一阶齐次线性微分方程(复数域)

$$y ^{\prime} + P y \ {=} \ 0$$

$\Gamma: \left\langle y \in \mathbb C,\ P \in \mathbb C \right\rangle$

$\mathbf{Solution}\quad \alpha_0: x \in \mathbb{C},\ x \Gamma y \quad \square \to \Gamma_0 \to \Gamma_1$

$\Gamma \to \Gamma_0 \ {=} \ \Gamma \mid \alpha_0$

改写导数形式

$${\color{Teal} \frac{\text{d} y}{\text{d} x}} + P {\color{Gray} y} \ {=} \ 0$$

$\left[\mathbb C :: \text{div}\right]$

$\alpha_1: \left\langle y \ne 0 \right\rangle \quad \square \to \Gamma_1$

$\_ : \left\langle y \ {=} \ 0 \right\rangle \quad \blacksquare$

$\Gamma \to \Gamma_0 \to \Gamma_1 \ {=} \ \Gamma_0 \mid \alpha_1$

除以 $y$

$${\color{Teal} \frac{1}{y}} \frac{\text{d} y }{\text{d} x} + {\color{Gray} P} {=} 0$$

$\mathbf{def}\quad \displaystyle I_1 \ {=} \ \int \text{d} {\color{Peach} x} \left\{ {\color{Peach} P} \right\}$

对 $x$ 积分

$${\color{Teal} \int} {\color{Cyan} \text{d} x} \left\{ \frac{1}{y} {\color{Gray} \frac{\text{d} y}{\text{d} x}}\right\} + {\color{Teal} I_1} \ {=} \ 0$$

$${\color{Grey} \int} {\color{Cyan} \text{d} y} {\color{Gray} \left\{ \frac{1}{y} \right\}} + I_1 \ {=} \ 0$$

$\left[\mathbb C :: \text{int}\right]\quad C \in \mathbb{C}$

这里 $\displaystyle \int \frac{\text{d}y}{y}$ 理解为复对数的一个分支，其导数为 $1/y$，不同分支相差 $2\pi i$，最终被常数 $C$ 吸收。

$${\color{Cyan} \ln} {\color{Teal} y} + I_1 \ {=} \ {\color{Teal} C}$$

取复指数

$${\color{Gray} y} \cdot {\color{Teal} e} ^{I_1} \ {=} \ {\color{Teal} e} ^C$$

$$y \ {=} \ {\color{Gray} e^C} \cdot e ^{{\color{Teal} -} I_1}$$

$\mathbf{overload}\quad {\color{Peach} C} \in \mathbb{C} / \{0\}$

$$y \ {=} \ {\color{Teal} C} e ^{- I_1}$$

$\blacksquare$

$\Gamma \to \Gamma_0 \rightleftharpoons \Gamma_1$

$\mathbf{overload}\quad {\color{Peach} C} \in \mathbb{C}$

$$y \ {=} \ {\color{Teal} C} e ^{- I_1}$$

$\blacksquare$

更笼统的情况

...

更特殊的情况

$y ^{\prime} + P y \ {=} \ 0$