积分因子公式

令 $\mu(x)=e^{\int P(x)\,dx}$;由链式法则求导,因指数 $\int P(x)\,dx$ 的导数为 $P(x)$,得 $\mu'(x)=P(x)\mu(x)$。

将 $\frac{dy}{dx}+P(x)y=Q(x)$ 两边乘以 $\mu(x)$:得到 $\mu(x)y' + \mu(x)P(x)y = \mu(x)Q(x)$。

由乘积法则,$\frac{d}{dx}[\mu(x)y(x)] = \mu(x)y'(x) + \mu'(x)y(x)$,又 $\mu'(x)=P(x)\mu(x)$,这恰好等于上式左边 $\mu(x)y' + \mu(x)P(x)y$。故方程化为 $\frac{d}{dx}[\mu(x)y(x)] = \mu(x)Q(x)$。

两边对 $x$ 积分,得 $\mu(x)y(x) = \int \mu(x)Q(x)\,dx + C$($C$ 为任意常数)。两边除以 $\mu(x)$(它是指数函数,永不为零)得到 $y(x) = \frac{1}{\mu(x)}\left(\int \mu(x) Q(x)\,dx + C\right)$。