# 一阶常微分方程

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=f(x(t),t)}$

## 一阶线性微分方程

${\displaystyle \forall t\in I,{\frac {\mathrm {d} x}{\mathrm {d} t}}=a(t)x(t)+b(t)}$

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=a(t)x(t)}$

${\displaystyle S=\left\{x:\;t\mapsto ce^{\int ^{t}a(u)\mathrm {d} u};\;\;c\in \mathbb {R} \right\}.}$

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=a(t)x(t)+b(t)}$

{\displaystyle {\begin{aligned}S'&=\left\{x:\;t\mapsto \left(c+\int ^{t}b(u)e^{-\int ^{u}a(v)\mathrm {d} v}\mathrm {d} u\right)e^{\int ^{t}a(u)\mathrm {d} u};\;\;c\in \mathbb {R} \right\}\\&=S+x^{*},\end{aligned}}}

${\displaystyle x^{*}:\;t\mapsto e^{\int ^{t}a(u)\mathrm {d} u}\int ^{t}b(u)e^{-\int ^{u}a(v)\mathrm {d} v}\mathrm {d} u=\int ^{t}b(u)e^{\int _{u}^{t}a(v)\mathrm {d} v}\mathrm {d} u}$

## 变量分离方程

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=a(t)b(x),}$

${\displaystyle {\frac {\mathrm {d} x}{b(x)}}=a(t)\mathrm {d} t}$

${\displaystyle \int ^{x}{\frac {1}{b(u)}}\mathrm {d} u=\int ^{t}a(s)\mathrm {d} s+c}$

## 恰当微分方程

${\displaystyle \mathrm {d} x=f(x,t)\mathrm {d} t}$

tx视为变量平等看待，可以将其看作是对称的一阶微分方程：

${\displaystyle P(x,t)\mathrm {d} x+Q(x,t)\mathrm {d} t=0}$

${\displaystyle P(x,t)\mathrm {d} x+Q(x,t)\mathrm {d} t=\mathrm {d} U(x,t)={\frac {\partial U}{\partial x}}(x,t)\mathrm {d} x+{\frac {\partial U}{\partial t}}(x,t)\mathrm {d} t,}$

${\displaystyle U(x,t)=c}$

${\displaystyle {\frac {\partial }{\partial t}}P(x,t)={\frac {\partial }{\partial x}}Q(x,t).}$

${\displaystyle U(x,t)=\int ^{x}P(u,t)\mathrm {d} u+\int \left[Q(x,s)-\left.{\frac {\partial }{\partial t}}\int ^{x}P(u,t)\mathrm {d} u\right|_{t=s}\right]\mathrm {d} s}$

### 积分因子

${\displaystyle P(x,t)\mathrm {d} x+Q(x,t)\mathrm {d} t=0}$

${\displaystyle \mu (x,t)P(x,t)\mathrm {d} x+\mu (x,t)Q(x,t)\mathrm {d} t=\mathrm {d} U(x,t)}$

## 解的存在性

${\displaystyle {\frac {\mathrm {d} x(t)}{\mathrm {d} t}}=f(x(t),t),\;\;x(t_{0})=x_{0}.}$

E为一个完备的有限维赋范向量空间UE中的一个开集I${\displaystyle \mathbb {R} }$中的一个区间。函数f是从U×I映射到E中的连续函数。柯西-利普希茨定理说明了，若函数fU中满足利普希茨条件，也就是说，

${\displaystyle \exists \kappa >0,\ \forall t\in I,\ \forall x,y\in U,\ \left|f(x,t)-f(y,t)\right|\leq \kappa \left|x-y\right|}$

## 参考来源

• 王高雄，周之铭，朱思铭，王寿松. 常微分方程. 高等教育出版社，第三版. 2006. ISBN 9787040193664.