二阶常系数齐次线性微分方程 :: 待定系数降阶法

$$\Gamma: y ^{\prime \prime} + p y ^{\prime} + q y \ {=} \ 0$$

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

$\mathbf{def}\quad \Delta \ {=} \ p^2 - 4q$

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

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

$\mathbf{await}\quad k \in \mathbb C,\ k ^{\prime} \ {=} \ 0$

$\mathbf{def}\quad u \in \mathbb C,\ {\color{Peach} y} \ {=} \ e ^{k {\color{Peach} x}} u$

$y {\color{Teal} ^{\prime}} \ {=} \ {\color{Teal} k} e ^{k x} u + e ^{k x} u {\color{Teal} ^{\prime}}$

$y ^{\prime {\color{Teal} {\prime}}} \ {=} \ k {\color{Teal} ^2} e ^{k x} u + {\color{Teal} 2} k e ^{k x} u ^{\prime} + e ^{k x} u ^{\prime {\color{Teal} \prime}}$

返回到方程中

$$\left({\color{Teal} k ^2 {\color{Cyan} e ^{k x}} u + 2 k {\color{Cyan} e ^{k x}} u ^{\prime} + {\color{Cyan} e ^{k x}} u ^{\prime \prime}}\right) + p \left({\color{Teal} k {\color{Cyan} e ^{k x}} u + {\color{Cyan} e ^{k x}} u ^{\prime}}\right) + q {\color{Teal} {\color{Cyan} e ^{k x}} u} \ {=} \ 0$$

$${\color{Gray} k ^2} u + {\color{Gray} 2 k} u ^{\prime} + u ^{\prime \prime} + {\color{Gray} p k} u + {\color{Gray} p} u ^{\prime} + {\color{Gray} q} u \ {=} \ 0$$

$$u ^{\prime \prime} + \left({\color{Teal} 2 k + p}\right) u ^{\prime} + {\color{Gray} \left({\color{Cyan} k ^2 + p k + q}\right) u} \ {=} \ 0$$

$\mathbf{echo}\quad {\color{Peach} k}: {\color{Peach} k ^2 + p k + q} \ {=} \ 0$

$$u ^{\prime \prime} + \left( 2 k + p\right) u ^{\prime} \ {=} \ 0$$

$\mathbf{use}$

$y : y ^{\prime {\color{Gray} \prime}} + a y ^{\prime} \ {=} \ 0$

$\mathbf{def}\quad C_1 \in \mathbb{C},\ C_2 \in \mathbb{C}$

$\mathbf{sync}\quad \alpha_1: \left\langle 2 k + p \ne 0 \right\rangle\quad \square \to \Gamma_1$

$\mathbf{sync}\quad \alpha_2: \left\langle 2 k + p \ {=} \ 0 \right\rangle \Rightarrow \left\langle k \ {=} \ - \frac{p}{2} \right\rangle\quad \square \to \Gamma_2$

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

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

$$\color{Tan} u \ {=} \ C_1 e ^{- \left(2 k + p\right) x} +C_2$$

$$y \ {=} \ {\color{Teal} \text{Re}} \left(C_1 e ^{{\color{Gray} - \left(k + p\right)} x} + C_2 {\color{Teal} e ^{{\color{Gray} k} x}}\right)$$

$\mathbf{overload}\quad r_1 \mid _{\left\langle a \right\rangle} \ {=} \ {\color{Peach} - \left(k ! _{\color{Black} \left\langle a \right\rangle} + p\right)},\ r_2 \mid _{\left\langle a \right\rangle} \ {=} \ {\color{Peach} k ! _{\color{Black} \left\langle a \right\rangle}}$

$${=} \ \text{Re} \left(C_1 e ^{{\color{Teal} r_1 \mid _{\left\langle a \right\rangle}} x} + C_2 e ^{{\color{Teal} r_2 \mid _{\left\langle a \right\rangle}} x}\right)$$

$\left[\mathbb R :: 一元二次方程\right]$

$\alpha_3: \left\langle \Delta > 0 \right\rangle \Rightarrow \left\langle p ^2 - 4 q < 0 \right\rangle \to \Gamma_3$

$\_: \left\langle \Delta \ {=} \ 0 \right\rangle \Rightarrow \left\langle 2 k + p \ {=} \ 0 \right\rangle \to \blacksquare$

$\alpha_4: \left\langle \Delta < 0 \right\rangle \Rightarrow \left\langle p ^2 - 4 q > 0 \right\rangle \to \Gamma_4$

$\Gamma \to \Gamma_0 :: \Gamma_1 :: \Gamma_3 \ {=} \ \Gamma_1 \mid \alpha_3$

$\mathbf{def}\quad \alpha \ {=} \ - \frac{p}{2},\ \beta \ {=} \ \frac{\sqrt{4 q - p ^2}}{2}$

$\mathbf{def}\quad r_1 \ {=} \ \alpha + \beta i,\ r_2 \ {=} \ \alpha - \beta i$

$$y \ {=} \ \text{Re} \left(C_1 {\color{Gray} e ^{{\color{Cyan} \left(\alpha + \beta i\right)} x}} + C_2 {\color{Gray} e ^{{\color{Cyan} \left(\alpha - \beta i\right)} x}}\right)$$

$${=} \ e ^{\alpha x} \text{Re} \left(C_1 {\color{Teal} \cos \left(\beta\right)} + C_1 {\color{Teal} i \sin \left(\beta\right)} + C_2 {\color{Teal} \cos \left({\color{Cyan} -} \beta\right)} {\color{Gray} +} C_2 {\color{Teal} i \sin \left({\color{Cyan} -} \beta\right)} \right)$$

$${=} \ e ^{\alpha x} \text{Re} \left(C_1 \cos \left(\beta\right) + C_1 i \sin \left(\beta\right) + C_2 \cos \left(\beta\right) {\color{Teal} -} C_2 i \sin \left(\beta\right) \right)$$

$${=} \ e ^{\alpha x} \text{Re} \left({\color{Teal} \left({\color{Black} C_1 + C_2}\right)} \cos \left(\beta\right) + i {\color{Teal} \left({\color{Black} C_1 - C_2}\right)} \sin \left(\beta\right)\right)$$

$${=} \ e ^{\alpha x} \left({\color{Teal}\text{Re} \left({\color{Black} C_1 + C_2}\right)} \cos \left(\beta\right) + {\color{Teal} \text{Re} \left({\color{Black}i \left(C_1 - C_2\right)}\right)} \sin \left(\beta\right)\right)$$

$\mathbf{overload}\quad C_1 \ {=} \ {\color{Peach} \text{Re} \left(C_1 + C_2\right)},\ C_2 \ {=} \ {\color{Peach} \text{Re} \left(i \left(C_1 - C_2\right)\right)}$

$${=} \ e ^{\alpha x} \left({\color{Teal} C_1} \cos \left(\beta\right) + {\color{Teal} C_2} \sin \left(\beta\right)\right)$$

$\blacksquare$

+ $\Gamma \to \Gamma_0 :: \Gamma_1 :: \Gamma_4 \ {=} \ \Gamma_1 \mid \alpha_4$

$\mathbf{def}\quad r_1 \ {=} \ \frac{- p + \sqrt{p^2 - 4 q}}{2},\ r_2 \ {=} \ \frac{- p - \sqrt{p^2 - 4 q}}{2}$

$$y \ {=} \ \left(C_1 e ^{{\color{Teal} r_1} x} + C_2 e ^{{\color{Teal} r_2} x}\right)$$

$\blacksquare$

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

$\mathbf{send}\quad y \ {=} \ C_1 x + C_2$

$$\color{Tan} u \ {=} \ C_1 x + C_2$$

$${\color{Teal} y} \ {=} \ e ^{- \frac{p}{2}} \left(C_1 x + C_2\right)$$

$\blacksquare$

$\Gamma \to \Gamma_0 \rightleftharpoons \Gamma_1 :: \Gamma_3 + \Gamma_1 :: \Gamma_4 + \Gamma_2$

$\mathbf{impl}\quad \alpha_1 + \Gamma_1 :: \alpha_4 \ {=} \ \left\langle \Delta > 0 \right\rangle$

$\mathbf{def}\quad r_1 \ {=} \ \frac{-p + \sqrt{\Delta}}{2},\ r_2 \ {=} \ \frac{-p - \sqrt{\Delta}}{2}$

$$y \ {=} \ C_1 e^{r_1 x} + C_2 e^{r_2 x}$$

$\mathbf{impl}\quad \alpha_2 \ {=} \ \left\langle \Delta \ {=} \ 0 \right\rangle$

$\mathbf{def}\quad r \ {=} \ -\frac{p}{2}$

$$y \ {=} \ e^{r x}(C_1 x + C_2)$$

$\mathbf{impl}\quad \alpha_1 + \Gamma_1 :: \alpha_3 \ {=} \ \left\langle \Delta < 0 \right\rangle$

$\mathbf{def}\quad \alpha \ {=} \ -\frac{p}{2},\ \beta \ {=} \ \frac{\sqrt{-\Delta}}{2}$

$$y \ {=} \ e^{\alpha x}\left(C_1 \cos\beta x + C_2 \sin\beta x\right)$$

$\blacksquare$