# 分離變數法

## 常微分方程

${\displaystyle {\frac {d}{dx}}f(x)=g(x)[h(f(x))]}$

${\displaystyle {\frac {dy}{dx}}=g(x)h(y)}$(1)

${\displaystyle {dy \over h(y)}={g(x)dx}}$

${\displaystyle \int {dy \over h(y)}=\int {g(x)dx}=k}$

### 第二種方法

${\displaystyle {\frac {1}{h(y)}}{\frac {dy}{dx}}=g(x)}$

${\displaystyle \int {\frac {1}{h(y)}}{\frac {dy}{dx}}\,dx=\int g(x)\,dx}$(2)

${\displaystyle \int {\frac {1}{h(y)}}\,dy=\int g(x)\,dx}$

### 實例 (I)

${\displaystyle {\frac {dy}{dx}}=y(1-y)}$(3)

${\displaystyle {\frac {dy}{y(1-y)}}=dx}$

${\displaystyle \int {\frac {dy}{y(1-y)}}=\int dx}$

${\displaystyle \ln |y|-\ln |1-y|=x+C}$

${\displaystyle y={\frac {1}{1+Be^{-x}}}}$

### 實例 (II)

${\displaystyle {\frac {dP}{dt}}=kP\left(1-{\frac {P}{K}}\right)}$

${\displaystyle \int {\frac {1}{P\left(1-{\frac {P}{K}}\right)}}{\frac {dp}{dt}}\,dt=\int k\,dt}$

${\displaystyle \int {\frac {dP}{P\left(1-{\frac {P}{K}}\right)}}=\int k\,dt}$

${\displaystyle P(t)={\frac {K}{1+Ae^{-kt}}}}$

## 偏微分方程

${\displaystyle F=F_{1}(x_{1})F_{2}(x_{2})\cdots F_{n}(x_{n})}$

${\displaystyle F=f_{1}(x_{1})+f_{2}(x_{2})+\cdots +f_{n}(x_{n})}$

### 實例 (III)

${\displaystyle {\frac {\partial F}{\partial x}}+{\frac {\partial F}{\partial y}}+{\frac {\partial F}{\partial z}}=0}$

${\displaystyle F(x,y,z)=X(x)+Y(y)+Z(z)}$

${\displaystyle {\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}}=0}$

${\displaystyle {\frac {dX}{dx}}=c_{1}}$
${\displaystyle {\frac {dY}{dy}}=c_{2}}$
${\displaystyle {\frac {dZ}{dz}}=c_{3}}$

${\displaystyle F(x,y,z)=c_{1}x+c_{2}y+c_{3}z+c_{4}}$

### 實例 (IV)

${\displaystyle \nabla ^{2}v+\lambda v={\partial ^{2}v \over \partial x^{2}}+{\partial ^{2}v \over \partial y^{2}}+\lambda v=0}$

${\displaystyle v=X(x)Y(y)}$

${\displaystyle {\partial ^{2} \over \partial x^{2}}[X(x)Y(y)]+{\partial ^{2} \over \partial y^{2}}[X(x)Y(y)]+\lambda X(x)Y(y)=0}$

${\displaystyle X''(x)Y(y)+X(x)Y''(y)+\lambda X(x)Y(y)=0}$

${\displaystyle {X''(x) \over X(x)}=-{Y''(y)+\lambda Y(y) \over Y(y)}}$

${\displaystyle {X''(x) \over X(x)}=k=-{Y''(y)+\lambda Y(y) \over Y(y)}}$

${\displaystyle X''(x)-kX(x)=0}$
${\displaystyle Y''(y)+(\lambda +k)Y(y)=0}$

${\displaystyle X(x)=A_{x}\cos({\sqrt {-k}}\ x+B_{x})}$
${\displaystyle Y(y)=A_{y}\cos({\sqrt {\lambda +k}}\ y+B_{y})}$

## 參考文獻

• A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9