# 常微分方程笔记（2）——一阶微分方程

## 常微分方程笔记（2）——一阶微分方程

Contents

### 分离变量

#### 变量可分离的方程

$$\frac{dy}{dx}=f(x)\cdot g(y),\label{4}$$

$$\frac{1}{g(y)}dy=f(x)dx,$$

$$\int\frac{1}{g(y)}dy=\int f(x)dx+C$$

#### 齐次方程

$$\frac{dy}{dx}=g\left(\frac{y}{x}\right),\label{7}$$

$$\frac{du}{dx}=\frac{g(u)-u}{x},$$

#### 可化为齐次的方程

$$\frac{dy}{dx}=h\left(\frac{ax+by+c}{Ax+By+C}\right),$$

$$\begin{cases} ax+by+c=0,\\ Ax+By+C=0. \end{cases}$$

$$\frac{du}{dv}=h\left(\frac{au+bv}{Au+Bv}\right),$$

$$\frac{du}{dx}=a+bh\left(\frac{u+c}{\lambda u+C}\right),$$

### 恰当方程和积分因子

$$P(x,y)dx+Q(x,y)dy=0.\label{2}$$

$$d\Phi(x,y)=P(x,y)dx+Q(x,y)dy,$$

$$\mu(x,y)P(x,y)dx+\mu(x,y)Q(x,y)dy=0$$

$$\frac{\partial P}{\partial y}(x,y)=\frac{\partial Q}{\partial x}(x,y).\label{5}$$

$$\int_{x_0}^xP(x,y)dx+\int_{y_0}^yQ(x_0,y)dy=C,$$

$\mu(x,y)$ 为方程的积分因子的充要条件为
$$\frac{\partial (\mu P)}{\partial y}(x,y)=\frac{\partial (\mu Q)}{\partial x}(x,y).$$

$$\frac{1}{Q(x,y)}\left(\frac{\partial P(x,y)}{\partial y}-\frac{\partial Q(x,y)}{\partial x}\right)=G(x)$$

$$\mu(x)=e^{\int G(x)}dx$$

$$\frac{1}{P(x,y)}\left(\frac{\partial Q(x,y)}{\partial x}-\frac{\partial P(x,y)}{\partial y}\right)=H(y)$$

$$\mu(y)=e^{\int H(x)}dy$$

$$\mu(x,y)P(x,y)dx+\mu(x,y)Q(x,y)dy=d\Phi(x,y),$$

$$\mu(x,y)g(\Phi(x,y))(P(x,y)dx+Q(x,y)dy)=d\int g(\Phi)d\Phi,$$

$$\mu(x,y)=\frac{1}{xP(x,y)+yQ(x,y)}$$

### 一阶显式方程

#### 一阶线性微分方程

$$\frac{dy}{dx}+p(x)y=q(x),\label{13}$$

$$\frac{d}{dx}\left(ye^{\int p(x)dx}\right)=q(x)e^{\int p(x)dx}.$$

$$ye^{\int p(x)dx}=\int q(x)e^{\int p(x)dx}dx+C.$$

$$y=e^{-\int p(x)dx}\left(\int q(x)e^{\int p(x)dx}dx+C\right).$$

#### Bernoulli方程

$$\frac{dy}{dx}+p(x)y=q(x)y^\alpha,\label{17}$$

$$\frac{dz}{dx}+(1-\alpha)p(x)z=(1-\alpha)q(x),$$

#### Riccati方程

$$\frac{dy}{dx}+g(x)y+h(x)y^2=k(x),\label{19}$$

$$u^\prime+gu+h(y^2-\phi^2)=0.$$

$$u^\prime+(g+2\phi h)u+hu^2=0,$$

### 一阶隐式方程

$$\dot{y}=\frac{dy}{dp}=\frac{dy}{dx}\cdot\frac{dx}{dp}=p\dot{x}.\label{6}$$

#### 一般情况

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

$$x=g(y,y^\prime),$$

$$y=h(x,y^\prime),$$

#### Clairaut方程

$$y=xy^\prime+g(y^\prime),$$

$$\dot{y}=p\dot{x}+x+\dot{g},$$

$$\begin{cases} x(p)=-\dot{g}(p),\\ y(p)=-p\dot{g}(p)+g(p). \end{cases}$$

$$y=cx+g(c).$$

#### D'Alembert方程

$$y=xf(y^\prime)+g(y^\prime),$$

$$\dot{y}=\dot{x}f+x\dot{f}+\dot{g},$$

$$\dot{x}=\frac{x\dot{f}(p)+\dot{g}(p)}{p-f(p)},$$

$$y=cx+d.$$