# 线性微分方程

## 维基百科，自由的百科全书

${\displaystyle {\mathcal {L))(y)=f\qquad \ldots \;\;(*)}$

## 简介

${\displaystyle {\mathcal {L))(y)=f\qquad \ldots \;\;(*)}$

${\displaystyle {\mathcal {L))(\lambda _{1}y_{1}+\lambda _{2}y_{2})=\lambda _{1}{\mathcal {L))(y_{1})+\lambda _{2}{\mathcal {L))(y_{2}).}$

${\displaystyle {\mathcal {L))(y)=0\qquad \ldots \;\;(**)}$

${\displaystyle y^{s))$是方程(*)的一个解函数。${\displaystyle y}$方程(**)的任意一个解函数。则它们的和${\displaystyle y^{s}+y}$仍然是(*)的解函数。另一方面，给定方程(*)的两个解函数：${\displaystyle y_{1}^{s))$${\displaystyle y_{2}^{s))$。则它们的差${\displaystyle y_{1}^{s}-y_{2}^{s))$会是方程(**)的解函数。这说明方程(*)的所有解函数都可以写成${\displaystyle y^{s}+y,\;y\in V}$的形式。其中V是方程(**)的解空间。所以方程(*)的所有解函数构成一个仿射空间V'，并且${\displaystyle V'=y^{s}+V}$

## 常系数齐次线性微分方程

${\displaystyle {\frac {d^{n}y}{dx^{n))}+A_{1}{\frac {d^{n-1}y}{dx^{n-1))}+\cdots +A_{n}y=0}$

${\displaystyle z^{n}e^{zx}+A_{1}z^{n-1}e^{zx}+\cdots +A_{n}e^{zx}=0.}$

${\displaystyle F(z)=z^{n}+A_{1}z^{n-1}+\cdots +A_{n}=0.\,}$

${\displaystyle {\frac {d^{k}y}{dx^{k))}\quad \quad (k=1,2,\dots ,n).}$

## 常系数非齐次线性微分方程

### 待定系数法

${\displaystyle {\frac {dy}{dx))=y+e^{2x}.\!}$

${\displaystyle {\frac {dy}{dx))=y.}$

${\displaystyle y=ce^{x}.\!}$

${\displaystyle y_{p}=Ae^{2x}.\!}$

${\displaystyle {\frac {d}{dx))\left(Ae^{2x}\right)=Ae^{2x}+e^{2x}\!}$
${\displaystyle 2Ae^{2x}=Ae^{2x}+e^{2x}\!}$
${\displaystyle 2A=A+1\,\!}$
${\displaystyle A=1.\,\!}$

${\displaystyle y=ce^{x}+e^{2x}.\!}$ (${\displaystyle c\in R}$)

### 常数变易法

${\displaystyle y^{\prime \prime }+py^{\prime }+qy=f(x)}$

${\displaystyle y=u_{1}y_{1}+u_{2}y_{2}.~~\mathrm {(1)} }$

${\displaystyle y'=u_{1}'y_{1}+u_{2}'y_{2}+u_{1}y_{1}'+u_{2}y_{2}'.}$

${\displaystyle u_{1}'y_{1}+u_{2}'y_{2}=0.~~\mathrm {(2)} }$

${\displaystyle y'=u_{1}y_{1}'+u_{2}y_{2}'.~~\mathrm {(3)} }$

${\displaystyle y''=u_{1}'y_{1}'+u_{2}'y_{2}'+u_{1}y_{1}''+u_{2}y_{2}''.~~\mathrm {(4)} }$

${\displaystyle u_{1}'y_{1}'+u_{2}'y_{2}'+u_{1}y_{1}''+u_{2}y_{2}''+pu_{1}y_{1}'+pu_{2}y_{2}'+qu_{1}y_{1}+qu_{2}y_{2}=f(x).}$

${\displaystyle u_{1}'y_{1}'+u_{2}'y_{2}'+(u_{1}y_{1}''+pu_{1}y_{1}'+qu_{1}y_{1})+(u_{2}y_{2}''+pu_{2}y_{2}'+qu_{2}y_{2})=f(x).}$

${\displaystyle u_{1}'y_{1}'+u_{2}'y_{2}'=f(x).~~\mathrm {(5)} }$

(2)和(5)联立起来，便得到了一个${\displaystyle u_{1}'}$${\displaystyle u_{2}'}$的方程组，便可得到${\displaystyle u_{1}'}$${\displaystyle u_{2}'}$的表达式；再积分，便可得到${\displaystyle u_{1))$${\displaystyle u_{2))$的表达式。

${\displaystyle u'_{j}=(-1)^{n+j}{\frac {W(y_{1},\ldots ,y_{j-1},y_{j+1}\ldots ,y_{n})_{0 \choose f)){W(y_{1},y_{2},\ldots ,y_{n}))).}$

## 变系数线性微分方程

n阶的变系数微分方程具有以下形式：

${\displaystyle p_{n}(x)y^{(n)}(x)+p_{n-1}(x)y^{(n-1)}(x)+\cdots +p_{0}(x)y(x)=r(x).}$

${\displaystyle x^{n}y^{(n)}(x)+a_{n-1}x^{n-1}y^{(n-1)}(x)+\cdots +a_{0}y(x)=0.}$

${\displaystyle \ Dy(x)+f(x)y(x)=g(x).}$

${\displaystyle Dy(x)e^{\int f(x)\,dx}+f(x)y(x)e^{\int f(x)\,dx}=g(x)e^{\int f(x)\,dx},}$

${\displaystyle D(y(x)e^{\int f(x)\,dx})=g(x)e^{\int f(x)\,dx))$

${\displaystyle y(x)e^{\int f(x)\,dx}=\int g(x)e^{\int f(x)\,dx}\,dx+c~,}$
${\displaystyle y(x)={\int g(x)e^{\int f(x)\,dx}\,dx+c \over e^{\int f(x)\,dx))~.}$

${\displaystyle y=e^{-a(x)}\left(\int r(x)e^{a(x)}\,dx+\kappa \right)}$

${\displaystyle a(x)=\int {p(x)\,dx}.}$

### 例子

${\displaystyle {\frac {dy}{dx))+by=1.}$

p(x) = b，r(x) = 1，因此微分方程的解为：

${\displaystyle y(x)=e^{-bx}\left({\frac {e^{bx)){b))+C\right)={\frac {1}{b))+Ce^{-bx}.}$

## 拉普拉斯变换解微分方程

${\displaystyle {\mathcal {L))\{f'\}=s{\mathcal {L))\{f\}-f(0)}$
${\displaystyle {\mathcal {L))\{f''\}=s^{2}{\mathcal {L))\{f\}-sf(0)-f'(0)}$
${\displaystyle {\mathcal {L))\{f^{(n)}\}=s^{n}{\mathcal {L))\{f\}-\Sigma _{i=1}^{n}s^{n-i}f^{(i-1)}(0).}$

${\displaystyle \sum _{i=0}^{n}a_{i}f^{(i)}(t)=\phi (t).}$

${\displaystyle \sum _{i=0}^{n}a_{i}{\mathcal {L))\{f^{(i)}(t)\}={\mathcal {L))\{\phi (t)\))$

${\displaystyle {\mathcal {L))\{f(t)\}=((\mathcal {L))\{\phi (t)\}+\sum _{i=1}^{n}a_{i}\sum _{j=1}^{i}s^{i-j}f^{(j-1)}(0) \over \sum _{i=0}^{n}a_{i}s^{i)).}$

f(t) 通过拉普拉斯反变换 ${\displaystyle {\mathcal {L))\{f(t)\))$ 求得。

## 参考文献

• Stanley J. Farlow(1994). An introduction to differential equations and their applications. McGraw-Hill, Inc. ISBN 0-07-020030-0. p.131-139, p.158-162.