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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,
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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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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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guys are equal.
So, in other words,
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the proof consists of,
what does this really mean?
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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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you draw a direction from the,
well, this being 2003,
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mostly computers draw them for
you.
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Nonetheless,
you do have to know a certain
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amount.
I've given you a couple of
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exercises where you have to draw
the direction field yourself.
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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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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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to waste, the computer method is
the naive one.
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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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that one: blah,
blah, blah, blah,
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blah, blah, blah,
equally spaced.
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And, at each point,
it computes f of x,
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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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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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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.
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They only like to see a little
bit of a half line.
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And, you can sometimes tell,
according to the program,
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tell the computer how long you
want that line to be,
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if you want it teeny or a
little bigger.
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Once in awhile you want you
want it narrower on it,
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but not right now.
Okay, that's what a computer
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does.
What does a human do?
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This is what it means to be
human.
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You use your intelligence.
From a human point of view,
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this stuff has been done in the
wrong order.
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And the reason it's been done
in the wrong order:
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because for each new point,
it requires a recalculation of
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f of (x, y).
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That is horrible.
The computer doesn't mind,
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but a human does.
So, for a human,
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the way to do it is not to
begin by picking the point,
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but to begin by picking the
slope that you would like to
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see.
So, you begin by taking the
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slope.
Let's call it the value of the
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slope, C.
So, you pick a number.
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C is two.
I want to see where are all the
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points in the plane where the
slope of that line element would
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be two?
Well, they will satisfy an
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equation.
The equation is f of (x,
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y) equals, in general,
it will be C.
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So, what you do is plot this,
plot the equation,
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plot this equation.
Notice, it's not the
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differential equation.
You can't exactly plot a
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differential equation.
It's a curve,
266
00:17:20 --> 00:17:26
an ordinary curve.
But which curve will depend;
267
00:17:24 --> 00:17:30
it's, in fact,
from the 18.02 point of view,
268
00:17:28 --> 00:17:34
the level curve of C,
sorry, it's a level curve of f
269
00:17:32 --> 00:17:38
of (x, y), the function f of x
and y corresponding to the level
270
00:17:37 --> 00:17:43
of value C.
But we are not going to call it
271
00:17:42 --> 00:17:48
that because this is not 18.02.
Instead, we're going to call it
272
00:17:48 --> 00:17:54
an isocline.
And then, you plot,
273
00:17:51 --> 00:17:57
well, you've done it.
So, you've got this isocline,
274
00:17:56 --> 00:18:02
except I'm going to use a
solution curve,
275
00:18:00 --> 00:18:06
solid lines,
only for integral curves.
276
00:18:03 --> 00:18:09
When we do plot isoclines,
to indicate that they are not
277
00:18:09 --> 00:18:15
solutions, we'll use dashed
lines for doing them.
278
00:18:15 --> 00:18:21
One of the computer things does
and the other one doesn't.
279
00:18:18 --> 00:18:24
But they use different colors,
also.
280
00:18:20 --> 00:18:26
There are different ways of
telling you what's an isocline
281
00:18:23 --> 00:18:29
and what's the solution curve.
So, and what do you do?
282
00:18:26 --> 00:18:32
So, these are all the points
where the slope is going to be
283
00:18:29 --> 00:18:35
C.
And now, what you do is draw in
284
00:18:32 --> 00:18:38
as many as you want of line
elements having slope C.
285
00:18:35 --> 00:18:41
Notice how efficient that is.
If you want 50 million of them
286
00:18:39 --> 00:18:45
and have the time,
draw in 50 million.
287
00:18:41 --> 00:18:47
If two or three are enough,
draw in two or three.
288
00:18:45 --> 00:18:51
You will be looking at the
picture.
289
00:18:47 --> 00:18:53
You will see what the curve
looks like, and that will give
290
00:18:51 --> 00:18:57
you your judgment as to how you
are to do that.
291
00:18:54 --> 00:19:00
So, in general,
a picture drawn that way,
292
00:18:57 --> 00:19:03
so let's say,
an isocline corresponding to C
293
00:18:59 --> 00:19:05
equals zero.
The line elements,
294
00:19:03 --> 00:19:09
and I think for an isocline,
for the purposes of this
295
00:19:07 --> 00:19:13
lecture, it would be a good idea
to put isoclines.
296
00:19:10 --> 00:19:16
Okay, so I'm going to put
solution curves in pink,
297
00:19:14 --> 00:19:20
or whatever this color is,
and isoclines are going to be
298
00:19:18 --> 00:19:24
in orange, I guess.
So, isocline,
299
00:19:21 --> 00:19:27
represented by a dashed line,
and now you will put in the
300
00:19:25 --> 00:19:31
line elements of,
we'll need lots of chalk for
301
00:19:28 --> 00:19:34
that.
So, I'll use white chalk.
302
00:19:32 --> 00:19:38
Y horizontal?
Because according to this the
303
00:19:34 --> 00:19:40
slope is supposed to be zero
there.
304
00:19:37 --> 00:19:43
And at the same way,
how about an isocline where the
305
00:19:40 --> 00:19:46
slope is negative one?
Let's suppose here C is equal
306
00:19:44 --> 00:19:50
to negative one.
Okay, then it will look like
307
00:19:47 --> 00:19:53
this.
These are supposed to be lines
308
00:19:49 --> 00:19:55
of slope negative one.
Don't shoot me if they are not.
309
00:19:53 --> 00:19:59
So, that's the principle.
So, this is how you will fill
310
00:19:56 --> 00:20:02
up the plane to draw a direction
field: by plotting the isoclines
311
00:20:01 --> 00:20:07
first.
And then, once you have the
312
00:20:04 --> 00:20:10
isoclines there,
you will have line elements.
313
00:20:07 --> 00:20:13
And you can draw a direction
field.
314
00:20:09 --> 00:20:15
Okay, so, for the next few
minutes, I'd like to work a
315
00:20:12 --> 00:20:18
couple of examples for you to
show how this works out in
316
00:20:15 --> 00:20:21
practice.
317
00:20:17 --> 00:20:23
318
319
320
00:20:34 --> 00:20:40
So, the first equation is going
to be y prime equals minus x
321
00:20:45 --> 00:20:51
over y.
Okay, first thing,
322
00:20:53 --> 00:20:59
what are the isoclines?
Well, the isoclines are going
323
00:21:03 --> 00:21:09
to be y.
Well, negative x over y is
324
00:21:08 --> 00:21:14
equal to C.
Maybe I better make two steps
325
00:21:12 --> 00:21:18
out of this.
Minus x over y is equal to C.
326
00:21:16 --> 00:21:22
But, of course,
nobody draws a curve in that
327
00:21:19 --> 00:21:25
form.
You'll want it in the form y
328
00:21:22 --> 00:21:28
equals minus one over
C times x.
329
00:21:26 --> 00:21:32
So, there's our isocline.
Why don't I put that up in
330
00:21:32 --> 00:21:38
orange since it's going to be,
that's the color I'll draw it
331
00:21:36 --> 00:21:42
in.
In other words,
332
00:21:38 --> 00:21:44
for different values of C,
now this thing is aligned.
333
00:21:42 --> 00:21:48
It's aligned,
in fact, through the origin.
334
00:21:45 --> 00:21:51
This looks pretty simple.
Okay, so here's our plane.
335
00:21:50 --> 00:21:56
The isoclines are going to be
lines through the origin.
336
00:21:54 --> 00:22:00
And now, let's put them in,
suppose, for example,
337
00:21:58 --> 00:22:04
C is equal to one.
Well, if C is equal to one,
338
00:22:06 --> 00:22:12
then it's the line,
y equals minus x.
339
00:22:14 --> 00:22:20
So, this is the isocline.
I'll put, down here,
340
00:22:23 --> 00:22:29
C equals minus one.
And, along it,
341
00:22:30 --> 00:22:36
no, something's wrong.
I'm sorry?
342
00:22:38 --> 00:22:44
C is one, not negative one,
right, thanks.
343
00:22:42 --> 00:22:48
Thanks.
So, C equals one.
344
00:22:44 --> 00:22:50
So, it should be little line
segments of slope one will be
345
00:22:50 --> 00:22:56
the line elements,
things of slope one.
346
00:22:54 --> 00:23:00
OK, now how about C equals
negative one?
347
00:23:00 --> 00:23:06
If C equals negative one,
then it's the line,
348
00:23:03 --> 00:23:09
y equals x.
And so, that's the isocline.
349
00:23:07 --> 00:23:13
Notice, still dash because
these are isoclines.
350
00:23:11 --> 00:23:17
Here, C is negative one.
And so, the slope elements look
351
00:23:15 --> 00:23:21
like this.
Notice, they are perpendicular.
352
00:23:19 --> 00:23:25
Now, notice that they are
always going to be perpendicular
353
00:23:23 --> 00:23:29
to the line because the slope of
this line is minus one over C.
354
00:23:30 --> 00:23:36
But, the slope of the line
element is going to be C.
355
00:23:33 --> 00:23:39
Those numbers,
minus one over C and C,
356
00:23:36 --> 00:23:42
are negative reciprocals.
And, you know that two lines
357
00:23:40 --> 00:23:46
whose slopes are negative
reciprocals are perpendicular.
358
00:23:44 --> 00:23:50
So, the line elements are going
to be perpendicular to these.
359
00:23:49 --> 00:23:55
And therefore,
I hardly even have to bother
360
00:23:52 --> 00:23:58
calculating, doing any more
calculation.
361
00:23:55 --> 00:24:01
Here's going to be a,
well, how about this one?
362
00:24:00 --> 00:24:06
Here's a controversial
isocline.
363
00:24:02 --> 00:24:08
Is that an isocline?
Well, wait a minute.
364
00:24:05 --> 00:24:11
That doesn't correspond to
anything looking like this.
365
00:24:10 --> 00:24:16
Ah-ha, but it would if I put C
multiplied through by C.
366
00:24:14 --> 00:24:20
And then, it would correspond
to C being zero.
367
00:24:18 --> 00:24:24
In other words,
don't write it like this.
368
00:24:21 --> 00:24:27
Multiply through by C.
It will read C y equals
369
00:24:25 --> 00:24:31
negative x.
And then, when C is zero,
370
00:24:29 --> 00:24:35
I have x equals zero,
which is exactly the y-axis.
371
00:24:35 --> 00:24:41
So, that really is included.
How about the x-axis?
372
00:24:38 --> 00:24:44
Well, the x-axis is not
included.
373
00:24:40 --> 00:24:46
However, most people include it
anyway.
374
00:24:43 --> 00:24:49
This is very common to be a
sort of sloppy and bending the
375
00:24:47 --> 00:24:53
edges of corners a little bit,
and hoping nobody will notice.
376
00:24:51 --> 00:24:57
We'll say that corresponds to C
equals infinity.
377
00:24:55 --> 00:25:01
I hope nobody wants to fight
about that.
378
00:24:58 --> 00:25:04
If you do, go fight with
somebody else.
379
00:25:02 --> 00:25:08
So, if C is infinity,
that means the little line
380
00:25:05 --> 00:25:11
segment should have infinite
slope, and by common consent,
381
00:25:10 --> 00:25:16
that means it should be
vertical.
382
00:25:12 --> 00:25:18
And so, we can even count this
as sort of an isocline.
383
00:25:17 --> 00:25:23
And, I'll make the dashes
smaller, indicate it has a lower
384
00:25:21 --> 00:25:27
status than the others.
And, I'll put this in,
385
00:25:25 --> 00:25:31
do this weaselly thing of
putting it in quotation marks to
386
00:25:29 --> 00:25:35
indicate that I'm not
responsible for it.
387
00:25:34 --> 00:25:40
Okay, now, we now have to put
it the integral curves.
388
00:25:39 --> 00:25:45
Well, nothing could be easier.
I'm looking for curves which
389
00:25:45 --> 00:25:51
are everywhere perpendicular to
these rays.
390
00:25:50 --> 00:25:56
Well, you know from geometry
that those are circles.
391
00:25:55 --> 00:26:01
So, the integral curves are
circles.
392
00:26:00 --> 00:26:06
And, it's an elementary
exercise, which I would not
393
00:26:04 --> 00:26:10
deprive you of the pleasure of.
Solve the ODE by separation of
394
00:26:08 --> 00:26:14
variables.
In other words,
395
00:26:10 --> 00:26:16
we've gotten the,
so the circles are ones with a
396
00:26:14 --> 00:26:20
center at the origin,
of course, equal some constant.
397
00:26:18 --> 00:26:24
I'll call it C1,
so it's not confused with this
398
00:26:22 --> 00:26:28
C.
They look like that,
399
00:26:24 --> 00:26:30
and now you should solve this
by separating variables,
400
00:26:28 --> 00:26:34
and just confirm that the
solutions are,
401
00:26:31 --> 00:26:37
in fact, those circles.
One interesting thing,
402
00:26:36 --> 00:26:42
and so I confirm this,
I won't do it because I want to
403
00:26:40 --> 00:26:46
do geometric and numerical
things today.
404
00:26:42 --> 00:26:48
So, if you solve it by
separating variables,
405
00:26:45 --> 00:26:51
one interesting thing to note
is that if I write the solution
406
00:26:49 --> 00:26:55
as y equals y1 of x, well,
407
00:26:52 --> 00:26:58
it'll look something like the
square root of C1 minus,
408
00:26:56 --> 00:27:02
let's make this squared because
that's the way people usually
409
00:27:00 --> 00:27:06
put the radius,
minus x squared.
410
00:27:03 --> 00:27:09
And so, a solution,
411
00:27:06 --> 00:27:12
a typical solution looks like
this.
412
00:27:09 --> 00:27:15
Well, what's the solution over
here?
413
00:27:11 --> 00:27:17
Well, that one solution will be
goes from here to here.
414
00:27:15 --> 00:27:21
If you like,
it has a negative side to it.
415
00:27:18 --> 00:27:24
So, I'll make,
let's say, plus.
416
00:27:21 --> 00:27:27
There's another solution,
which has a negative value.
417
00:27:25 --> 00:27:31
But let's use the one with the
positive value of the square
418
00:27:29 --> 00:27:35
root.
My point is this,
419
00:27:32 --> 00:27:38
that that solution,
the domain of that solution,
420
00:27:35 --> 00:27:41
really only goes from here to
here.
421
00:27:38 --> 00:27:44
It's not the whole x-axis.
It's just a limited piece of
422
00:27:42 --> 00:27:48
the x-axis where that solution
is defined.
423
00:27:45 --> 00:27:51
There's no way of extending it
further.
424
00:27:48 --> 00:27:54
And, there's no way of
predicting, by looking at the
425
00:27:52 --> 00:27:58
differential equation,
that a typical solution was
426
00:27:56 --> 00:28:02
going to have a limited domain
like that.
427
00:28:01 --> 00:28:07
In other words,
you could find a solution,
428
00:28:04 --> 00:28:10
but how far out is it going to
go?
429
00:28:07 --> 00:28:13
Sometimes, it's impossible to
tell, except by either finding
430
00:28:12 --> 00:28:18
it explicitly,
or by asking a computer to draw
431
00:28:16 --> 00:28:22
a picture of it,
and seeing if that gives you
432
00:28:19 --> 00:28:25
some insight.
It's one of the many
433
00:28:22 --> 00:28:28
difficulties in handling
differential equations.
434
00:28:26 --> 00:28:32
You don't know what the domain
of a solution is going to be
435
00:28:31 --> 00:28:37
until you've actually calculated
it.
436
00:28:36 --> 00:28:42
Now, a slightly more
complicated example is going to
437
00:28:40 --> 00:28:46
be, let's see, y prime
equals one plus x minus y.
438
00:28:43 --> 00:28:49
It's not a lot more
439
00:28:46 --> 00:28:52
complicated, and as a computer
exercise, you will work with,
440
00:28:51 --> 00:28:57
still, more complicated ones.
But here, the isoclines would
441
00:28:56 --> 00:29:02
be what?
Well, I set that equal to C.
442
00:29:00 --> 00:29:06
Can you do the algebra in your
head?
443
00:29:02 --> 00:29:08
An isocline will have the
equation: this equals C.
444
00:29:07 --> 00:29:13
So, I'm going to put the y on
the right hand side,
445
00:29:11 --> 00:29:17
and that C on the left hand
side.
446
00:29:13 --> 00:29:19
So, it will have the equation y
equals one plus x minus C,
447
00:29:19 --> 00:29:25
or a nicer way to
write it would be x plus one
448
00:29:23 --> 00:29:29
minus C.
I guess it really doesn't
449
00:29:28 --> 00:29:34
matter.
So there's the equation of the
450
00:29:31 --> 00:29:37
isocline.
Let's quickly draw the
451
00:29:34 --> 00:29:40
direction field.
And notice, by the way,
452
00:29:36 --> 00:29:42
it's a simple equation,
but you cannot separate
453
00:29:39 --> 00:29:45
variables.
So, I will not,
454
00:29:41 --> 00:29:47
today at any rate,
be able to check the answer.
455
00:29:44 --> 00:29:50
I will not be able to get an
analytic answer.
456
00:29:47 --> 00:29:53
All we'll be able to do now is
get a geometric answer.
457
00:29:50 --> 00:29:56
But notice how quickly,
relatively quickly,
458
00:29:53 --> 00:29:59
one can get it.
So, I'm feeling for how the
459
00:29:56 --> 00:30:02
solutions behave to this
equation.
460
00:30:00 --> 00:30:06
All right, let's see,
what should we plot first?
461
00:30:05 --> 00:30:11
I like C equals one,
no, don't do C equals one.
462
00:30:10 --> 00:30:16
Let's do C equals zero,
first.
463
00:30:13 --> 00:30:19
C equals zero.
That's the line.
464
00:30:16 --> 00:30:22
y equals x plus 1.
465
00:30:19 --> 00:30:25
Okay, let me run and get that
chalk.
466
00:30:23 --> 00:30:29
So, I'll isoclines are in
orange.
467
00:30:27 --> 00:30:33
If so, when C equals zero,
y equals x plus one.
468
00:30:32 --> 00:30:38
So, let's say it's this curve.
C equals zero.
469
00:30:38 --> 00:30:44
How about C equals negative
one?
470
00:30:42 --> 00:30:48
Then it's y equals x plus two.
471
00:30:47 --> 00:30:53
It's this curve.
Well, let's label it down here.
472
00:30:53 --> 00:30:59
So, this is C equals negative
one.
473
00:30:57 --> 00:31:03
C equals negative two would be
y equals x, no,
474
00:31:02 --> 00:31:08
what am I doing?
C equals negative one is y
475
00:31:08 --> 00:31:14
equals x plus two.
That's right.
476
00:31:12 --> 00:31:18
Well, how about the other side?
If C equals plus one,
477
00:31:16 --> 00:31:22
well, then it's going to go
through the origin.
478
00:31:20 --> 00:31:26
It looks like a little more
room down here.
479
00:31:24 --> 00:31:30
How about, so if this is going
to be C equals one,
480
00:31:28 --> 00:31:34
then I sort of get the idea.
C equals two will look like
481
00:31:34 --> 00:31:40
this.
They're all going to be
482
00:31:37 --> 00:31:43
parallel lines because all
that's changing is the
483
00:31:42 --> 00:31:48
y-intercept, as I do this thing.
So, here, it's C equals two.
484
00:31:47 --> 00:31:53
That's probably enough.
All right, let's put it in the
485
00:31:53 --> 00:31:59
line elements.
All right, C equals negative
486
00:31:57 --> 00:32:03
one.
These will be perpendicular.
487
00:32:00 --> 00:32:06
C equals zero,
like this.
488
00:32:04 --> 00:32:10
C equals one.
Oh, this is interesting.
489
00:32:06 --> 00:32:12
I can't even draw in the line
elements because they seem to
490
00:32:10 --> 00:32:16
coincide with the curve itself,
with the line itself.
491
00:32:14 --> 00:32:20
They write y along the line,
and that makes it hard to draw
492
00:32:18 --> 00:32:24
them in.
How about C equals two?
493
00:32:20 --> 00:32:26
Well, here, the line elements
will be slanty.
494
00:32:23 --> 00:32:29
They'll have slope two,
so a pretty slanty up.
495
00:32:26 --> 00:32:32
And, I can see if a C equals
three in the same way.
496
00:32:31 --> 00:32:37
There are going to be even more
slantier up.
497
00:32:34 --> 00:32:40
And here, they're going to be
even more slanty down.
498
00:32:37 --> 00:32:43
This is not very scientific
terminology or mathematical,
499
00:32:41 --> 00:32:47
but you get the idea.
Okay, so there's our quick
500
00:32:45 --> 00:32:51
version of the direction field.
All we have to do is put in
501
00:32:49 --> 00:32:55
some integral curves now.
Well, it looks like it's doing
502
00:32:53 --> 00:32:59
this.
It gets less slanty here.
503
00:32:55 --> 00:33:01
It levels out,
has slope zero.
504
00:32:59 --> 00:33:05
And now, in this part of the
plain, the slope seems to be
505
00:33:03 --> 00:33:09
rising.
So, it must do something like
506
00:33:06 --> 00:33:12
that.
This guy must do something like
507
00:33:08 --> 00:33:14
this.
I'm a little doubtful of what I
508
00:33:11 --> 00:33:17
should be doing here.
Or, how about going from the
509
00:33:15 --> 00:33:21
other side?
Well, it rises,
510
00:33:17 --> 00:33:23
gets a little,
should it cross this?
511
00:33:20 --> 00:33:26
What should I do?
Well, there's one integral
512
00:33:23 --> 00:33:29
curve, which is easy to see.
It's this one.
513
00:33:26 --> 00:33:32
This line is both an isocline
and an integral curve.
514
00:33:32 --> 00:33:38
It's everything,
except drawable,
515
00:33:35 --> 00:33:41
[LAUGHTER] so,
you understand this is the same
516
00:33:41 --> 00:33:47
line.
It's both orange and pink at
517
00:33:45 --> 00:33:51
the same time.
But I don't know what
518
00:33:49 --> 00:33:55
combination color that would
make.
519
00:33:53 --> 00:33:59
It doesn't look like a line,
but be sympathetic.
520
00:34:00 --> 00:34:06
Now, the question is,
what's happening in this
521
00:34:04 --> 00:34:10
corridor?
In the corridor,
522
00:34:06 --> 00:34:12
that's not a mathematical word
either, between the isoclines
523
00:34:12 --> 00:34:18
for, well, what are they?
They are the isoclines for C
524
00:34:18 --> 00:34:24
equals two, and C equals zero.
How does that corridor look?
525
00:34:23 --> 00:34:29
Well: something like this.
Over here, the lines all look
526
00:34:29 --> 00:34:35
like that.
And here, they all look like
527
00:34:33 --> 00:34:39
this.
The slope is two.
528
00:34:36 --> 00:34:42
And, a hapless solution gets in
there.
529
00:34:39 --> 00:34:45
What's it to do?
Well, do you see that if a
530
00:34:43 --> 00:34:49
solution gets in that corridor,
an integral curve gets in that
531
00:34:49 --> 00:34:55
corridor, no escape is possible.
It's like a lobster trap.
532
00:34:54 --> 00:35:00
The lobster can walk in.
But it cannot walk out because
533
00:34:58 --> 00:35:04
things are always going in.
How could it escape?
534
00:35:03 --> 00:35:09
Well, it would have to double
back, somehow,
535
00:35:06 --> 00:35:12
and remember,
to escape, it has to be,
536
00:35:10 --> 00:35:16
to escape on the left side,
it must be going horizontally.
537
00:35:17 --> 00:35:23
But, how could it do that
without doubling back first and
538
00:35:20 --> 00:35:26
having the wrong slope?
The slope of everything in this
539
00:35:24 --> 00:35:30
corridor is positive,
and to double back and escape,
540
00:35:28 --> 00:35:34
it would at some point have to
have negative slope.
541
00:35:32 --> 00:35:38
It can't do that.
Well, could it escape on the
542
00:35:35 --> 00:35:41
right-hand side?
No, because at the moment when
543
00:35:39 --> 00:35:45
it wants to cross,
it will have to have a slope
544
00:35:42 --> 00:35:48
less than this line.
But all these spiky guys are
545
00:35:46 --> 00:35:52
pointing; it can't escape that
way either.
546
00:35:50 --> 00:35:56
So, no escape is possible.
It has to continue on,
547
00:35:53 --> 00:35:59
there.
But, more than that is true.
548
00:35:56 --> 00:36:02
So, a solution can't escape.
Once it's in there,
549
00:36:01 --> 00:36:07
it can't escape.
It's like, what do they call
550
00:36:04 --> 00:36:10
those plants,
I forget, pitcher plants.
551
00:36:07 --> 00:36:13
All they hear is they are going
down.
552
00:36:10 --> 00:36:16
So, it looks like that.
And so, the poor little insect
553
00:36:14 --> 00:36:20
falls in.
They could climb up the walls
554
00:36:17 --> 00:36:23
except that all the hairs are
going the wrong direction,
555
00:36:22 --> 00:36:28
and it can't get over them.
Well, let's think of it that
556
00:36:26 --> 00:36:32
way: this poor trap solution.
So, it does what it has to do.
557
00:36:32 --> 00:36:38
Now, there's more to it than
that.
558
00:36:35 --> 00:36:41
Because there are two
principles involved here that
559
00:36:39 --> 00:36:45
you should know,
that help a lot in drawing
560
00:36:43 --> 00:36:49
these pictures.
Principle number one is that
561
00:36:46 --> 00:36:52
two integral curves cannot cross
at an angle.
562
00:36:50 --> 00:36:56
Two integral curves can't
cross, I mean,
563
00:36:53 --> 00:36:59
by crossing at an angle like
that.
564
00:36:56 --> 00:37:02
I'll indicate what I mean by a
picture like that.
565
00:37:02 --> 00:37:08
Now, why not?
This is an important principle.
566
00:37:05 --> 00:37:11
Let's put that up in the white
box.
567
00:37:08 --> 00:37:14
They can't cross because if two
integral curves,
568
00:37:12 --> 00:37:18
are trying to cross,
well, one will look like this.
569
00:37:16 --> 00:37:22
It's an integral curve because
it has this slope.
570
00:37:20 --> 00:37:26
And, the other integral curve
has this slope.
571
00:37:24 --> 00:37:30
And now, they fight with each
other.
572
00:37:27 --> 00:37:33
What is the true slope at that
point?
573
00:37:32 --> 00:37:38
Well, the direction field only
allows you to have one slope.
574
00:37:36 --> 00:37:42
If there's a line element at
that point, it has a definite
575
00:37:40 --> 00:37:46
slope.
And therefore,
576
00:37:41 --> 00:37:47
it cannot have both the slope
and that one.
577
00:37:44 --> 00:37:50
It's as simple as that.
So, the reason is you can't
578
00:37:48 --> 00:37:54
have two slopes.
The direction field doesn't
579
00:37:51 --> 00:37:57
allow it.
Well, that's a big,
580
00:37:53 --> 00:37:59
big help because if I know,
here's an integral curve,
581
00:37:57 --> 00:38:03
and if I know that none of
these other pink integral curves
582
00:38:01 --> 00:38:07
are allowed to cross it,
how else can I do it?
583
00:38:06 --> 00:38:12
Well, they can't escape.
They can't cross.
584
00:38:09 --> 00:38:15
It's sort of clear that they
must get closer and closer to
585
00:38:13 --> 00:38:19
it.
You know, I'd have to work a
586
00:38:16 --> 00:38:22
little to justify that.
But I think that nobody would
587
00:38:20 --> 00:38:26
have any doubt of it who did a
little experimentation.
588
00:38:24 --> 00:38:30
In other words,
all these curves joined that
589
00:38:28 --> 00:38:34
little tube and get closer and
closer to this line,
590
00:38:32 --> 00:38:38
y equals x.
And there, without solving the
591
00:38:37 --> 00:38:43
differential equation,
it's clear that all of these
592
00:38:42 --> 00:38:48
solutions, how do they behave?
As x goes to infinity,
593
00:38:47 --> 00:38:53
they become asymptotic to,
they become closer and closer
594
00:38:52 --> 00:38:58
to the solution,
x.
595
00:38:54 --> 00:39:00
Is x a solution?
Yeah, because y equals x is an
596
00:38:58 --> 00:39:04
integral curve.
Is x a solution?
597
00:39:02 --> 00:39:08
Yeah, because if I plug in y
equals x, I get what?
598
00:39:07 --> 00:39:13
On the right-hand side,
I get one.
599
00:39:10 --> 00:39:16
And on the left-hand side,
I get one.
600
00:39:14 --> 00:39:20
One equals one.
So, this is a solution.
601
00:39:18 --> 00:39:24
Let's indicate that it's a
solution.
602
00:39:21 --> 00:39:27
So, analytically,
we've discovered an analytic
603
00:39:26 --> 00:39:32
solution to the differential
equation, namely,
604
00:39:31 --> 00:39:37
Y equals X, just by this
geometric process.
605
00:39:37 --> 00:39:43
Now, there's one more principle
like that, which is less
606
00:39:41 --> 00:39:47
obvious.
But you do have to know it.
607
00:39:44 --> 00:39:50
So, you are not allowed to
cross.
608
00:39:46 --> 00:39:52
That's clear.
But it's much,
609
00:39:49 --> 00:39:55
much, much, much,
much less obvious that two
610
00:39:52 --> 00:39:58
integral curves cannot touch.
That is, they cannot even be
611
00:39:57 --> 00:40:03
tangent.
Two integral curves cannot be
612
00:40:00 --> 00:40:06
tangent.
613
00:40:02 --> 00:40:08
614
615
616
00:40:10 --> 00:40:16
I'll indicate that by the word
touch, which is what a lot of
617
00:40:19 --> 00:40:25
people say.
In other words,
618
00:40:23 --> 00:40:29
if this is illegal,
so is this.
619
00:40:28 --> 00:40:34
It can't happen.
You know, without that,
620
00:40:33 --> 00:40:39
for example,
it might be,
621
00:40:35 --> 00:40:41
I might feel that there would
be nothing in this to prevent
622
00:40:39 --> 00:40:45
those curves from joining.
Why couldn't these pink curves
623
00:40:43 --> 00:40:49
join the line,
y equals x?
624
00:40:45 --> 00:40:51
You know, it's a solution.
They just pitch a ride,
625
00:40:49 --> 00:40:55
as it were.
The answer is they cannot do
626
00:40:52 --> 00:40:58
that because they have to just
get asymptotic to it,
627
00:40:55 --> 00:41:01
ever, ever closer.
They can't join y equals x
628
00:40:59 --> 00:41:05
because at the point where they
join, you have that situation.
629
00:41:05 --> 00:41:11
Now, why can't you to have
this?
630
00:41:09 --> 00:41:15
That's much more sophisticated
than this, and the reason is
631
00:41:17 --> 00:41:23
because of something called the
Existence and Uniqueness
632
00:41:24 --> 00:41:30
Theorem, which says that there
is through a point,
633
00:41:31 --> 00:41:37
x zero y zero,
that y prime equals f of
634
00:41:38 --> 00:41:44
(x, y) has only one,
635
00:41:43 --> 00:41:49
and only one solution.
One has one solution.
636
00:41:49 --> 00:41:55
In mathematics speak,
that means at least one
637
00:41:53 --> 00:41:59
solution.
It doesn't mean it has just one
638
00:41:56 --> 00:42:02
solution.
That's mathematical convention.
639
00:41:59 --> 00:42:05
It has one solution,
at least one solution.
640
00:42:02 --> 00:42:08
But, the killer is,
only one solution.
641
00:42:06 --> 00:42:12
That's what you have to say in
mathematics if you want just
642
00:42:10 --> 00:42:16
one, one, and only one solution
through the point
643
00:42:15 --> 00:42:21
x zero y zero.
So, the fact that it has one,
644
00:42:18 --> 00:42:24
that is the existence part.
The fact that it has only one
645
00:42:23 --> 00:42:29
is the uniqueness part of the
theorem.
646
00:42:26 --> 00:42:32
Now, like all good mathematical
theorems, this one does have
647
00:42:31 --> 00:42:37
hypotheses.
So, this is not going to be a
648
00:42:35 --> 00:42:41
course, I warn you,
those of you who are
649
00:42:39 --> 00:42:45
theoretically inclined,
very rich in hypotheses.
650
00:42:44 --> 00:42:50
But, hypotheses for those one
or that f of (x,
651
00:42:48 --> 00:42:54
y) should be a
continuous function.
652
00:42:52 --> 00:42:58
Now, like polynomial,
signs, should be continuous
653
00:42:57 --> 00:43:03
near, in the vicinity of that
point.
654
00:43:02 --> 00:43:08
That guarantees existence,
and what guarantees uniqueness
655
00:43:08 --> 00:43:14
is the hypothesis that you would
not guess by yourself.
656
00:43:14 --> 00:43:20
Neither would I.
What guarantees the uniqueness
657
00:43:19 --> 00:43:25
is that also,
it's partial derivative with
658
00:43:24 --> 00:43:30
respect to y should be
continuous, should be continuous
659
00:43:30 --> 00:43:36
near x zero y zero.
660
00:43:35 --> 00:43:41
Well, I have to make a
decision.
661
00:43:38 --> 00:43:44
I don't have time to talk about
Euler's method.
662
00:43:43 --> 00:43:49
I'll refer you to the,
there's one page of notes,
663
00:43:49 --> 00:43:55
and I couldn't do any more than
just repeat what's on those
664
00:43:55 --> 00:44:01
notes.
So, I'll trust you to read
665
00:43:59 --> 00:44:05
that.
And instead,
666
00:44:02 --> 00:44:08
let me give you an example
which will solidify these things
667
00:44:09 --> 00:44:15
in your mind a little bit.
I think that's a better course.
668
00:44:17 --> 00:44:23
The example is not in your
notes, and therefore,
669
00:44:22 --> 00:44:28
remember, you heard it here
first.
670
00:44:27 --> 00:44:33
Okay, so what's the example?
So, there is that differential
671
00:44:34 --> 00:44:40
equation.
Now, let's just solve it by
672
00:44:38 --> 00:44:44
separating variables.
Can you do it in your head?
673
00:44:42 --> 00:44:48
dy over dx, put all the y's on
the left.
674
00:44:44 --> 00:44:50
It will look like dy over one
minus y.
675
00:44:48 --> 00:44:54
Put all the dx's on the left.
So, the dx here goes on the
676
00:44:52 --> 00:44:58
right, rather.
That will be dx.
677
00:44:54 --> 00:45:00
And then, the x goes down into
the denominator.
678
00:44:57 --> 00:45:03
So now, it looks like that.
And, if I integrate both sides,
679
00:45:03 --> 00:45:09
I get the log of one minus y,
I guess, maybe with a,
680
00:45:08 --> 00:45:14
I never bothered with that,
but you can.
681
00:45:12 --> 00:45:18
It should be absolute values.
All right, put an absolute
682
00:45:17 --> 00:45:23
value, plus a constant.
And now, if I exponentiate both
683
00:45:23 --> 00:45:29
sides, the constant is positive.
So, this is going to look like
684
00:45:29 --> 00:45:35
y.
One minus y equals x
685
00:45:33 --> 00:45:39
And, the constant will be e to
686
00:45:36 --> 00:45:42
the C1.
And, I'll just make that a new
687
00:45:39 --> 00:45:45
constant, Cx.
And now, by letting C be
688
00:45:42 --> 00:45:48
negative, that's why you can get
rid of the absolute values,
689
00:45:45 --> 00:45:51
if you allow C to have negative
values as well as positive
690
00:45:49 --> 00:45:55
values.
Let's write this in a more
691
00:45:51 --> 00:45:57
human form.
So, y is equal to one minus Cx.
692
00:45:53 --> 00:45:59
Good, all right,
693
00:45:55 --> 00:46:01
let's just plot those.
So, these are the solutions.
694
00:46:00 --> 00:46:06
It's a pretty easy equation,
pretty easy solution method,
695
00:46:05 --> 00:46:11
just separation of variables.
What do they look like?
696
00:46:11 --> 00:46:17
Well, these are all lines whose
intercept is at one.
697
00:46:16 --> 00:46:22
And, they have any slope
whatsoever.
698
00:46:19 --> 00:46:25
So, these are the lines that
look like that.
699
00:46:24 --> 00:46:30
Okay, now let me ask,
existence and uniqueness.
700
00:46:29 --> 00:46:35
Existence: through which points
of the plane does the solution
701
00:46:35 --> 00:46:41
go?
Answer: through every point of
702
00:46:39 --> 00:46:45
the plane, through any point
here, I can find one and only
703
00:46:44 --> 00:46:50
one of those lines,
except for these stupid guys
704
00:46:48 --> 00:46:54
here on the stalk of the flower.
Here, for each of these points,
705
00:46:53 --> 00:46:59
there is no existence.
There is no solution to this
706
00:46:57 --> 00:47:03
differential equation,
which goes through any of these
707
00:47:02 --> 00:47:08
wiggly points on the y-axis,
with one exception.
708
00:47:07 --> 00:47:13
This point is oversupplied.
At this point,
709
00:47:10 --> 00:47:16
it's not existence that fails.
It's uniqueness that fails:
710
00:47:14 --> 00:47:20
no uniqueness.
There are lots of things which
711
00:47:18 --> 00:47:24
go through here.
Now, is that a violation of the
712
00:47:21 --> 00:47:27
existence and uniqueness
theorem?
713
00:47:24 --> 00:47:30
It cannot be a violation
because the theorem has no
714
00:47:28 --> 00:47:34
exceptions.
Otherwise, it wouldn't be a
715
00:47:31 --> 00:47:37
theorem.
So, let's take a look.
716
00:47:34 --> 00:47:40
What's wrong?
We thought we solved it modulo,
717
00:47:37 --> 00:47:43
putting the absolute value
signs on the log.
718
00:47:40 --> 00:47:46
What's wrong?
The answer: what's wrong is to
719
00:47:43 --> 00:47:49
use the theorem you must write
the differential equation in
720
00:47:48 --> 00:47:54
standard form,
in the green form I gave you.
721
00:47:51 --> 00:47:57
Let's write the differential
equation the way we were
722
00:47:54 --> 00:48:00
supposed to.
It says dy / dx equals one
723
00:47:57 --> 00:48:03
minus y divided by x.
724
00:48:02 --> 00:48:08
And now, I see,
the right-hand side is not
725
00:48:05 --> 00:48:11
continuous, in fact,
not even defined when x equals
726
00:48:09 --> 00:48:15
zero, when along the y-axis.
And therefore,
727
00:48:12 --> 00:48:18
the existence and uniqueness is
not guaranteed along the line,
728
00:48:16 --> 00:48:22
x equals zero of the y-axis.
And, in fact,
729
00:48:20 --> 00:48:26
we see that it failed.
Now, as a practical matter,
730
00:48:23 --> 00:48:29
it's the way existence and
uniqueness fails in all ordinary
731
00:48:28 --> 00:48:34
life work with differential
equations is not through
732
00:48:32 --> 00:48:38
sophisticated examples that
mathematicians can construct.
733
00:48:38 --> 00:48:44
But normally,
because f of (x,
734
00:48:40 --> 00:48:46
y) will fail to be
defined somewhere,
735
00:48:43 --> 00:48:49
and those will be the bad
points.
736
00:48:46 --> 00:48:52
Thanks.