# 全微分方程

## 定义

${\displaystyle I(x,y)\,\mathrm {d} x+J(x,y)\,\mathrm {d} y=0,\,\!}$

${\displaystyle {\frac {\partial F}{\partial x}}(x,y)=I}$

${\displaystyle {\frac {\partial F}{\partial y}}(x,y)=J.}$

“全微分方程”的命名指的是函数的全导数。对于函数${\displaystyle F(x_{0},x_{1},...,x_{n-1},x_{n})}$ ，全导数为：

${\displaystyle {\frac {\mathrm {d} F}{\mathrm {d} x_{0}}}={\frac {\partial F}{\partial x_{0}}}+\sum _{i=1}^{n}{\frac {\partial F}{\partial x_{i}}}{\frac {\mathrm {d} x_{i}}{\mathrm {d} x_{0}}}.}$

### 例子

${\displaystyle F(x,y):={\frac {1}{2}}(x^{2}+y^{2})}$

${\displaystyle xx'+yy'=0.\,}$

## 势函数的存在

${\displaystyle I(x,y)\,dx+J(x,y)\,dy=0,\,\!}$

${\displaystyle {\frac {\partial I}{\partial y}}(x,y)={\frac {\partial J}{\partial x}}(x,y).}$

## 全微分方程的解

${\displaystyle F(x,f(x))=c.\,}$

${\displaystyle y(x_{0})=y_{0}\,}$

${\displaystyle F(x,y)=\int _{x_{0}}^{x}I(t,y_{0})dt+\int _{y_{0}}^{y}J(x,t)dt.}$

${\displaystyle F(x,y)=c\,}$

## 参考文献

• Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley, 1986.
• Ross, C. C. §3.3 in Differential Equations. New York: Springer-Verlag, 2004.
• Zwillinger, D. Ch. 62 in Handbook of Differential Equations. San Diego, CA: Academic Press, 1997.