日: 2014年2月10日

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．