全微分方程

$\int_M \mathrm{d}\omega = \oint_{\partial M} \omega$

定义

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

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

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

“全微分方程”的命名指的是函数的全导数。对于函数$F(x_0, x_1,...,x_{n-1},x_n)$，全导数为：

$\frac{\mathrm{d}F}{\mathrm{d}x_0}=\frac{\partial F}{\partial x_0}+\sum_{i=1}^{n}\frac{\partial F}{\partial x_i}\frac{\mathrm{d}x_i}{\mathrm{d}x_0}.$

例子

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

$xx' + yy' = 0.\,$

势函数的存在

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

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

全微分方程的解

$F(x, f(x)) = c.\,$

$y(x_0) = y_0\,$

$F(x,y) = \int_{x_0}^x I(t,y_0) dt + \int_{y_0}^y J(x,t) dt.$

$F(x,y) = c\,$

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.