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• OK, let's get started. Now... I'm assuming that, A, you went recitation yesterday,

• B, that even if you didn't, you know how to separate variables, and you know how to construct simple

• models, solve physical problems with differential equations, and possibly even solve them.

• So, you should have learned that either in high school, or 18.01 here, or... yeah.

• So, I'm going to start from that point, assume you know that. I'm not going to tell you what differential

• equations are, or what modeling is. If you still are uncertain about those things, the

• book has a very long and good explanation of it. Just read that stuff. So,

• we are talking about first order ODEs.

• ODE: I'll only use three ... two acronyms. ODE is ordinary differential equations. I think all of MIT

• knows that, whether they've been taking the course or not. So, we are talking about first-order ODEs.

• which in standard form, are written, you isolate the derivative of y with respect

• to, x, let's say, on the left-hand side, and on the right-hand side you write everything

• else. You can't always do this very well, but for today, I'm going to assume that it

• has been done and it's doable. So, for example, some of the ones that will be considered either

• today or in the problem set are things like

• oh... y' = x / y

• That's pretty simple. The problem set has y' = ...let's see...

• x - y^2.

• And, it also has y' = y - x^2.

• There are others, too. Now, when you look at this, this, of course, you can solve by

• separating variables. So, this is solvable. This one is-- and neither of these can you

• separate variables. And they look extremely similar. But they are extremely dissimilar.

• The most dissimilar about them is that this one is easily solvable. And you will learn,

• if you don't know already, next time next Friday how to solve this one

• This one, which looks almost the same, is unsolvable in a certain sense. Namely, there

• are no elementary functions which you can write down, which will give a solution of

• that differential equation. So, right away, one confronts the most significant fact that

• even for the simplest possible differential equations, those which only involve the first

• derivative, it's possible to write down extremely looking simple guys.

• I'll put this one up in blue to indicate that it's bad. Whoops, sorry, I mean, not really

• bad, but recalcitrant. It's not solvable in the ordinary sense in which you think of an

• equation is solvable. And, since those equations are the rule rather than the exception, I'm

• going about this first day to not solving a single differential equation, but indicating

• to you what you do when you meet a blue equation like that.

• What do you do with it? So, this first day is going to be devoted to geometric ways of

• looking at differential equations and numerical. At the very end, I'll talk a little bit about

• numerical ways. And you'll work on both of those for the first problem set. So,

• what's our geometric view of differential equations?

• Well, it's something that's contrasted with

• the usual procedures, by which you solve things and find elementary functions which solve

• them. I'll call that the analytic method. So, on the one hand, we have the analytic

• ideas, in which you write down explicitly the equation, y' = f(x,y).

• And, you look for certain functions, which are called its solutions. Now, so there's

• the ODE. And, y1 of x, notice I don't use a separate letter. I don't use g or h or something

• like that for the solution because the letters multiply so quickly, that is, multiply in

• the sense of rabbits, that after a while, if you keep using different letters for each

• new idea, you can't figure out what you're talking about.

• So, I'll use y1 means, it's a solution of this differential equation. Of course, the

• differential equation has many solutions containing an arbitrary constant. So, we'll call this

• the solution. Now, the geometric view,

• the geometric guy that corresponds to this version

• of writing the equation, is something called a direction field.

• And, the solution is, from

• the geometric point of view, something called an integral curve.

• So, let me explain if you

• don't know what the direction field is. I know for some of you, I'm reviewing what you

• learned in high school. Those of you who had the BC syllabus in high school should know

• these things. But, it never hurts to get a little more practice. And, in any event, I

• think the computer stuff that you will be doing on the problem set, a certain amount

• of it should be novel to you.

• It was novel to me, so why not to you? So, what's a direction field? Well, the direction

• field is, you take the plane, and in each point of the plane-- of course, that's an

• impossibility. But, you pick some points of the plane. You draw what's called a little

• line element. So, there is a point. It's a little line, and the only thing which distinguishes

• it outside of its position in the plane, so here's the point, (x,y), at which we are drawing

• this line element, is its slope. And, what is its slope? Its slope is to be f(x,y).

• And now, You fill up the plane with these things until you're tired of putting then in. So,

• I'm going to get tired pretty quickly.

• So, I don't know, let's not make them all go the same way. That sort of seems cheating.

• How about here? Here's a few randomly chosen line elements that I put in, and I putted

• the slopes at random since I didn't have any particular differential equation in mind.

• Now, the integral curve, so those are the line elements. The integral curve is a curve,

• which goes through the plane, and at every point is tangent to the line element there.

• So, this is the integral curve. Hey, wait a minute, I thought tangents were the line

• element there didn't even touch it. Well, I can't fill up the plane with line elements.

• Here, at this point, there was a line element, which I didn't bother drawing in. And, it

• was tangent to that. Same thing over here: if I drew the line element here, I would find

• that the curve had exactly the right slope there.

• So, the point is the integral, what distinguishes the integral curve is that everywhere it has

• the direction, that's the way I'll indicate that it's tangent, has the direction of the

• field everywhere at all points on the curve, of course, where it doesn't go. It doesn't

• have any mission to fulfill. Now, I say that this integral curve is the graph of the solution

• to the differential equation. In other words, writing down analytically the differential

• equation is the same geometrically as drawing this direction field, and solving analytically

• for a solution of the differential equation is the same thing as geometrically drawing

• an integral curve. So, what am I saying?

• I say that an integral curve,

• all right, let me write it this way. I'll make a little theorem

• out of it, that y1(x) is a solution to the differential equation

• if, and only if,

• the graph, the curve associated with this, the graph of y1 of x is an integral curve.

• Integral curve of what? Well, of the direction field associated with that equation. But there isn't

• quite enough room to write that on the board. But, you could put it in your notes, if you

• take notes. So, this is the relation between the two, the integral curves of the graphs

• or solutions.

• Now, why is that so? Well, in fact, all I have to do to prove this, if you can call

• it a proof at all, is simply to translate what each side really means. What does it

• really mean to say that a given function is a solution to the differential equation? Well,

• it means that if you plug it into the differential equation, it satisfies it. Okay, what is that?

• So, how do I plug it into the differential equation and check that it satisfies it?

• Well, doing it in the abstract, I first calculate its derivative. And then, how will it look

• after I plugged it into the differential equation? Well, I don't do anything to the x, but wherever

• I see y, I plug in this particular function. So, in notation, that would be written this

• way. So, for this to be a solution means this,

• that that equation is satisfied. Okay, what

• does it mean for the graph to be an integral curve? Well, it means that at each point,

• the slope of this curve, it means that the slope of y1 of x should be, at each point,

• (x1,y1). It should be equal to the slope of the direction field at that point.

• And then, what is the slope of the direction field at that point? Well, it is f of that

• particular, well, at the point, (x,y1). If you like, you can put a subscript, one, on

• there, send a one here or a zero there, to indicate that you mean a particular point.

• But, it looks better if you don't. But, there's some possibility of confusion. I admit to

• that. So, the slope of the direction field, what is that slope? Well, by the way, I calculated

• the direction field. Its slope at the point was to be x, whatever the value of x was,

• and whatever the value of y1(x) was, substituted into the right-hand side of the equation.

• So, what the slope of this function of that curve of the graph should be equal to the

• slope of the direction field. Now, what does this say?

• Well, what's the slope of y1(x)? That's y1'(x). That's from the first day of 18.01, calculus.

• What's the slope of the direction field? This? Well, it's this. And, that's with the right

• hand side. So, saying these two guys are the same or equal, is exactly, analytically, the

• same as saying these two guys are equal. So, in other words, the proof consists of, what

• does this really mean? What does this really mean? And after you see what both really mean,

• you say, yeah, they're the same.

• So, I don't how to write that. It's okay: same, same, how's that? This is the same as that.

• Okay, well, this leaves us the interesting question of how do you draw a direction from

• the, well, this being 2003, mostly computers draw them for you. Nonetheless, you do have

• to know a certain amount. I've given you a couple of exercises where you have to draw

• the direction field yourself. This is so you get a feeling for it, and also because humans

• don't draw direction fields the same way computers do. So, let's first of all, how did computers

• do it? They are very stupid. There's no problem.

• Since they go very fast and have unlimited amounts of energy to waste, the computer method

• is the naive one. You pick the point. You pick a point, and generally, they are usually

• equally spaced. You determine some spacing, that one: blah, blah, blah, blah, blah, blah,

• blah, equally spaced. And, at each point, it computes f(x, y) at the point, finds, meets,

• and computes the value of f of (x, y), that function, and the next thing is, on the screen,

• it draws, at (x, y), the little line element having slope f(x, y). In other words, it does

• what the differential equation tells it to do.

• And the only thing that it does is you can, if you are telling the thing to draw the direction

• field, about the only option you have is telling what the spacing should be, and sometimes

• people don't like to see a whole line. They only like to see a little bit of a half line.

• And, you can sometimes tell, according to the program, tell the computer how long you

• want that line to be, if you want it teeny or a little bigger. Once in awhile you want

• you want it narrower on it, but not right now.

• Okay, that's what a computer does. What does a human do? This is what it means to be human.

• You use your intelligence. From a human point of view, this stuff has been done in the wrong

• order. And the reason it's been done in the wrong order: because for each new point, it

• requires a recalculation of f(x, y). That is horrible. The computer doesn't mind, but

• a human does. So, for a human, the way to do it is not to begin by picking the point,

• but to begin by picking the slope that you would like to see. So, you begin by taking

• the slope. Let's call it the value of the slope, C. So, you pick a number. C is two.

• I want to see where are all the points in the plane where the slope of that line element

• would be two? Well, they will satisfy an equation.

• The equation is f(x,y) = C, in general. So, what you do is plot this, plot the equation,

• plot this equation. Notice, it's not the differential equation. You can't exactly plot a differential

• equation. It's a curve, an ordinary curve. But which curve will depend; it's, in fact,

• from the 18.02 point of view, the level curve of C, sorry, it's a level curve of f of (x,

• y), the function f of x and y corresponding to the level of value C

• But we are not going to call it that because this is not 18.02. Instead, we're going to