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

$\Gamma: \left\langle y,\ a,\ a ^{\prime} \ {=} \ 0 \right\rangle$

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

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

$\mathbf{def}\quad u \ {=} \ {\color{Peach} y ^{\prime}}$

$${\color{Teal} u} ^{\prime} + a {\color{Teal} u} \ {=} \ 0$$

$\mathbf{use}$

$y ^{\prime} + a y \ {=} \ 0,\ a ^{\prime} \ {=} \ 0 \Rightarrow y \ {=} \ C e ^{- a x}$

$\mathbf{def}\quad C$

$${\color{Gray} u} \ {=} \ {\color{Teal} C} e ^{- a x}$$

$${\color{Teal} y} \ {=} \ {\color{Teal} \int \text{d} x \left\{{\color{Gray} C} {\color{Black} e ^{- a x}}\right\}}$$

$${=} \ {\color{Teal} C} \int \text{d} x \left\{ e ^{- a x}\right\}$$

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

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

$\alpha_2 : \left\langle a \ {=} \ 0 \right\rangle \quad \square \to \Gamma_2$

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

$$y \ {=} \ {\color{Cyan} \frac{\color{Gray} C}{- a}} \int \text{d} \left({\color{Teal} - a} x\right) \left\{ e ^{- a x}\right\}$$

$\mathbf{def}\quad C \ {=} \ {\color{Peach} \frac{C}{- a}}$

$${=} \ {\color{Teal} C} {\color{Gray} \int \text{d} \left(- a x\right) \left\{ e ^{- a x}\right\}}$$

$\left[\mathbb R :: \text{int}\right]\quad \mathbf{def}\quad C_0$

$${=} \ C {\color{Teal} \left(- a e ^{- a x} + C_0\right)}$$

$${=} \ {\color{Gray} - a C} e ^{- a x} + {\color{Gray} C C_0}$$

$\mathbf{def}\quad C_1 \ {=} \ {\color{Peach} - a C},\ C_2 \ {=} \ {\color{Peach} C C_0}$

$${=} \ {\color{Teal} C_1} e ^{- a x} + {\color{Teal} C_2}$$

$\blacksquare$

$\Gamma \to \Gamma_0 \to \Gamma_2 \ {=} \ \Gamma_0 \mid \alpha_2$

$$y \ {=} \ C {\color{Gray} \int \text{d} x}$$

$\left[\mathbb R :: \text{int}\right]\quad \mathbf{def}\quad C_0$

$${=} \ C {\color{Teal} \left(x + C_0\right)}$$

$${=} \ {\color{Gray} C} x + {\color{Gray} C C_0}$$

$\mathbf{def}\quad C_1 \ {=} \ {\color{Peach} C},\ C_2 \ {=} \ {\color{Peach} C C_0}$

$${=} \ {\color{Teal} C_1} x + {\color{Teal} C_2}$$

$\blacksquare$

$\Gamma \to \Gamma_0 \rightleftharpoons \Gamma_1 + \Gamma_2$

$\mathbf{def}\quad C_1,\ C_2$

$\mathbf{impl}\quad \alpha_1$

$$y \ {=} \ C_1 e ^{- a x} +C_2$$

$\mathbf{impl}\quad \alpha_2$

$$y \ {=} \ C_1 x + C_2$$

更笼统的情况

...

更特殊的情况

...

在复数域的情况

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