lib/2026/4/17/1 :: \(\beta_2 : \left\langle Q \ne 0 \right\rangle\)

$T_2 {=} \Gamma_1 \mid \beta_2$

$\mathbf{def}\quad \displaystyle u: {\color{Peach} y} \ {=} \ u {\color{Peach} e ^{- I_1}} \quad \square \to u ? \to T_2$

$T_2 \to \Phi_u \ {=} \ T_2 \mid u$

在 $x$ 上求导得到

$$y {\color{Teal} ^{\prime}} \ {=} \ u {\color{Teal} ^{\prime}} e ^{- I_1} {\color{Teal} +} u {\color{Gray} \left(e ^{- I_1}\right)^{\prime}}$$

$$y ^{\prime} \ {=} \ u ^{\prime} e ^{- I_1} - {\color{Cyan} I_1 ^\prime} {\color{Gray} u e ^{- I_1}}$$

$$y ^{\prime} \ {=} \ u ^{\prime} e ^{- I_1} - P y$$

带入到 $T_2$

$${\color{Teal} u ^{\prime} e ^{- I_1}} {\color{Cyan} - P y} {\color{Gray} + P y} {=} f$$

$$u ^{\prime} e ^{{\color{Gray} -} I_1} {=} f$$

$$u ^{\prime} \ {=} \ f e ^{I_1}$$

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

$\mathbf {def} \quad C$

$$u \ {=} \ {\color{Teal} \int \text{d} x \left\{{\color{Black} f e ^{I_1}}\right\} + C}$$

$\blacksquare$

$T_2 \rightleftharpoons \Psi_u$

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