分离变量法

常微分方程

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

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

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

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

第二种方法

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

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

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

实例 (I)

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

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

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

$\ln |y| - \ln |1 - y|=x+C$

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

实例 (II)

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

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

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

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

偏微分方程

$F = F_1(x_1) F_2(x_2) \cdots F_n(x_n)$

$F = f_1(x_1) + f_2(x_2) + \cdots + f_n(x_n)$

实例 (III)

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

$F(x,y,z) = X(x) + Y(y) + Z(z)$

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

$\frac{dX}{dx} = c_1$
$\frac{dY}{dy} = c_2$
$\frac{dZ}{dz} = c_3$

$F(x,y,z) = c_1 x + c_2 y + c_3 z + c_4$

实例 (IV)

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

$v = X(x)Y(y)$

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

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

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

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

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

$X(x)=A_x\cos(\sqrt{- k}\ x+B_x)$
$Y(y)=A_y\cos(\sqrt{\lambda+k}\ y+B_y)$

参考文献

• 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