微分方程

可分离变量的微分方程

指数变化率

如果y以正比与当前数量的速率变化\(\frac{\mathrm{d}y}{\mathrm{d}t}=ky\)并且当t=0时\(y=y_0\),则

\[y=y_0\mathrm{e}^{kt}\]

k为速率常数,k>0表示增长, k<0表示衰减

Proof:

Solve differential equations with separable varaibles

1.separate varaibles

\[\begin{align} \frac{\mathrm{d}y}{\mathrm{d}t}&=ky\\ \frac{1}{y}\mathrm{dy}&=k\mathrm{d}t \end{align}\]

2.Intefral at both ends of the equation

\[\begin{align} \int \frac{1}{y}\mathrm{dy}&=\int k\mathrm{d}t\\ ln|y|&=kt+C \end{align}\]

3.Simplify the equation

\[y=y_0\mathrm{e}^{kt}\]

线性一阶微分方程

线性一阶微分方程的解

线性方程 \(\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\) 的解为

\[y=\frac{1}{v(x)}\int v(x)Q(x)\mathrm{d}x\]

其中

\[v(x)=\mathrm{e}^{\int P(x)\mathrm{d}x}\]

Proof:

Inspired by the Deravitive Product Rule

\[\frac{\mathrm{d}y}{\mathrm{d}x}(vy)=y'v+yv'\]

Product v(x) to the both ends of the equation, turn it to the form of DPR

\[\begin{align} v(x)\frac{\mathrm{d}y}{\mathrm{d}x}+v(x)P(x)y&=v(x)Q(x)\\ \frac{\mathrm{d}}{\mathrm{d}x}(v(x)y)&=v(x)Q(x)\\ y&=\frac{1}{v(x)}\int v(x)Q(x)\mathrm{d}x \end{align}\]

For v(x)

\[\frac{\mathrm{dv}}{\mathrm{d}x}=v(x)P(x)\\ v(x)=\mathrm{e}^{\int P(x)\mathrm{d}x}\]