each
other.
00:37:27.000 --> 00:37:33.000
What is the true slope at that
point?
00:37:32.000 --> 00:37:38.000
Well, the direction field only
allows you to have one slope.
00:37:36.000 --> 00:37:42.000
If there's a line element at
that point, it has a definite
00:37:40.000 --> 00:37:46.000
slope.
And therefore,
00:37:41.000 --> 00:37:47.000
it cannot have both the slope
and that one.
00:37:44.000 --> 00:37:50.000
It's as simple as that.
So, the reason is you can't
00:37:48.000 --> 00:37:54.000
have two slopes.
The direction field doesn't
00:37:51.000 --> 00:37:57.000
allow it.
Well, that's a big,
00:37:53.000 --> 00:37:59.000
big help because if I know,
here's an integral curve,
00:37:57.000 --> 00:38:03.000
and if I know that none of
these other pink integral curves
00:38:01.000 --> 00:38:07.000
are allowed to cross it,
how else can I do it?
00:38:06.000 --> 00:38:12.000
Well, they can't escape.
They can't cross.
00:38:09.000 --> 00:38:15.000
It's sort of clear that they
must get closer and closer to
00:38:13.000 --> 00:38:19.000
it.
You know, I'd have to work a
00:38:16.000 --> 00:38:22.000
little to justify that.
But I think that nobody would
00:38:20.000 --> 00:38:26.000
have any doubt of it who did a
little experimentation.
00:38:24.000 --> 00:38:30.000
In other words,
all these curves joined that
00:38:28.000 --> 00:38:34.000
little tube and get closer and
closer to this line,
00:38:32.000 --> 00:38:38.000
y equals x.
And there, without solving the
00:38:37.000 --> 00:38:43.000
differential equation,
it's clear that all of these
00:38:42.000 --> 00:38:48.000
solutions, how do they behave?
As x goes to infinity,
00:38:47.000 --> 00:38:53.000
they become asymptotic to,
they become closer and closer
00:38:52.000 --> 00:38:58.000
to the solution,
x.
00:38:54.000 --> 00:39:00.000
Is x a solution?
Yeah, because y equals x is an
00:38:58.000 --> 00:39:04.000
integral curve.
Is x a solution?
00:39:02.000 --> 00:39:08.000
Yeah, because if I plug in y
equals x, I get what?
00:39:07.000 --> 00:39:13.000
On the right-hand side,
I get one.
00:39:10.000 --> 00:39:16.000
And on the left-hand side,
I get one.
00:39:14.000 --> 00:39:20.000
One equals one.
So, this is a solution.
00:39:18.000 --> 00:39:24.000
Let's indicate that it's a
solution.
00:39:21.000 --> 00:39:27.000
So, analytically,
we've discovered an analytic
00:39:26.000 --> 00:39:32.000
solution to the differential
equation, namely,
00:39:31.000 --> 00:39:37.000
Y equals X, just by this
geometric process.
00:39:37.000 --> 00:39:43.000
Now, there's one more principle
like that, which is less
00:39:41.000 --> 00:39:47.000
obvious.
But you do have to know it.
00:39:44.000 --> 00:39:50.000
So, you are not allowed to
cross.
00:39:46.000 --> 00:39:52.000
That's clear.
But it's much,
00:39:49.000 --> 00:39:55.000
much, much, much,
much less obvious that two
00:39:52.000 --> 00:39:58.000
integral curves cannot touch.
That is, they cannot even be
00:39:57.000 --> 00:40:03.000
tangent.
Two integral curves cannot be
00:40:00.000 --> 00:40:06.000
tangent.
00:40:10.000 --> 00:40:16.000
I'll indicate that by the word
touch, which is what a lot of
00:40:19.000 --> 00:40:25.000
people say.
In other words,
00:40:23.000 --> 00:40:29.000
if this is illegal,
so is this.
00:40:28.000 --> 00:40:34.000
It can't happen.
You know, without that,
00:40:33.000 --> 00:40:39.000
for example,
it might be,
00:40:35.000 --> 00:40:41.000
I might feel that there would
be nothing in this to prevent
00:40:39.000 --> 00:40:45.000
those curves from joining.
Why couldn't these pink curves
00:40:43.000 --> 00:40:49.000
join the line,
y equals x?
00:40:45.000 --> 00:40:51.000
You know, it's a solution.
They just pitch a ride,
00:40:49.000 --> 00:40:55.000
as it were.
The answer is they cannot do
00:40:52.000 --> 00:40:58.000
that because they have to just
get asymptotic to it,
00:40:55.000 --> 00:41:01.000
ever, ever closer.
They can't join y equals x
00:40:59.000 --> 00:41:05.000
because at the point where they
join, you have that situation.
00:41:05.000 --> 00:41:11.000
Now, why can't you to have
this?
00:41:09.000 --> 00:41:15.000
That's much more sophisticated
than this, and the reason is
00:41:17.000 --> 00:41:23.000
because of something called the
Existence and Uniqueness
00:41:24.000 --> 00:41:30.000
Theorem, which says that there
is through a point,
00:41:31.000 --> 00:41:37.000
x zero y zero,
that y prime equals f of
00:41:38.000 --> 00:41:44.000
(x, y) has only one,
00:41:43.000 --> 00:41:49.000
and only one solution.
One has one solution.
00:41:49.000 --> 00:41:55.000
In mathematics speak,
that means at least one
00:41:53.000 --> 00:41:59.000
solution.
It doesn't mean it has just one
00:41:56.000 --> 00:42:02.000
solution.
That's mathematical convention.
00:41:59.000 --> 00:42:05.000
It has one solution,
at least one solution.
00:42:02.000 --> 00:42:08.000
But, the killer is,
only one solution.
00:42:06.000 --> 00:42:12.000
That's what you have to say in
mathematics if you want just
00:42:10.000 --> 00:42:16.000
one, one, and only one solution
through the point
00:42:15.000 --> 00:42:21.000
x zero y zero.
So, the fact that it has one,
00:42:18.000 --> 00:42:24.000
that is the existence part.
The fact that it has only one
00:42:23.000 --> 00:42:29.000
is the uniqueness part of the
theorem.
00:42:26.000 --> 00:42:32.000
Now, like all good mathematical
theorems, this one does have
00:42:31.000 --> 00:42:37.000
hypotheses.
So, this is not going to be a
00:42:35.000 --> 00:42:41.000
course, I warn you,
those of you who are
00:42:39.000 --> 00:42:45.000
theoretically inclined,
very rich in hypotheses.
00:42:44.000 --> 00:42:50.000
But, hypotheses for those one
or that f of (x,
00:42:48.000 --> 00:42:54.000
y) should be a
continuous function.
00:42:52.000 --> 00:42:58.000
Now, like polynomial,
signs, should be continuous
00:42:57.000 --> 00:43:03.000
near, in the vicinity of that
point.
00:43:02.000 --> 00:43:08.000
That guarantees existence,
and what guarantees uniqueness
00:43:08.000 --> 00:43:14.000
is the hypothesis that you would
not guess by yourself.
00:43:14.000 --> 00:43:20.000
Neither would I.
What guarantees the uniqueness
00:43:19.000 --> 00:43:25.000
is that also,
it's partial derivative with
00:43:24.000 --> 00:43:30.000
respect to y should be
continuous, should be continuous
00:43:30.000 --> 00:43:36.000
near x zero y zero.
00:43:35.000 --> 00:43:41.000
Well, I have to make a
decision.
00:43:38.000 --> 00:43:44.000
I don't have time to talk about
Euler's method.
00:43:43.000 --> 00:43:49.000
I'll refer you to the,
there's one page of notes,
00:43:49.000 --> 00:43:55.000
and I couldn't do any more than
just repeat what's on those
00:43:55.000 --> 00:44:01.000
notes.
So, I'll trust you to read
00:43:59.000 --> 00:44:05.000
that.
And instead,
00:44:02.000 --> 00:44:08.000
let me give you an example
which will solidify these things
00:44:09.000 --> 00:44:15.000
in your mind a little bit.
I think that's a better course.
00:44:17.000 --> 00:44:23.000
The example is not in your
notes, and therefore,
00:44:22.000 --> 00:44:28.000
remember, you heard it here
first.
00:44:27.000 --> 00:44:33.000
Okay, so what's the example?
So, there is that differential
00:44:34.000 --> 00:44:40.000
equation.
Now, let's just solve it by
00:44:38.000 --> 00:44:44.000
separating variables.
Can you do it in your head?
00:44:42.000 --> 00:44:48.000
dy over dx, put all the y's on
the left.
00:44:44.000 --> 00:44:50.000
It will look like dy over one
minus y.
00:44:48.000 --> 00:44:54.000
Put all the dx's on the left.
So, the dx here goes on the
00:44:52.000 --> 00:44:58.000
right, rather.
That will be dx.
00:44:54.000 --> 00:45:00.000
And then, the x goes down into
the denominator.
00:44:57.000 --> 00:45:03.000
So now, it looks like that.
And, if I integrate both sides,
00:45:03.000 --> 00:45:09.000
I get the log of one minus y,
I guess, maybe with a,
00:45:08.000 --> 00:45:14.000
I never bothered with that,
but you can.
00:45:12.000 --> 00:45:18.000
It should be absolute values.
All right, put an absolute
00:45:17.000 --> 00:45:23.000
value, plus a constant.
And now, if I exponentiate both
00:45:23.000 --> 00:45:29.000
sides, the constant is positive.
So, this is going to look like
00:45:29.000 --> 00:45:35.000
y.
One minus y equals x
00:45:33.000 --> 00:45:39.000
And, the constant will be e to
00:45:36.000 --> 00:45:42.000
the C1.
And, I'll just make that a new
00:45:39.000 --> 00:45:45.000
constant, Cx.
And now, by letting C be
00:45:42.000 --> 00:45:48.000
negative, that's why you can get
rid of the absolute values,
00:45:45.000 --> 00:45:51.000
if you allow C to have negative
values as well as positive
00:45:49.000 --> 00:45:55.000
values.
Let's write this in a more
00:45:51.000 --> 00:45:57.000
human form.
So, y is equal to one minus Cx.
00:45:53.000 --> 00:45:59.000
Good, all right,
00:45:55.000 --> 00:46:01.000
let's just plot those.
So, these are the solutions.
00:46:00.000 --> 00:46:06.000
It's a pretty easy equation,
pretty easy solution method,
00:46:05.000 --> 00:46:11.000
just separation of variables.
What do they look like?
00:46:11.000 --> 00:46:17.000
Well, these are all lines whose
intercept is at one.
00:46:16.000 --> 00:46:22.000
And, they have any slope
whatsoever.
00:46:19.000 --> 00:46:25.000
So, these are the lines that
look like that.
00:46:24.000 --> 00:46:30.000
Okay, now let me ask,
existence and uniqueness.
00:46:29.000 --> 00:46:35.000
Existence: through which points
of the plane does the solution
00:46:35.000 --> 00:46:41.000
go?
Answer: through every point of
00:46:39.000 --> 00:46:45.000
the plane, through any point
here, I can find one and only
00:46:44.000 --> 00:46:50.000
one of those lines,
except for these stupid guys
00:46:48.000 --> 00:46:54.000
here on the stalk of the flower.
Here, for each of these points,
00:46:53.000 --> 00:46:59.000
there is no existence.
There is no solution to this
00:46:57.000 --> 00:47:03.000
differential equation,
which goes through any of these
00:47:02.000 --> 00:47:08.000
wiggly points on the y-axis,
with one exception.
00:47:07.000 --> 00:47:13.000
This point is oversupplied.
At this point,
00:47:10.000 --> 00:47:16.000
it's not existence that fails.
It's uniqueness that fails:
00:47:14.000 --> 00:47:20.000
no uniqueness.
There are lots of things which
00:47:18.000 --> 00:47:24.000
go through here.
Now, is that a violation of the
00:47:21.000 --> 00:47:27.000
existence and uniqueness
theorem?
00:47:24.000 --> 00:47:30.000
It cannot be a violation
because the theorem has no
00:47:28.000 --> 00:47:34.000
exceptions.
Otherwise, it wouldn't be a
00:47:31.000 --> 00:47:37.000
theorem.
So, let's take a look.
00:47:34.000 --> 00:47:40.000
What's wrong?
We thought we solved it modulo,
00:47:37.000 --> 00:47:43.000
putting the absolute value
signs on the log.
00:47:40.000 --> 00:47:46.000
What's wrong?
The answer: what's wrong is to
00:47:43.000 --> 00:47:49.000
use the theorem you must write
the differential equation in
00:47:48.000 --> 00:47:54.000
standard form,
in the green form I gave you.
00:47:51.000 --> 00:47:57.000
Let's write the differential
equation the way we were
00:47:54.000 --> 00:48:00.000
supposed to.
It says dy / dx equals one
00:47:57.000 --> 00:48:03.000
minus y divided by x.
00:48:02.000 --> 00:48:08.000
And now, I see,
the right-hand side is not
00:48:05.000 --> 00:48:11.000
continuous, in fact,
not even defined when x equals
00:48:09.000 --> 00:48:15.000
zero, when along the y-axis.
And therefore,
00:48:12.000 --> 00:48:18.000
the existence and uniqueness is
not guaranteed along the line,
00:48:16.000 --> 00:48:22.000
x equals zero of the y-axis.
And, in fact,
00:48:20.000 --> 00:48:26.000
we see that it failed.
Now, as a practical matter,
00:48:23.000 --> 00:48:29.000
it's the way existence and
uniqueness fails in all ordinary
00:48:28.000 --> 00:48:34.000
life work with differential
equations is not through
00:48:32.000 --> 00:48:38.000
sophisticated examples that
mathematicians can construct.
00:48:38.000 --> 00:48:44.000
But normally,
because f of (x,
00:48:40.000 --> 00:48:46.000
y) will fail to be
defined somewhere,
00:48:43.000 --> 00:48:49.000
and those will be the bad
points.
00:48:46.000 --> 00:48:52.000
Thanks.