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Laws of vector algebra and unit vectors

Laws of vector algebra

If $\bold{A}$ , $\bold{B}$ and $\bold{C}$ are vectors, and $m$ and $n$ are scalars, then
$\begin{array}{lll}1. & \bold{A} + \bold{B} = \bold{B} + \bold{A} & Communicative Law for Addition\\ 2. & \bold{A} + (\bold{B} + \bold{C}) = (\bold{A} + \bold{B}) + \bold{C} & Associative Law for Addition\\ 3. & m(n\bold{A}) = (mn)\bold{A} = n(m\bold{A}) & Associative Law for Multiplication\\ 4. & (m + n)\bold{A} = m\bold{A} + n\bold{A} & Distributive Law\\ 5. & m(\bold{A} + \bold{B}) = m\bold{A} + m\bold{B} & Distributive Law \end{array}$
Note that in these laws only multiplication of a vector by one or more scalars is defined.

Unit vectors

Unit vectors are vectors having unit length. If $\bold{A}$ is any vector with length $A > 0$ , then $\bold{A}/A$ is a unit vector, denoted by $\bold{a}$ , having the same direction as $\bold{A}$ .

ベクトル代数の法則と単位ベクトル

ベクトル代数の法則

　 $\bold{A}$ , $\bold{B}$ および $\bold{C}$ がベクトルであり $m$ および $n$ がスカラーであるとすると

$\begin{array}{lll}1. & \bold{A} + \bold{B} = \bold{B} + \bold{A} & Communicative Law for Addition\\ 2. & \bold{A} + (\bold{B} + \bold{C}) = (\bold{A} + \bold{B}) + \bold{C} & Associative Law for Addition\\ 3. & m(n\bold{A}) = (mn)\bold{A} = n(m\bold{A}) & Associative Law for Multiplication\\ 4. & (m + n)\bold{A} = m\bold{A} + n\bold{A} & Distributive Law\\ 5. & m(\bold{A} + \bold{B}) = m\bold{A} + m\bold{B} & Distributive Law \end{array}$

上記の法則は一つのベクトルと一つ以上のスカラーの積算に適用されることを強調しておきます．

単位ベクトル

　単位ベクトルは単位長を有します．仮に $\bold{A}$ が長さが $A > 0$ のベクトルなら $\bold{A}/A$ は単位ベクトルであり， $\bold{a}$ と記述し， $\bold{A}$ と同じ方向を有します．

Vector algebra

The operations of addition, subtraction and multiplication familiar in the algebra of numbers are, with suitable definition, capable of extension to an algebra of vectors. The following definitions are fundamental.

Two vectors $\bold{A}$ and $\bold{B}$ are equal if they have the same magnitude and direction regardless of their initial points.

A vector having direction opposite to that of vector $\bold{A}$ but with the same magnitude is denoted by $-\bold{A}$ .

The sum or resultant of vectors $\bold{A}$ and $\bold{B}$ is a vector $\bold{C}$ formed by placing the initial point of $\bold{B}$ on the terminal point of $\bold{A}$ and joining the initial point of $\bold{A}$ to the terminal point of $\bold{B}$ . The sum $\bold{C}$ is written $\bold{C} = \bold{A} + \bold{B}$ . The definition here is equivalent to the parallelogram law for vector addition.

The difference of vectors $\bold{A}$ and $\bold{B}$ , represented by $\bold{A} - \bold{B}$ , is that vector $\bold{C}$ which added to $\bold{B}$ gives $\bold{A}$ . Equivalently, $\bold{A} - \bold{B}$ may be defined as $\bold{A} + (-\bold{B})$ . If $\bold{A} = \bold{B}$ , then $\bold{A} - \bold{B}$ is defined as the null or zero vector and is represented by the symbol $\bold{0}$ . This has a magnitude of zero but its direction is not defined.

Multiplication of vector $\bold{A}$ by a scalar m produces a vector $m\bold{A}$ with magnitude $|m|$ times the magnitude of $\bold{A}$ and direction the same as or opposite to that of $\bold{A}$ according as $m$ is positive or negative. If $m = 0$ , $m\bold{A} = \bold{0}$ , the null vector.

ベクトル代数

　実数の代数においておなじみの加算，減算，積算の演算は，適切に定義すればベクトル代数にも拡張可能です．下記の定義は基本的なものです．

二つのベクトル $\bold{A}$ と $\bold{B}$ が同じ大きさと方向を有するなら，始点が異なっても 等しい．
あるベクトル $\bold{A}$ と反対の方向を有するが大きさの同じベクトルは $-\bold{A}$ と記述する．
ベクトル $\bold{A}$ と $\bold{B}$ の和または結果がベクトル $\bold{C}$ であり， $\bold{B}$ の始点を $\bold{A}$ の終点に置き，また $\bold{A}$ の始点を $\bold{B}$ の終点に結合して得られる．和 $\bold{C}$ は $\bold{C} = \bold{A} + \bold{B}$ と記述される．ここでの定義はベクトル加算の 平行四辺形の法則 に等しい．
ベクトル $\bold{A}$ と $\bold{B}$ との減算は $\bold{A} - \bold{B}$ と表現し，ベクトル $\bold{B}$ に $\bold{C}$ を加算すると $\bold{A}$ が得られることである．同様に， $\bold{A} - \bold{B}$ は $\bold{A} + (-\bold{B})$ として定義される．仮に $\bold{A} = \bold{B}$ の時， $\bold{A} - \bold{B}$ はヌルまたは 零ベクトル と定義され，記号 $\bold{0}$ で表現される．これは大きさがゼロで方向は定義されていない．
ベクトル $\bold{A}$ にスカラー m を積算する処理はベクトル $m\bold{A}$ であり大きさがベクトル $\bold{A}$ の $|m|$ 倍であり，方向がベクトル $\bold{A}$ と同じか正反対であり， $m$ が正負いずれを取るのかに依存する．仮に $m = 0$ の時は $m\bold{A} = \bold{0}$ となり，ヌルベクトルである．

Vectors and scalars

There are quantities in physics characterized by both magnitude and direction, such as displacement, velocity force and acceleration. To describe such quantities, we introduce the concept of a vector as a directed line segment $\overrightarrow{PQ}$ from one point P called the initial point to another point Q called the terminal point. We denote vectors by bold faced letters or letters with an arrow over them. Thus $\overrightarrow{PQ}$ is denoted by $\bold A$ or $\vec{A}$ . The magnitude or length of the vector is then denoted by $|\overrightarrow{PQ}|$ , $\overline{PQ}$ , $|\bold{A}|$ or $|\overrightarrow{A}|$ .

Other quantities in physics are characterized by magnitude only, such as mass, length and temperature. Such quantities are often called scalars to distinguish them from vectors, but it must be emphasized that apart from units such as feet, degrees, etc., they are nothing more than real numbers. We can thus denote them by ordinary letters as usual.

ベクトルとスカラー

　物理学において量とは，大きさと方向が特徴です．例えば変位，速度，力，加速度など．これらの量を記述するためにベクトルという概念を導入しましょう．有向線分 $\overrightarrow{PQ}$ はある点 P を始点と呼び，もう一方の点 Q を終点と呼びます．ベクトルを太字の文字または上に矢印のついた文字で記述しましょう．ゆえに $\overrightarrow{PQ}$ は $\bold A$ または $\vec{A}$ とも記述できます．ゆえにベクトルの 大きさ や長さは $|\overrightarrow{PQ}|$ , $\overline{PQ}$ , $|\bold{A}|$ または $|\overrightarrow{A}|$ と記述されます．

　物理学における他の量は大きさだけという特徴があります．例えば質量や長さ，温度など．それらの量はしばしば スカラー と呼ばれ，ベクトルとは区別されます．一つ強調しておかないといけないのは，フィートや度といった単位とは別に，それらは実数に限らないということです．ゆえにそれらを普通の文字で一般的に記述できます．

General linear differential equation of order n

The general linear differential equation of order n has the form
$\displaystyle a_0(x)\frac{d^ny}{dx^n} + a_1(x)\frac{d^{n-1}y}{dx^{n-1}} + \cdots + a_{n-1}(x)\frac{dy}{dx} + a_n(x)y = R(x)\ \ \ (1)$
A differential equation which cannot be written in this form is called nonlinear.

$\displaystyle x\frac{d^2y}{dx^2} + 3\frac{dy}{dx} - 2xy = \sin x$ is a second order linear equation. $\displaystyle y\frac{d^2y}{dx^2} - x\left(\frac{dy}{dx}\right)^2 + x^2y = e^{-x}$ is a second order nonlinear equation.

If R(x), the right side of (1), is replaced by zero the resulting equation is called the complementary, reduced or homogeneous equation. If R(x) ≠ 0, the equation is called the complete or nonhomogeneous equation.

If $\displaystyle x\frac{d^2y}{dx^2} + 3\frac{dy}{dx} -2xy =\sin x$ is the complete equation, then $\displaystyle x\frac{d^2y}{dx^2} + 3\frac{dy}{dx} - 2xy = 0$ is the corresponding complementary, reduced or homogeneous equation.

If $a_0(x)\, \cdots \,a_n(x)$ are all constants, (1) is said to have constant coefficient, otherwise it is said to have variable coefficients.

n階の一般線形微分方程式

　n 階の一般的な線形微分方程式は次の形を取ります．

$\displaystyle a_0(x)\frac{d^ny}{dx^n} + a_1(x)\frac{d^{n-1}y}{dx^{n-1}} + \cdots + a_{n-1}(x)\frac{dy}{dx} + a_n(x)y = R(x)\ \ \ (1)$

　この形で書けない微分方程式は非線形と呼ばれます．

$\displaystyle x\frac{d^2y}{dx^2} + 3\frac{dy}{dx} - 2xy = \sin x$ は 2 階の線形微分方程式です． $\displaystyle y\frac{d^2y}{dx^2} - x\left(\frac{dy}{dx}\right)^2 + x^2y = e^{-x}$ は 2 階の非線形微分方程式です．

　仮に式 (1) の右辺 R(x) をゼロに置換した場合その方程式は相補，縮約または同次方程式と呼ばれます．仮に R(x) ≠ 0 の時，その方程式は完全または 非同次 方程式と呼ばれます．

　仮に $\displaystyle x\frac{d^2y}{dx^2} + 3\frac{dy}{dx} -2xy =\sin x$ が完全方程式の時，ゆえに $\displaystyle x\frac{d^2y}{dx^2} + 3\frac{dy}{dx} - 2xy = 0$ は対応する相補，縮約または同次方程式です．

　仮に $a_0(x)\, \cdots \,a_n(x)$ が全て定数の時， (1) は constant coefficient を持つと言われ，そうでなければ 変数係数 を持つと言われます．

Numerical methods for solving differential equations

Given the boudary-value problem
$\displaystyle dy/dx = f(x, y)\ \ \ y(x_0) = y_0\ \ \ (1)$
it may not be possible to obtain an exact solution. In such case various methods are available for obtaining an approximate or numerical solution. In the following we list several methods.

Step by step or Euler method

Taylor series method

Picard’s method

Runge-Kutta method

1. Step by step or Euler method

In this method we replace the differential equation of (1) by the approximation
$\displaystyle \frac{y(x_0 + h)-y(x_0)}{h} = f(x_0, y_0)\ \ \ (2)$
so that
$\displaystyle y(x_0 + h) = y(x_0) + hf(x_0, y_0)\ \ \ (3)$
By continuity in this manner we can then find $y(x_0 + 2h),\ y(x_0 + 3h)$ , etc. We choose $h$ sufficiently small so as to obtain good approximations.

A modified procedure of this method can also be used.

2. Taylor series method

By successive differentiation of the differential equation in (1) we can find $y'(x_0),\ y''(x_0),\ y'''(x_0),\cdots$ . Then the solution is given by the Taylor series
$\displaystyle y(x) = y(x_0) + y'(x_0)(x - x_0) + \frac{y''(x_0)(x - x_0)^2}{2!} + \cdots\ \ \ (4)$
assuming that the series converges. If it does we can obtain $y(x_0 + h)$ to any desired accuracy.

3. Picard’s method

By integrating the differential equation in (1) and using the boundary condition, we find
$\displaystyle y(x) = y_0 + \int^{x}_{x_0}\! f(u, y)du\ \ \ (5)$
Assuming the approximation $y_1(x) = y_0$ , we obtain from (5) a new approximation.
$\displaystyle y_2(x) = y_0 + \int^{x}_{x_0}\! f(u, y_1)du\ \ \ (6)$
Using this in (5) we obtain another approximation.
$\displaystyle y_3(x) = y_0 + \int^{x}_{x_0}\! f(u, y_2)du\ \ \ (7)$
Continuing in this manner we obtain a sequence of approximations $y_1, y_2, y_3,\cdots$ . The limit of this sequence, if it exists, is the required solution. However, by carrying out the procedure a few times, good approximations can be obtained.

4. Runge-Kutta method

This method consists of computing
$\displaystyle \left.\begin{array}{rcl}k_1 & = & hf(x_0, y_0) \\ k_2 & = & hf(x_0 + \frac{1}{2}h, y_0 + \frac{1}{2}k_1) \\ k_3 & = & hf(x_0 + \frac{1}{2}h, y_0 + \frac{1}{2}k_2 \\ k_4 & = & hf(x_0 + \frac{1}{2}h, y_0 + \frac{1}{2}k_3) \end{array} \right\}\ \ \ (8)$
Then
$\displaystyle y(x_0 + h) = y_0 + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)\ \ \ (9)$
These methods can also be adapted for higher order differential equations by writing them as several first order equations.

微分方程式を解くための数値解法

　次の境界値問題が与えられたとします．

$\displaystyle dy/dx = f(x, y)\ \ \ y(x_0) = y_0\ \ \ (1)$

恐らく正確な解を得ることはできないでしょう．このような場合，様々な方法で近似値や数値解法が得られます．いくつかの方法を列挙します．

逐次近似法またはオイラー法
テイラー級数法
ピカール法
ルンゲクッタ法

1. 逐次近似法またはオイラー法

　この手法では微分方程式 (1) を次の近似式で置換します．

$\displaystyle \frac{y(x_0 + h)-y(x_0)}{h} = f(x_0, y_0)\ \ \ (2)$

そのため，

$\displaystyle y(x_0 + h) = y(x_0) + hf(x_0, y_0)\ \ \ (3)$

このような連続性により $y(x_0 + 2h),\ y(x_0 + 3h)$ 等を見出すことができます．十分に小さな $h$ を選ぶことで良い近似が得られます．

　この方法の変法もまた用いられています．

2. テイラー級数法

　(1) における微分方程式を連続して微分することで $y'(x_0),\ y''(x_0),\ y'''(x_0),\cdots$ が得られます．そしてその解は，その級数が収束することを前提に，次のテイラー級数で得られます．

$\displaystyle y(x) = y(x_0) + y'(x_0)(x - x_0) + \frac{y''(x_0)(x - x_0)^2}{2!} + \cdots\ \ \ (4)$

級数が収束するならいかなる精度でも $y(x_0 + h)$ が得られます．

3. ピカール法

　(1) の微分方程式を積分し，境界条件を用いることで次式が得られます．

$\displaystyle y(x) = y_0 + \int^{x}_{x_0}\! f(u, y)du\ \ \ (5)$

近似式 $y_1(x) = y_0$ を前提として (5) から次の新しい近似式が得られます．

$\displaystyle y_2(x) = y_0 + \int^{x}_{x_0}\! f(u, y_1)du\ \ \ (6)$

(5) においてこれを用いると別の近似式が得られます．

$\displaystyle y_3(x) = y_0 + \int^{x}_{x_0}\! f(u, y_2)du\ \ \ (7)$

　このように連続して一連の近似式 $y_1, y_2, y_3,\cdots$ を得ます．この一連の近似式の極限は，もし存在するなら，求められる解です．しかしながら数回の手順を行うことで良い近似が得られます．

4. ルンゲクッタ法

　この手法は次の計算を含みます．

$\displaystyle \left.\begin{array}{rcl}k_1 & = & hf(x_0, y_0) \\ k_2 & = & hf(x_0 + \frac{1}{2}h, y_0 + \frac{1}{2}k_1) \\ k_3 & = & hf(x_0 + \frac{1}{2}h, y_0 + \frac{1}{2}k_2 \\ k_4 & = & hf(x_0 + \frac{1}{2}h, y_0 + \frac{1}{2}k_3) \end{array} \right\}\ \ \ (8)$

　ゆえに

$\displaystyle y(x_0 + h) = y_0 + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)\ \ \ (9)$

　これらの手法もまた数個の1階の微分方程式として記述することで高階の微分方程式に適合しています．

Special first order equations and solutions

Any first order differential equation can be put into the form
$\displaystyle \frac{dy}{dx} = f(x,y)$
or
$\displaystyle M(x,y)dx + N(x,y)dy = 0$
and the general solution of such an equation contains one arbitrary constant. Many special devices are available for finding general solutions of various types of first order differential equations. In the following list some of types are given.

Separation of variables

Exact equation

Integrating factor

Linear equation

Homogeneous equation

Bernoulli’s equation

Equation solvable for y

Clairaut’s equation

Miscellaneous equations

1. Separation of variables

If differential equation is given as below,
$\displaystyle f_1(x)g_1(y)dx + f_2(x)g_2(y)dy = 0$
divide by $g_1(y)f_2(x) \ne 0$ and integrate to obtain general solution
$\displaystyle \int\frac{f_1(x)}{f_2(x)}dx + \int\frac{g_2(y)}{g_1(y)}dy = c$
2. Exact equation

If differential equation is given as below,
$\displaystyle M(x, y)dx + N(x, y)dy = 0$
where $\displaystyle \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$

The equation can be written as
$\displaystyle Mdx + Ndy = dU(x, y) = 0$
where dU is an exact differential. Thus the solution is $U(x, y) = c$ or equivalently
$\displaystyle \int M\partial x + \int\left(N - \frac{\partial}{\partial y}\int M\partial x\right)dy = c$
where δx indicates that the integration is to be performed with respect to x keeping y constant.

3. Integrating factor

If differential equation is given as below,
$\displaystyle M(x, y)dx + N(x, y)dy = 0$
where
$\displaystyle \frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$
The equation can be written as an exact differential equation
$\displaystyle \mu M dx + \mu N dy = 0$
where μ is an appropriate integrating factor.

The following combination are often useful in finding integration factors.

$\displaystyle \frac{xdy - ydx}{x^2} = d\left(\frac{y}{x}\right)$
$\displaystyle \frac{xdy - ydx}{y^2} = -d\left(\frac{x}{y}\right)$
$\displaystyle \frac{xdy - ydx}{x^2 + y^2} = d\left(\tan^{-1}\frac{y}{x}\right)$
$\displaystyle \frac{xdy - ydx}{x^2 - y^2} = \frac{1}{2}d\left(\ln\frac{x - y}{x + y}\right)$
$\displaystyle \frac{xdx + ydy}{x^2 + y^2} = \frac{1}{2}d\{\ln(x^2 + y^2)\}$

4. Linear equation

If differential equation is given as below,
$\displaystyle \frac{dy}{dx} + P(x)y = Q(x)$
An integrating factor is given by
$\displaystyle \mu = e^{\int P(x)dx}$
and the equation can then be written
$\displaystyle \frac{d}{dx}(\mu y) = \mu Q$
with solution
$\displaystyle \mu y = \int \mu Qdx + c$
or
$\displaystyle ye^{\int Pdx} = \int Qe^{\int Pdx}dx + c$
5. Homogeneous equation

If differential equation is given as below,
$\displaystyle \frac{dy}{dx} = F\left(\frac{y}{x}\right)$
Let $y/x = v$ or $y = vx$ , and the equation becomes
$\displaystyle v + x\frac{dv}{dx} + F(x)$
or
$\displaystyle xdv + (F(x) - v)dx = 0$
which is of Type 1 and has the solution
$\displaystyle \ln x = \int \frac{dv}{F(v) - v} + c$
where $v = y/x$ . If $F(v) = v$ , the solution is $y = cx$ .

6. Bernoulli’s equation

If differential equation is given as below,
$\displaystyle \frac{dy}{dx} + P(x)y = Q(x)y^n,\ n \neq 0, 1$
Letting $v = y^{1 - n}$ , the equation reduces to Type 4 with solution
$\displaystyle ve^{(1-n)\int Pdx} = (1 - n)\int Qe^{(1-n)\int Pdx}dx + c$
If n = 0, the equation is of Type 4. If n = 1, it is of Type 1.

7. Equation solvable for y

If differential equation is given as below,
$\displaystyle y = g(x, y)$
where
$\displaystyle p = y'$
Differentiate both sides of the equation with respect to x to obtain
$\displaystyle \frac{dy}{dx} = \frac{dg}{dx} = \frac{\partial g}{\partial x} + \frac{\partial g}{\partial p}\frac{\partial p}{\partial x}$
or
$\displaystyle p = \frac{\partial g}{\partial x} + \frac{\partial g}{\partial p}\frac{\partial p}{\partial x}$
Then solve this last equation to obtain $G(x, p, c) = 0$ . The required solution is obtained by eliminating p between $G(x, p, c) = 0$ and $y = g(x, p)$ .

An analogous method exists if the equation is solvable for x.

8. Clairaut’s equation

If differential equation is given as below,
$\displaystyle y = px + F(p)$
where
$\displaystyle p = y'$
The equation is of Type 7 and has solution
$\displaystyle y = cx + F(c)$
The equation will also have a singular solution in general.

9. Miscellaneous equations

If differential equation is given as below,
$\displaystyle (a) \frac{dy}{dx} = F(\alpha x + \beta y)\\\vspace{0.2 in} (b) \frac{dy}{dx} = F\left(\frac{\alpha_1 x + \beta_1 y + \gamma_1}{\alpha_2 x + \beta_2 y + \gamma_2}\right)$
(a)Letting $\alpha x + \beta y = v$ , the equation reduces Type 1.

(b)Let $x = X +h,\ y = Y + k$ and choose constants h and k so that the equation reduces to Type 5. This is possible if and only if $\alpha_1/\alpha_2 \neq \beta_1/\beta_2$ . If $\alpha_1/\alpha_2 = \beta_1/\beta_2$ , the equation reduces to Type 9(a).

特殊な1階常微分方程式とその解

　いかなる 1 階の微分方程式も次の形に置き換えることができます．

$\displaystyle \frac{dy}{dx} = f(x,y)$

または

$\displaystyle M(x,y)dx + N(x,y)dy = 0$

そしてそれらの方程式の一般解は一つの任意定数を持ちます．様々な種類の 1 階微分方程式の一般解の発見には多くの特殊な装置が有用です．下記のリストにはその幾つかを示してあります．

変数分離法
完全微分方程式
積分因子
線形微分方程式
同次方程式
ベルヌーイの方程式
y について解ける方程式
クレローの方程式
その他の方程式

1. 変数分離法

　微分方程式が下記のようである場合は

$\displaystyle f_1(x)g_1(y)dx + f_2(x)g_2(y)dy = 0$

一般解を得るには $g_1(y)f_2(x) \ne 0$ で除し，積分します．

$\displaystyle \int\frac{f_1(x)}{f_2(x)}dx + \int\frac{g_2(y)}{g_1(y)}dy = c$

2. 完全微分方程式

　微分方程式が下記のようである場合は

$\displaystyle M(x, y)dx + N(x, y)dy = 0$

ここで $\displaystyle \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$

　この方程式は次のように書き換えられます．

$\displaystyle Mdx + Ndy = dU(x, y) = 0$

ここで dU は完全微分方程式です．ゆえにその解は $U(x, y) = c$ または同等の解として

$\displaystyle \int M\partial x + \int\left(N - \frac{\partial}{\partial y}\int M\partial x\right)dy = c$

ここで δx は y を定数とし x による積分を行うことを示します．

3. 積分因子

　微分方程式が下記のようである場合，

$\displaystyle M(x, y)dx + N(x, y)dy = 0$

ここで

$\displaystyle \frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$

　この方程式は次のように完全微分方程式に書き換えられます．

$\displaystyle \mu M dx + \mu N dy = 0$

ここで μ は適切な積分因子です．

　下記の組み合わせは積分因子を発見するのにしばしば有用です．

$\displaystyle \frac{xdy - ydx}{x^2} = d\left(\frac{y}{x}\right)$
$\displaystyle \frac{xdy - ydx}{y^2} = -d\left(\frac{x}{y}\right)$
$\displaystyle \frac{xdy - ydx}{x^2 + y^2} = d\left(\tan^{-1}\frac{y}{x}\right)$
$\displaystyle \frac{xdy - ydx}{x^2 - y^2} = \frac{1}{2}d\left(\ln\frac{x - y}{x + y}\right)$
$\displaystyle \frac{xdx + ydy}{x^2 + y^2} = \frac{1}{2}d\{\ln(x^2 + y^2)\}$

4. 線形微分方程式

　微分方程式が下記のようである場合，

$\displaystyle \frac{dy}{dx} + P(x)y = Q(x)$

　積分因子は次のように得られます．

$\displaystyle \mu = e^{\int P(x)dx}$

また方程式は次のように書き換えられます．

$\displaystyle \frac{d}{dx}(\mu y) = \mu Q$

解は以下のようです．

$\displaystyle \mu y = \int \mu Qdx + c$

または

$\displaystyle ye^{\int Pdx} = \int Qe^{\int Pdx}dx + c$

5. 同次方程式

微分方程式が下記のようである場合，

$\displaystyle \frac{dy}{dx} = F\left(\frac{y}{x}\right)$

　 $y/x = v$ または $y = vx$ とします．すると方程式は次のようになります．

$\displaystyle v + x\frac{dv}{dx} + F(x)$

または

$\displaystyle xdv + (F(x) - v)dx = 0$

ここで Type 1 と同じになり，解は次のようになります．

$\displaystyle \ln x = \int \frac{dv}{F(v) - v} + c$

ここで $v = y/x$ です．もし $F(v) = v$ なら解は $y = cx$ となります．

6. ベルヌーイの方程式

微分方程式が下記のようである場合，

$\displaystyle \frac{dy}{dx} + P(x)y = Q(x)y^n,\ n \neq 0, 1$

　 $v = y^{1 - n}$ とします．すると方程式は Type 4 に置き換えられ，解は次のようになります．

$\displaystyle ve^{(1-n)\int Pdx} = (1 - n)\int Qe^{(1-n)\int Pdx}dx + c$

　仮に n = 0 なら方程式は Type 4 と同じであり， n = 1 なら Type 1 と同じです．

7. y について解ける方程式

　微分方程式が下記のようである場合，

$\displaystyle y = g(x, y)$

ここで

$\displaystyle p = y'$

　方程式の両辺を x について微分すると次が得られます．

$\displaystyle \frac{dy}{dx} = \frac{dg}{dx} = \frac{\partial g}{\partial x} + \frac{\partial g}{\partial p}\frac{\partial p}{\partial x}$

または

$\displaystyle p = \frac{\partial g}{\partial x} + \frac{\partial g}{\partial p}\frac{\partial p}{\partial x}$

　そしてこの最後の方程式を解くと $G(x, p, c) = 0$ が得られます．必要な解は $G(x, p, c) = 0$ と $y = g(x, p)$ の間の p を消去して得られます．

　その方程式を x について解くための類似の方法が存在します．

8. クレローの方程式

　微分方程式が下記のようである場合，

$\displaystyle y = px + F(p)$

ここで

$\displaystyle p = y'$

　この方程式は Type 7 に置き換えられ，解は次の通りです．

$\displaystyle y = cx + F(c)$

　この方程式もまた一般解のうち単解を有します．

9. その他の方程式

微分方程式が下記のようである場合，

$\displaystyle (a) \frac{dy}{dx} = F(\alpha x + \beta y)\\\vspace{0.2 in} (b) \frac{dy}{dx} = F\left(\frac{\alpha_1 x + \beta_1 y + \gamma_1}{\alpha_2 x + \beta_2 y + \gamma_2}\right)$

(a) $\alpha x + \beta y = v$ とすると，方程式は Type 1 に置き換えられます．

(b) $x = X +h,\ y = Y + k$ とし，定数 h と k を方程式が Type 5 に置き換えられるように定めます．この処理は以下の場合，すなわち $\alpha_1/\alpha_2 \neq \beta_1/\beta_2$ の場合に限り可能です．仮に $\alpha_1/\alpha_2 = \beta_1/\beta_2$ の場合，方程式は Type 9(a) に置き換えられます．

Ordinary differential equations

Definition of a differential equation

A differential equation is an equation involving derivatives or differentials.

Equations involving only one independent variable are called ordinary differential equations. Equations with two or more independent variables are called partial differential equation.

Order of a differential equation

An equation having a derivative of nth order but no higher is called an nth order differential equation.

Arbitrary constants

An arbitrary constant, often denoted by a letter at the beginning of the alphabet such as A, B, C, c₁, c₂, etc., may assume values independently of the variables involved. For example in $y = x^2 + c_1x + c_2$ , c₁ and c₂ are arbitrary constants.

The relation of $y = Ae^{-4x + B}$ which can be written $y = Ae^Be^{-4x} = Ce^{-4x}$ actually involves only one arbitrary constant. It’s always assumed that the minimum number of constants is present, i.e. the arbitrary constants are essential.

Solution of a differential equation

A solution of a differential equation is a relation between the variables which is free of derivatives and which satisfies the differential equation identically. $y = x^2 + c_1x + c_2$ is a solution of $y'' = 2$ since by substitution the identity 2 = 2.

A general solution of an nth order differential equation is one involving n (essential) arbitrary constants. Since $y = x^2 + c_1x + c_2$ has two arbitrary constants and satisfies the second order differential equation $y'' = 2$ , it is a general solution of $y'' = 2$ .

A particular solution is a solution obtained from the general solution by assigning specific values to the arbitrary constants. $y = x^2 - 3x + 2$ is a particular solution of $y'' = 2$ and is obtained from the general solution $y = x^2 + c_1x + c_2$ by putting c₁ = -3 and c₂ = 2.

A singular solution is a solution which cannot be obtained from the general solution by specifying values of the arbitrary constants. The general solution of $y = xy' - y'^2$ is $y = cx - c^2$ . However, as seen by substitution another solution is $y = x^2/4$ which cannot be obtained from the general solution for any constant c. This second solution is a singular solution.

Differential equation of a family of curves

A general solution of an nth order differential equation has n arbitrary constants (or parameters) and represents geometrically an n parameter family of curves. Conversely a relation with n arbitrary constants (sometimes called a primitive) has associated with it a differential equation of order n (of which it is a general solution) called the differential equation of the family. This differential equation is obtained by differentiating the primitive n times and then eliminating the n arbitrary constants among the n + 1 resulting equations.

常微分方程式

微分方程式の定義

　微分方程式とは微分を含む方程式のことです．

　ただ一つの独立変数を含む方程式を常微分方程式と呼びます．2つ以上の独立変数を含む方程式を偏微分方程式と呼びます．

微分方程式の階数

　n次の微分を持つ方程式がそれ以上高階の微分を持たない時，n階微分方程式と言います．

任意定数

　任意定数とは，しばしば1文字で記述され A, B, C, c₁, c₂ などのようにアルファベットで始まりますが，関与する変数とは独立しているのを前提にしています．例えば $y = x^2 + c_1x + c_2$ という関数においては c₁ と c₂ が任意定数です．

　 $y = Ae^{-4x + B}$ の式の関係は $y = Ae^Be^{-4x} = Ce^{-4x}$ と記述され，事実上ただ一つの任意定数を伴っています．定数の最小数が存在することを前提にしています．すなわち任意定数は不可欠です．

微分方程式の解

　微分方程式の解は変数間の関係であり，微分を持たず同一の微分方程式を満たすものを言います． $y = x^2 + c_1x + c_2$ は $y'' = 2$ の解であり，2 = 2 の同一性を置換したものです．

　n 階の微分方程式の一般解は唯一の n 個の（不可欠な）任意定数を伴います． $y = x^2 + c_1x + c_2$ が 2 つの任意定数を有し，2 階の微分方程式 $y'' = 2$ を満たすため，それは $y'' = 2$ の一般解です．

　特殊解は一般解の任意定数に特殊な値を割り付けることで得られます． $y = x^2 - 3x + 2$ は $y'' = 2$ の特殊解であり，一般解 $y = x^2 + c_1x + c_2$ に c₁ = -3 および c₂ = 2 を代入することで得られます．

　単解は任意定数の値を特定しても一般解からは得られません． $y = xy' - y'^2$ の一般解は $y = cx - c^2$ です．しかし別の置換法によって見てみると $y = x^2/4$ はいかなる定数 c によっても一般解からは得られません．この後者が単解です．

曲線族の微分方程式

　n 階の微分方程式の一般解は n 個の任意定数（または変数）をもち，n 変数の曲線族を幾何学的に表しています．逆に n 個の任意定数との関連は（時に原始関数とも呼ばれますが） n 階の微分方程式と関連付けられており（故に一般解なのですが）族の微分方程式と呼ばれます．この微分方程式は原始関数を n 回微分することにより得られ，その結果 n + 1 個の方程式の中で n 個の任意定数は消えます．

Complex numbers

Complex numbers arose in order to solve polynomial equations such as $x^2 + 1 = 0$ or $x^2 + x + 1 = 0$ which are not satisfied by real numbers. It’s assumed that a complex number has the form a + bi where a, b are real numbers and i, called imaginary unit, has the property that i² = -1. Complex numbers are defined as follows.

Addition.
$\displaystyle (a + bi) + (c + di) = (a + c) + (b + d)i$

Subtraction.
$\displaystyle (a + bi) - (c + di) = (a - c) + (b - d)i$

Multiplication.
$\displaystyle (a + bi)\times(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$

Division.
$\displaystyle \frac{a + bi}{c + di} = \frac{a + bi}{c + di}\times\frac{c - di}{c - di} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i$

The ordinary rules of algebra has been used except that replaces i² by -1 wherever it occurs. The commutative, associative and distributive laws also apply to complex numbers. It’s called a and b of a + bi the real and imaginary parts, respectively. Two complex numbers are equal if and only if their real and imaginary parts are respectively equal.

A complex number z = x + iy can be considered as a point P with coordinates (x, y) on a rectangular xy plane called in this case the complex plane or Argand diagram. If the line would be constructed from origin O to P and let ρ be the distance OP and φ the angle made by OP with the positive x axis, you could have from Figure

$\displaystyle x = \rho \cos\phi,\ y = \rho\sin\phi,\ \rho = \sqrt{x^2 + y^2}$
and could write the complex number in so-called polar form as
$\displaystyle z = x + iy = \rho(\cos\phi + i\sin\phi) = \rho cis\phi$
It’s often called that ρ the modulus or absolute value of of z and denote it by |z|. The angle φ is called the amplitude or argument of z abbreviated arg z. It could be also written $\rho = \sqrt{z\bar{z}}$ where $\bar{z} = x - iy$ is called the conjugate of $z = x + iy$ .

If you write two complex numbers in polar form as
$\displaystyle z_1 = \rho_1(\cos\phi_1 + i\sin\phi_1),\ z_2 = \rho_2(\cos\phi_2 + i\sin\phi_2)$
then
$\displaystyle z_1z_2 = \rho_1\rho_2[\cos(\phi_1 + \phi_2) + i\sin(\phi_1 + \phi_2)]\\\vspace{0.2 in} \frac{z_1}{z_2} = \frac{\rho_1}{\rho_2}[\cos(\phi_1 - \phi_2) + i\sin(\phi_1 - \phi_2)]$
Also if n is any real number, you have
$\displaystyle z^n = [\rho(\cos\phi + i\sin\phi)]^n = \rho^n(\cos n\phi + i\sin n\phi)$
which is often called De Moivre’s theorem. You can use this to determine roots of complex numbers. For example if n is a positive integer,
$\displaystyle z^{\frac{1}{n}} = [\rho(\cos\phi + i\sin\phi)]^\frac{1}{n} = \rho^{\frac{1}{n}}\left\{\cos\left(\frac{\phi + 2k\pi}{n}\right) + i\sin\left(\frac{\phi + 2k\pi}{n}\right)\right\}\\\vspace{0.2 in} \ \ k = 0,\ 1,\ 2,\ \cdots,\ n-1$
Using the series for e^x, sin x, cos x, you are led to define
$\displaystyle e^{i\phi} = \cos\phi + i\sin\phi,\ e^{-i\phi} = \cos\phi - i\sin\phi$
which are called Euler’s formulas and which enable you to rewrite equations in terms of exponentials.

複素数

　複素数は $x^2 + 1 = 0$ や $x^2 + x + 1 = 0$ といった実数解を持たない整方程式を解く際に出現します．複素数は a + bi という形をしており，a, b は実数部であり，i は虚数部と呼ばれ，i² = -1 という性質を持ちます．複素数は以下のように定義されます．

加算
$\displaystyle (a + bi) + (c + di) = (a + c) + (b + d)i$
減算
$\displaystyle (a + bi) - (c + di) = (a - c) + (b - d)i$
乗算
$\displaystyle (a + bi)\times(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$
除算
$\displaystyle \frac{a + bi}{c + di} = \frac{a + bi}{c + di}\times\frac{c - di}{c - di} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i$

　いかなる状況でも i² を -1 で置換することを除けば，代数の通常の定義が用いられます．交換法則，結合法則，分配法則もまた複素数に適用されます．a + bi の a および b はそれぞれ実数部および虚数部と呼ばれます．二つの複素数が等しいとは，実数部と虚数部とがそれぞれ等しい時に限られます．

　ある複素数 z = x + iy は xy 直交平面上の (x, y) 座標の点 P とみなすことができ，この場合この平面を複素平面またはアルガン図と呼びます．仮に原点 O から点 P への直線を引き，OP 間の距離を ρ とし，OP と x 軸のなす角度を φ とすると，次の図を得ます．

$\displaystyle x = \rho \cos\phi,\ y = \rho\sin\phi,\ \rho = \sqrt{x^2 + y^2}$

また複素数をいわゆる極座標系式として以下のように記述できます．

$\displaystyle z = x + iy = \rho(\cos\phi + i\sin\phi) = \rho cis\phi$

　しばしば ρ は z のモジュラスまたは絶対値と呼ばれ，|z| と記述します．角 φ は z の amplitude または argument と呼ばれ，arg z と略記します． $\rho = \sqrt{z\bar{z}}$ とも記述できます．ただし $\bar{z} = x - iy$ は $z = x + iy$ の共役複素数と呼ばれます．

　極座標で複素数を記述すると以下のようになります．

$\displaystyle z_1 = \rho_1(\cos\phi_1 + i\sin\phi_1),\ z_2 = \rho_2(\cos\phi_2 + i\sin\phi_2)$

すると

$\displaystyle z_1z_2 = \rho_1\rho_2[\cos(\phi_1 + \phi_2) + i\sin(\phi_1 + \phi_2)]\\\vspace{0.2 in} \frac{z_1}{z_2} = \frac{\rho_1}{\rho_2}[\cos(\phi_1 - \phi_2) + i\sin(\phi_1 - \phi_2)]$

　仮に n を任意の実数とすると以下を得ます．

$\displaystyle z^n = [\rho(\cos\phi + i\sin\phi)]^n = \rho^n(\cos n\phi + i\sin n\phi)$

これはド・モアブルの定理として知られています．これを用いて複素数の根を定義できます．例えば仮に n を正の整数と仮定すると，

$\displaystyle z^{\frac{1}{n}} = [\rho(\cos\phi + i\sin\phi)]^\frac{1}{n} = \rho^{\frac{1}{n}}\left\{\cos\left(\frac{\phi + 2k\pi}{n}\right) + i\sin\left(\frac{\phi + 2k\pi}{n}\right)\right\}\\\vspace{0.2 in} \ \ k = 0,\ 1,\ 2,\ \cdots,\ n-1$

e^x, sin x, cos x のための級数を用いると以下の定義に導かれます．

$\displaystyle e^{i\phi} = \cos\phi + i\sin\phi,\ e^{-i\phi} = \cos\phi - i\sin\phi$

これらはオイラーの公式と呼ばれ，方程式を指数関数の観点から書き直すことができます．

Leibnitz’s rule for differentiating an integral

Let
$\displaystyle I(a) = \int_{a}^{b}f(x, \alpha)dx$
where f is supposed continuous and differentiable. Then Leibnitz’s rule states that if a and b are differentiable functions of α,
$\displaystyle \frac{dI}{d\alpha} = \int_{a}^{b}\frac{\partial f}{\partial \alpha}dx + f(b, \alpha)\frac{db}{d\alpha} - f(a, \alpha)\frac{da}{d\alpha}$

積分を微分するためのライプニッツの法則

　今，

$\displaystyle I(a) = \int_{a}^{b}f(x, \alpha)dx$

であるとし，f が連続で微分可能であると仮定します．するとライプニッツの法則により，もし a と b とが共に α の関数を微分可能である時には以下のことが言えます．

$\displaystyle \frac{dI}{d\alpha} = \int_{a}^{b}\frac{\partial f}{\partial \alpha}dx + f(b, \alpha)\frac{db}{d\alpha} - f(a, \alpha)\frac{da}{d\alpha}$

Method of Lagrange multipliers

Sometimes you may wish to find the relative maxima or minima of f(x, y) = 0 subject to some constraint condition φ(x, y) = 0. To do this, form the function h(x, y) = f(x, y) + λφ(x, y) and set
$\displaystyle \partial h/\partial x = 0,\ \partial h/\partial y = 0$
The constant λ is called a Lagrange multiplier and the method is called the method of Lagrange multipliers.

ラグランジュの未定乗数法

　 φ(x, y) = 0 という制約条件のもとで f(x, y) = 0 の極大と極小を求めたくなるかもしれません．このために h(x, y) = f(x, y) + λφ(x, y) という関数を置き，次のようにします．

$\displaystyle \partial h/\partial x = 0,\ \partial h/\partial y = 0$

　定数 λ はラグランジュ未定乗数と呼ばれ，この方法をラグランジュ未定乗数法と呼びます．

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