人工智能教程 - 数学基础课程1.2 - 数学分析(三)-3

一阶线性

First-order linear

$a(x)y'+b(x)y=c(x)$

linear: $ay_1+by_2=c$

齐次(homogeneous) (c=0)

Standard linear from y’+ p(x)y = q(x)

Models
温度-浓度模型
mixing
Decay, bank some motion
传导(Conducion)

Newton cooling law:

$\frac{dT}{dt}=k(T_e-T)$

k>0 <— conductivity
T(0) = $T_0$

扩散(Diffusion)

C = salt concentration inside
$C_e$ = salt conc. outside
半透明膜(membrare wall)

$\frac{dC}{dt}=k_1(C_e-C)$

$k_1$> 0

General linear equation:

$\frac{dT}{dt}+kT=kT_e$

Method: y’+ py = q

0. standard linear form
1. Calc $e^{\int pdx}$ : int factor
2. Mulit both sides by $e^{\int pdx}$
3. Integrate

Temp: $\frac{dT}{dt}+kT=kT_e)$

p: $e^{ kt}$ Multi by $e^{ kt}$
$(e^{ kt}T)' = k.T_e.e^{ kt}$

T(0) = $T_0$ ; C = $T_0$
k>0

Integral: $e^{ kt}T$ = $\int kT_e(t)e^{kt}+c$

$T = e^{ -kt}\int _0^1kT_e(t_1)e^{kt_1}dt_1+ce^{-kt}$

稳定态(steady state solution)

来源：CSDN

作者：KuFun人工智能

链接：https://blog.csdn.net/fsdaewrq/article/details/104201896