✍内容

当微分方程 ${\displaystyle P(x,y)dx+Q(x,y)dy=0}$ 是恰当微分方程时，即有 ${\displaystyle \frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}}$，这个时候对于方程的解 ${\displaystyle u(x,y)}$，有 ${\displaystyle \frac{\partial u}{\partial y}=P,\frac{\partial u}{\partial x}=Q}$，于是积分得到 ${\displaystyle u=\int P(x,y)dy+\varphi (x)}$，带回 ${\displaystyle \frac{\partial u}{\partial x}=Q}$ 解得 ${\displaystyle \varphi (x)}$，就有了方程的解. 但是有时候微分方程不是恰当微分方程，即 ${\displaystyle \frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x}\ne 0}$，这时候要引入积分因子 ${\displaystyle \mu =\mu (x,y)}$，凑出恰当微分方程 ${\displaystyle \mu Pdx+\mu Qdy=0}$. 即有 ${\displaystyle \frac{\partial \mu P}{\partial y}=\frac{\partial \mu Q}{\partial x}}$，即 ${\displaystyle Q\frac{\partial \mu }{\partial x}-P\frac{\partial \mu }{\partial y}=(\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x})\mu }$. 我们希望 ${\displaystyle \mu }$ 只与一个变量有关，这样便于计算出 ${\displaystyle \mu }$，以 $x$ 为例，${\displaystyle \mu }$ 只与 $x$ 有关的充要条件是 ${\displaystyle \frac{\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x}}{Q}=\psi (x)}$，于是解出 ${\displaystyle \mu =e^{\int \psi (x)dx}}$. 同理，${\displaystyle \mu }$ 只与 ${\displaystyle y}$ 有关的充要条件为 ${\displaystyle {\displaystyle \frac{\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x}}{-P}=\varphi (y)}}$，解得 ${\displaystyle \mu =e^{\int \varphi (y)dy}}$.

积分因子不唯一，在 ${\displaystyle \mu =\mu (x,y)}$ 的情况下很难求解，不过也有一些特殊情况如下：

对于齐次微分方程 ${\displaystyle P(x,y)dx+Q(x,y)dy=0}$，若 ${\displaystyle xP+yQ\ne 0}$，则有积分因子 ${\displaystyle \mu =\frac{1}{xP+yQ}}$.

形如 $\frac{dy}{dx}=f\left(\frac{y}{x}\right)$ 的一阶微分方程称为齐次微分方程，如 ${\displaystyle dy=\frac{x^2+y^2}{xy}dx}$.