一阶非齐次线性微分方程

$$\Gamma: y ^{\prime} + P y {=} Q$$

$\Gamma: \left\langle y,\ P,\ Q\right\rangle$

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

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

$\left[\text{常数变易法}\right]$

$\mathbf{impl}$

$\beta_1: \left\langle Q \ {=} \ 0 \right\rangle$

$$\beta_1: y \ {=} \ {\color{Teal} C e ^{- I_1}}$$

$\mathbf{impl}$

$\beta_2 : \left\langle Q \ne 0 \right\rangle$

$\mathbf{overload}\quad C$

$$\beta_2: y \ {=} \ e ^{- I_1} {\color{Teal} \left(\int \text{d} x \left\{Q e ^{I_1}\right\} + C\right)}$$

注意到 $\Gamma_0 \mid \beta_1$ 适用于 $\Gamma_0 \mid \beta_2$ 的形式

$$y \ {=} \ e ^{- I_1} \left(\int \text{d} x \left\{Q e ^{I_1}\right\} + C\right)$$

$\blacksquare$

更特殊的情况

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