User:EGM6321.f09.team1.Zhichao Gong/Mtg8

Mtg 8: Thu, 10 Sep 09

page8-1

Application: General non-homogenous L1_ODE_VC

$${\color{red}(1)} \ P(x){y}^{'}+Q(x)y=R(x)$$

$$If \ P(x) \neq 0 \ \forall \ \Rightarrow$$

$${\color{red}(2)}$$

$${\color{blue} \underset{ \overset{ \uparrow}{N(x,y)}}}+{\color{blue} \underset{M(x,y)}{ \underbrace}} \ {\color{blue}p.4-2} \ {\color{red}Eq.(3)}$$

$${\color{blue} \underset{ \overset{||}{1}}}({\color{blue} \underset{ \overset{||}{0}}}-{\color{blue} \underset{ \overset{||}{{a}_{0}(x)}}})=-f(x) \ {\color{red}Case1}$$

$$\Rightarrow$$

$$\Rightarrow$$

page8-2

$${\color{blue}p.8-1} \ {\color{red}Eq(2)} \ {y}^{'}+{a}_{0}y=b \ {\color{red}(1)}$$

$$Mult \ by \ h: \ h({y}^{'}+{a}_{0}y)=hb \ {\color{red}(2)}$$

$$Recall \ \frac{{h}_{x}}{h}= \underset{ \overset{||}{{a}_{0}}}{f} \ {\color{blue}p.6-3} \ {\color{red}(2)}$$

$$\Rightarrow \ h{a}_{0}={h}_{x}={h}^{'} \ {\color{red}(3)}$$

$${\color{red}Eqs \ (3) \ and \ (2)} \ {\color{blue} \underset{{(hy)}^{'}}{ \underbrace}}$$

$$\Rightarrow$$

$$\color{red}(4)$$

Application: Non-homoge L1.ODE.VC

$${y}^{'}+{\color{blue} \underset{{a}_{0}}{ \underbrace}}={\color{blue} \underset{b(x)}{ \underbrace}} \ {\color{red}(5)}$$

$${\color{blue}HW:} \ h(x)=x \ {\color{red}(6)}$$

page8-3

$$Mult \ (5) \ by \ h(x): \ {\color{blue} \underset{\frac{d}{dx}(xy)}{ \underbrace}}+y={x}^{3}$$

$${\color{blue}\underline{HW:}} \ y=\frac{{x}^{3}}{4}+\frac{C}{x}$$

$$C=constant$$

Recall HW on p.4-2 A class ofN1_ODEs

p,6-4 Eq(2): particular case 1.

$$One \ possibility$$

$$\ N(x,y)=N(x)$$

$$\ {N}_{x}=b(x)$$

$$\ {M}_{y}=a(x)$$

$${M}_{y}(x,y)=a(x) \ \Rightarrow \ M(x,y)=a(x)y+k$$

$${N}_{x}=b(x) \ \Rightarrow \ N(x,y)=\int_{}^{x}b(s)ds=c(s)$$

page8-4

$$\ Mdx + Ndy = [a(x)y + k]dx + c(x)dy = 0$$

Exact L1_ODE_VC