一阶常系数齐次线性微分方程(复数域)
$$y ^{\prime} + a y \ {=} \ 0,\ a ^{\prime} \ {=} \ 0$$
$\Gamma: \left\langle y \in \mathbb C,\ a \in \mathbb C \right\rangle$
$\mathbf{Solution}\quad \alpha_0: x \in \mathbb{C},\ x \Gamma y \quad \square \to \Gamma_0 \to \Gamma_1$
$\Gamma \to \Gamma_0 \ {=} \ \Gamma \mid \alpha_0$
改写导数形式
$${\color{Teal} \frac{\text{d} y}{\text{d} x}} + a {\color{Gray} y} \ {=} \ 0$$
$\left[\mathbb C :: \text{div}\right]$
$\alpha_1: \left\langle y \ne 0 \right\rangle \quad \square \to \Gamma_1$
$\_ : \left\langle y \ {=} \ 0 \right\rangle \quad \blacksquare$
$\Gamma \to \Gamma_0 \to \Gamma_1 \ {=} \ \Gamma_0 \mid \alpha_1$
除以 $y$
$${\color{Teal} \frac{1}{y}} \frac{\text{d} y }{\text{d} x} + a {=} 0$$
$\mathbf{def}\quad \displaystyle I_1 \ {=} \ \int \text{d} {\color{Peach} x} \left\{ {\color{Peach} a} \right\}$
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