Differential EquationsLecture #1

Suggested Questions

What is the standard form of the first order ODE?

What is the geometric view? Compare it with the analytic view.

How do you draw a direction field?

What is the second worked-out example?

What is the existence and uniqueness theorem?

Words and Phrases From the Lecture

Introduction

recitation

separate variables

18.01 (= Single Variable Calculus)

First Order ODEs

ODE (ordinary differential equation)

acronym

isolate the derivative of y with respect to x, let’s say,

solve by separating variables

The most dissimilar about them is

elementary functions

confront the most significant fact

I mean, not really bad, but recalcitrant.

those equations are the rule rather than the exception

devote this first day to not solving

geometric way of looking at equations

numerical ways

Geometric View of OEDs

analytic method

write down explicitly the equation

y’=f(x, y) (y prime equals f of x, y)

letters multiply so quickly

a solution to this differential equation

containing a arbitrary constant

direction field

integral curve

It never hurts to get a little more practice.

in any event

the computer stuff that you’ll be doing on the problem set

to a certain amount should be a novelty to you

take the plan

line element

slope

you fill out the plane with these things

until you are tired putting them in

Let’s not make them all go the same way.

That sort of seems cheating

tangent to the line element there

a graph of the solution

I’ll make a little theorem out of that.

if and only if

the curve associated with this

plug it into the differential equation

calculate its derivative

This leaves us an interesting question.

How to Draw a Direction Field

they have unlimited amounts of energy to waste

a naïve one

from a 18.02 point of view

a level curve of f(x, y) corresponding to the level of value C

isocline

we’d use dashed lines for doing them

Don’t shoot me if they are not.

Examples

I’d better make two steps out of this.

It’s a line, in fact, through the origin.

Wait a minute. Something’s wrong.

Notice they are always going to be perpendicular.

Tow lines whose slopes are negative reciprocals are perpendicular.

which is exactly the y axis

C equals infinity

by common consent

vertical

It’s an elementary exercise, of which I would not deprive you of your pleasure.

The circles are the ones with the center at the origin, of course.

radius

the domain of that solution

it’s not the whole x-axis

some feeling for how the solution will behave to this equation

They are all gonna be parallel lines.

pretty slanty up

Here it levels out, has slope zero.

What’s happening in this corridor?

It’s like a lobster trap.

All these spiky guys are pointing, it can’t escape that way, either.

pitcher plant

when x goes to infinity, they become asymptotic to the solution x.

hitch a ride, as it were

existence and uniqueness theorem

in mathematics-speak

those of you who are theoretically inclined

continuous function

the partial derivative with respect to y

Euler’s method

solidify these things on your mind a little bit

x goes to the denominator

integrate both sides

absolute value

exponentiate both sides

These are all lines whose intercept is 1.

[笠原xx]は笠原皓司著『微分積分学』サイエンス社のxxページを指す。

recitation 授業に対応した演習。以下のアドレスから入手できる。

1801 here = “Single Variable Calculus”(MITの１変数微積の授業のこと)

The book has a very long and good explanation of it. この授業の教科書は以下の通り。

アマゾンで約8500円。

Edwards and Penny. 2004. Elementary Differential Equations with Boundary Value Problems

5th edition. Pearson Education.

ordinary differential equations (ODE’s) 常微分方程式 cf. partial differential equations 偏微分方程式

the standard form of first order ODEs １階微分方程式の標準形

dy/dx=f(x,y)

いわゆる正規型微分方程式[笠原218]

You isolate the derivative (１階導関数) of y with respect to x on the left hand side

problem set 数回分の授業の復習用練習問題。下のアドレスから入手可。

y’(y prime) the (first-order) derivative of y

y’= f(x,y) (y prime equals f of x and y)

y’= x/y (y prime equals x over y) You can solve this by separating the valuables.

y’= x−y2 (y prime equals x minus y squared)
y’= y−x2 (y prime equals y minus x squared)
In neither of these, you can separate the variables and they look extremely similar. But they are extremely dissimilar. The most dissimilar about them is that this one (y’= y−x2) is easily solvable. This one (y’= x−y2), 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.

elementary functions 初等関数[笠原70]

not really bad but recalcitrant扱いにくい

Those equations are rule rather than exception.

So this first day is going to be devoted to geometric ways of looking at differential equations and, 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.

numerical ways 数値解析を指す

analytic view of ODEs geometric view of ODEs [笠原223]
(= solving ODEs by finding elementary functions)
(1)explicitly writing down the equation  drawing the direction field
(2)finding an elementary function that solves the equation  drawing an integral curve

y1(x) (y one 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, like rabbits, multiply in the sense of rabbits and that after a while, if you keep using different letters for each new idea, you can’t figure out what you’re talking about.

Of course, differential equations have many solutions containing an arbitrary constant(任意定数).

direction field 方向場[笠原223]

Those of you who had PC syllabus in high school (高校のコンピューターの授業) should know these things.

In any event, I think the computer stuff that you’re doing in the problem set is, in a certain amount, it should be novelty to you. It’s a novelty to me, so, why not, to you.

slope (接線の)傾き

Let’s not make them all go the same way. That seems to be sort of cheating.

An integral curve is a curve which goes through the plane, and at every point tangent to the line element there.

The integral curve (solution curveとも)has the direction of the field everywhere at all points (on the curve).

I will make a little theorem(定理) out of that.

y1(x) is a solution to the ODE y’=f(x,y) (if and only if) the graph of y1(x) is an integral curve (of the direction field associated with that equation)

P if and only if Q (必要十分条件の表し方；iffと略記)

plug in 代入する

If you plug it into the differential equation, it satisfies it.

This leaves us with an interesting question.

So this is for you to get a feeling for it.

How to Draw Direction Field
computer
  1. Pick the point (x,y) (equally spaced).
  2. Compute the value of f(x,y).
  3. Draw at a point (x,y) the line element having the slope of f(x,y).
human
  1. Pick the value of the slope C.
  2. Plot the equation (f(x,y)=C). (isocline represented by a dashed line)
  3. Draw in line elements having the slope C on the isocline.

They (=computers) go very fast and have an unlimited amount of energy to waste.

This is what it means to be a human.

You use your intelligence. From a human point of view, this stuff has been done in the wrong order, because for each new point, it requires the recalculation of f(x,y). That is horrible.

It’s an ordinary curve. But which curve will depend. It’s in fact from a 1802 (1801の次の微積の授業)point of view, it’s the level curve of the function f(x,y) corresponding to the level value of C.

level curve f(x,y)が一定の値(＝C)をとる時f(x,y)=Cを満たす点(x,y)のx-y平面上のグラフ

Instead, we’re going to call it isocline. 等高線[笠原224；等傾斜法]

dashed line 破線 isoclineを表す

solid line 実線 integral curveを表す

So for the next few minutes, I’d like to work on a few examples for you to show how this works out for practice.

Example 1
y’= −x/y
What are the isoclines? −x/y = c y = −1/c x
Draw the isoclines for c=1, −1
The isocline is y-axis (x = 0) when c = 0.
x-axis is not included.
The integral curves are circles.

Maybe, I’d better make two steps out of this.

y= −1/c x (y equals negative one over c times x)

The isocline is going to be a line through the origin(原点を通る直線).

C= −1 (C equals negative one)

Wait a minute. Something’s wrong. I’m sorry? (もう一度言ってくれ)

Notice they(=isoclineとその上に書き込まれた傾きcのline element) are perpendicular(直交している).

Those numbers, –1/c and c, are negative reciprocals(積が－１になる). And you know the two lines whose slopes are negative reciprocals are perpendicular.

reciprocal 逆数

Here’s a controversial isocline. Is that (=y-軸) an isocline? That doesn’t correspond to anything looking like this (= −1/c x). But it would if I put c multiplied through by c and then it will correspond to c being 0. In other words, don’t write like this. Multiplied through by c, we’ll read cy = −x and then when c is 0, I had x = 0, which is exactly y-axis. So that really is included.

y axis y-軸

x axis x-軸

infinity 無限大

The x-axis is not included. However, most people include it (=x-軸) anyway. This is very common to be a sort of sloppy, bendingthe edges of corners a little bit(厳密性を貫かないこと), you know, hoping nobody will notice. We’ll say that corresponds to c = infinity.

So if c is infinity, that means that the little line segments should have infinite slope, by common consent, it should be vertical.

And it’s an elementary exercise of which I will not deprive you of the pleasure. Solve the ODEs by separation of variables.

radius 半径

√c12－x2 square root of c one squared minus x squared

the domain(定義域) of the solution

where the solution is defined

There’s no way of predicting by looking at a differential equation that the typical solution is going to have a limited domain like that.

You don’t know what the domain of the solution is going to be until you actually calculated it.

Example 2
y’=1＋ｘ－y
The isocline will have the equation y=x＋(1－c)
When c=0, y＝x＋1
y=xは解のひとつ。他の解はy=xを漸近線にもつ。
As x goes to infinity, y(x) becomes asymptotic to x. しだいにｘに近づく(cf. asymptote 漸近線)

I sort of get the idea.

They are all parallel lines(平行線).

The line elements are slanty(傾きが急), have the slope of 2, pretty slanty up

There’s one integral curve which is easy to see,

It’s everything, except drawable.

What’s happening in this corridor (between the isoclines of C=2 and C=0)?

And a hapless solution gets in there. What’s it to do

If a solution gets in the corridor, no escape is possible.It’s like a lobster trap.

pitcher plant ハエジゴク

There are two principles involved here that you should know.

Two principles on integral curves
Two integral curves can’t cross at an angle.
Two integral curves cannot be tangent. (微分方程式の解の一意性による)
Existence and Uniqueness Theorem
The existence and uniqueness theorem says that y’=f(x,y) has one and only one solutionthrough the point of (x0,y0).
解の存在と一意性の定理[笠原225－228]

It’s as simple as that.

They hitch a ride, as it were.

That’s a mathematical convention.

The killer(驚くべき、このすごいこと) is that you have only one solution.

polynomial多項式

continuous function連続関数

the partial derivative with respect to y yに関する偏導関数

Euler’s methodオイラー方(近似解を求める方法；数値解析の入門書参照)

denominator 分母

absolute value 絶対値

If I integrate the both sides(両辺を積分すると), I get the log of 1－y.

自然対数(natural logarithm)をlnで表す。

If I exponentiate both sides, (a=bからea=ebを得ること)

stalk(茎) of the flower

Example 3
xy’=y－1(変数分離形で解はy=Cx＋1)
isoclineは傾きをcとして、y’= (y－1)/x=cより,直線y=cx＋1
この微分方程式のstandard form(正規形)はy’= (y－1)/xだから、右辺はx=0で定義されてない。したがって、このとき(y軸上)は解の存在と一意性は成り立たない。

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