一阶常微分方程

(1) 可分离变量方程

$$y^{\prime}=f(x)g(y)\qquad \int\frac{\mathrm{d}y}{g(y)}=\int f(x)\mathrm{d}x$$

(2) 齐次方程

$$y^{\prime}=f(x,y)=g\left(\frac{y}{x}\right)$$

令$z=\frac{y}{x},y=xz$有

$$\frac{\mathrm{d}y}{\mathrm{d}x}=z+x\frac{\mathrm{d}z}{\mathrm{d}x}\qquad\frac{\mathrm{d}z}{\mathrm{d}x}=\frac{g(z)-z}{x}\qquad\int\frac{\mathrm{d}z}{g(z)-z}=\int\frac{\mathrm{d}x}{x}$$

(3) 准齐次方程

形如

$$\frac{\mathrm{d}y}{\mathrm{d}x}=f\left(\frac{a_{1}x+b_{1}y+c_{1}}{a_{2}x+b_{2}y+c_{2}}\right)$$

(a) $\Delta=\begin{vmatrix} a_{1}&b_{1}\ a_{2}&b_{2} \end{vmatrix}\neq 0\qquad \begin{aligned}&a_{1}x+b_{1}y+c_{1}=0\ &a_{2}x+b_{2}y+c_{2}=0\end{aligned} \qquad(x,y)=(\alpha,\beta)$

$$\begin{aligned}&x=X+\alpha\ &y=Y+\beta \end{aligned}\qquad \frac{\mathrm{d}Y}{\mathrm{d}X}=f\left(\frac{a_{1}X+b_{1}Y}{a_{2}X+b_{2}Y}\right)$$

(b) $\Delta=0\qquad\frac{a_{2}}{a_{1}}=\frac{b_{2}}{b_{1}}=\lambda$，令$z=a_{1}x+b_{1}y$有

$$\frac{\mathrm{d}z}{\mathrm{d}x}=a_{1}+b_{1}f\left(\frac{z+c_{1}}{\lambda z+c_{2}}\right)$$

(4) 一阶线性方程

形如

$$y^{\prime}+P(x)y=Q(x)$$

则有

$$\begin{aligned}y^{\prime}+P(x)y&=Q(x)\ \left(ye^{\int P\mathrm{d}x}\right)^{\prime}&=e^{\int P\mathrm{d}x}Q\ ye^{\int P\mathrm{d}x}&=\int e^{\int P\mathrm{d}x}Q\mathrm{d}x+C\ y&=e^{-\int P\mathrm{d}x}\left[\int e^{\int P\mathrm{d}x}Q\mathrm{d}x+C\right]\end{aligned}$$

(5) Bernoulli方程

形如

$$y^{\prime}+P(x)y=Q(x)y^{\prime\prime}$$

当$n\neq 0,1$时，令$z=y^{1-n}$，则有：

$$z^{\prime}+(1-n)Pz=(1-n)Q$$

高阶微分方程

(1) 纯导型方程

$$\begin{aligned} y^{(n)}&=f(x)\\ y^{(n-1)}&=\int f(x)\mathrm{d}+C\\ \vdots&\qquad\vdots \end{aligned}$$

(2) 缺$y$型方程

$$y^{\prime\prime}=f(x,y^{\prime})$$

设$y^{\prime}=p$，则

$$p^{\prime}=f(x,p)$$

(3) 缺$x$型方程

$$y^{\prime\prime}=f(y,y^{\prime})$$

设$y^{\prime}=p$，则

$$\begin{aligned}y^{\prime\prime}&=\frac{\mathrm{d}p}{\mathrm{d}x}=\frac{\mathrm{d}p}{\mathrm{d}y}\cdot\frac{\mathrm{d}y}{\mathrm{d}x}=p\frac{\mathrm{d}p}{\mathrm{d}y}\\ p\frac{\mathrm{d}p}{\mathrm{d}y}&=f(y,p)\end{aligned}$$

二阶线性方程(常系数线性方程)

(1) 二阶齐次方程

$$y^{\prime\prime}+py^{\prime}+qy=0\qquad y=e^{rx}$$

特征方程

$$r^{2}+pr+q=0$$

(a) 当特征根是不相等的实数$r_{1},r_{2}$时

$$y=C_{1}e^{r_{1}x}+C_{2}e^{r_{2}x}$$

(b) 当特征根是重根$r_{1}=r_{2}$时

$$y=\left(C_{1}+C_{2}x\right)e^{rx}$$

$$y=e^{\alpha x}\left(C_{1}\cos\beta x+C_{2}\sin\beta x\right)$$

(2) 二阶非齐次方程

$$y^{\prime\prime}=py^{\prime}+qy=f$$

待定系数法得特解

(a) $f(x)=e^{\lambda x}P_{m}(x)$，则$y=x^{k}Q_{m}(x)e^{\lambda x}$

$$k=\begin{cases}&0&\qquad\lambda\text{不是特征方程的根}\\ &1&\qquad\lambda\text{是特征方程的单根}\\ &2&\qquad\lambda\text{是特征方程的重根}\end{cases}$$

(b) $f(x)=e^{\lambda x}\left[P_{l}\cos\omega x+Q_n(x)\sin\omega x\right]$，则$y=x^{k}e^{\lambda x}\left[A_{m}(x)\cos\omega x+B_{m}(x)\sin\omega x\right]$

$$k=\begin{cases}&0&\qquad\lambda\pm i\omega\text{不是特征方程的根}\\ &1&\qquad\lambda\pm i\omega\text{便是特征方程的根}\end{cases}$$