常微分方程

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微积分学
\text{e} = \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n
函数 · 导数 · 微分 · 积分

数学分析中,常微分方程ordinary differential equation,簡稱ODE)是未知函数只含有一个自变量的微分方程。对于微积分的基本概念,请参见微积分微分学积分学等条目。

很多科学问题都可以表示为常微分方程,例如根据牛顿第二运动定律物体的作用下的位移 s时间 t 的关系就可以表示为如下常微分方程:

m\frac{d^2s}{dt^2}=f(s)

其中 m 是物体的质量f(s) 是物体所受的力,是位移的函数。所要求解的未知函数是位移 s,它只以时间 t 为自变量。

精确解总结[编辑]

一些微分方程有精确封闭形式的解,这里给出几个重要的类型。

在下表中,P(x), Q(x), P(y), Q(y), 和M(x,y), N(x,y)是任何x, y的可积函数,b, c是给定的实常数,C1, C2,... 是任意常数(一般为复数)。这些微分方程的等价或替代形式通过积分可以得到解。

在积分求中,λ和ε是虚拟的变量(求和下表的连续类似物),记号∫xF(λ)dλ只表示F(λ)对λ积分,在积分以后λ = x替换,无需加常数(明确说明)。

微分方程 解法 通解
可分离方程
一阶, 变量可分离于xy (一般情况, 下面有特殊情况)[1]

 P_1(x)Q_1(y) + P_2(x)Q_2(y)\,\frac{dy}{dx} = 0 \,\!

 P_1(x)Q_1(y)\,dx + P_2(x)Q_2(y)\,dy = 0 \,\!

分离变量(除以P2Q1).  \int^x \frac{P_1(\lambda)}{P_2(\lambda)}\,d\lambda + \int^y \frac{Q_2(\lambda)}{Q_1(\lambda)}\,d\lambda = C \,\!
一阶, 变量可分离于x[2]

\frac{dy}{dx} = F(x)\,\!

dy= F(x) \, dx\,\!

直接积分. y= \int^x F(\lambda) \, d\lambda + C \,\!
一阶, 自治, 变量可分离于y[2]

\frac{dy}{dx} = F(y)\,\!

dy= F(y) \, dx\,\!

分离变量 (除以F). x=\int^y \frac{d\lambda}{F(\lambda)}+C\,\!
一阶, 变量可分离于xy[2]

P(y)\frac{dy}{dx} + Q(x)= 0\,\!

P(y)\,dy + Q(x)\,dx =0\,\!

Integrate throughout. \int^y P(\lambda)\,{d\lambda} + \int^x Q(\lambda)\,d\lambda = C\,\!
一般一阶微分方程
一阶, 齐次[2]

\frac{dy}{dx} = F \left( \frac{y}{x} \right ) \,\!

y = ux, 然后通过分离变量ux求解.  \ln (Cx) = \int^{y/x} \frac{d\lambda}{F(\lambda) - \lambda} \, \!
一阶, 可分离变量[1]

 yM(xy) + xN(xy)\,\frac{dy}{dx} = 0 \,\!

 yM(xy)\,dx + xN(xy)\,dy = 0 \,\!

分离变量(除以xy).

 \ln (Cx) = \int^{xy} \frac{N(\lambda)\,d\lambda}{\lambda [N(\lambda)-M(\lambda)] } \,\!

如果N = M, 解为xy = C.

恰当微分, 一阶[2]

 M(x,y) \frac{dy}{dx} + N(x,y) = 0 \,\!

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

其中  \frac{\partial M}{\partial x} = \frac{\partial N}{\partial y} \, \!

Integrate throughout.  \begin{align}
F(x,y) & = \int^y M(x,\lambda)\,d\lambda + \int^x N(\lambda,y)\,d\lambda \\
 & + Y(y) + X(x) = C 
\end{align} \,\!

where Y(y) and X(x) are functions from the integrals rather than constant values, which are set to make the final function F(x, y) satisfy the initial equation.

反常微分, 一阶[2]

 M(x,y) \frac{dy}{dx} + N(x,y) = 0 \,\!

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

其中 \frac{\partial M}{\partial x} \neq \frac{\partial N}{\partial y} \, \!

积分变量 μ(x, y) 满足

 \frac{\partial (\mu M)}{\partial x} = \frac{\partial (\mu N)}{\partial y} \, \!

如果 μ(x, y) can be found:

 \begin{align}
F(x,y) & = \int^y \mu(x,\lambda)M(x,\lambda)\,d\lambda + \int^x \mu(\lambda,y)N(\lambda,y)\,d\lambda \\
& + Y(y) + X(x) = C \\
\end{align} \, \!

一般二阶微分方程
二阶, 自治[3]

\frac{d^2y}{dx^2} = F(y) \,\!

原方程乘以 2dy/dx, 代换2 \frac{dy}{dx}\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right)^2 \,\!, 然后两次积分.  x = \pm \int^y \frac{ d \lambda}{\sqrt{2 \int^\lambda F(\epsilon) \, d \epsilon + C_1}} + C_2 \, \!
线性方程 (最高到n阶)
一阶, 线性, 非齐次的函数系数[2]

\frac{dy}{dx} + P(x)y=Q(x)\,\!

积分因子: e^{\int^x P(\lambda)\,d\lambda}. y = e^{- \int^x P(\lambda) \, d\lambda}\left[\int^x e^{\int^\lambda P(\epsilon) \, d\epsilon}Q(\lambda) \, {d\lambda} +C \right]
二阶, 线性, 非齐次的常系数[4]

\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = r(x)\,\!

余函数 yc: 设 yc = eαx, substitute and solve polynomial in α, to find the linearly independent 函数 e^{\alpha_j x}.

Particular integral yp: in general the method of variation of parameters, though for very simple r(x) inspection may work.[2]

y=y_c+y_p

如果 b2 > 4c, 则:

y_c=C_1e^{ \left ( -b+\sqrt{b^2 - 4c} \right )\frac{x}{2}} + C_2e^{-\left ( b+\sqrt{b^2 - 4c} \right )\frac{x}{2}}\,\!

如果 b2 = 4c, 则:

y_c = (C_1x + C_2)e^{-bx/2}\,\!

如果 b2 < 4c, 则:

 y_c = e^{ -b\frac{x}{2}} \left [ C_1 \sin{\left ( \sqrt{\left | b^2-4c \right |}\frac{x}{2} \right )} + C_2\cos{\left ( \sqrt{\left | b^2-4c \right |}\frac{x}{2} \right )} \right ]  \,\!

n阶, 线性, 非齐次, 常系数[4]

 \sum_{j=0}^n b_j \frac{d^j y}{dx^j} = r(x)\,\!

Complementary function yc: assume yc = eαx, substitute and solve polynomial in α, to find the linearly independent functions e^{\alpha_j x}.

Particular integral yp: in general the method of variation of parameters, though for very simple r(x) inspection may work.[2]

y=y_c+y_p

Since αj are the solutions of the polynomial of degree n:  \prod_{j=1}^n \left ( \alpha - \alpha_j \right ) = 0 \,\!, then:

for αj all different,

 y_c = \sum_{j=1}^n C_j e^{\alpha_j x} \,\!

for each root αj repeated kj times,

 y_c = \sum_{j=1}^n \left( \sum_{\ell=1}^{k_j} C_\ell x^{\ell-1}\right )e^{\alpha_j x} \,\!

for some αj complex, then setting α = χj + iγj, and using Euler's formula, allows some terms in the previous results to be written in the form

 C_je^{\alpha_j x} = C_j e^{\chi_j x}\cos(\gamma_j x + \phi_j)\,\!

where ϕj is an arbitrary constant (phase shift).

参见[编辑]

参考资料[编辑]

  1. ^ 1.0 1.1 Mathematical Handbook of Formulas and Tables (3rd edition), S. Lipschutz, M.R. Spiegel, J. Liu, Schuam's Outline Series, 2009, ISC_2N 978-0-07-154855-7
  2. ^ 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Elementary Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986, ISBN 0-471-83824-1
  3. ^ Further Elementary Analysis, R. Porter, G.Bell & Sons (London), 1978, ISBN 0-7135-1594-5
  4. ^ 4.0 4.1 Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISC_2N 978-0-521-86153-3