User:EGM6321.f09.team1.Zhichao Gong/Mtg6

Mtg 6: Tue, 3 Sep 09

page6-1

Reverse engineering to invert billions and billions of exact nonlinear 1st order ODEs:

$$Just \ invert \ any \ \phi(x,y) \ {\color{blue}(nonlinear \ wrt \ (x,y))}$$

$$\phi(x,y)=6{x}^{4}+2{y}^{\frac{3}{2}}$$

$${\color{blue}p.5-3} \ {\color{red}Eq(1):} \ F=\frac{d}{dx} \phi(x,y(x))$$

$$ \begin{cases} & M={\phi}_{x} \\ & N={\phi}_{y} \end{cases}$$ $$\Rightarrow$$ $$M+N{y}^{'}=0$$ HW: Complete details and invent 3 more

page6-2

What if exactness condition 2 (p.6-1) not satisfied ? Euler intergate factor method: Assume p.5-2 Eq(2) not exact, then  mult it w/ h(x,y) yetr unkonwn(to <P>be determined)</P>

$$h(x,y)[M(x,y)dx+N(x,y)dy]=0$$

$$\Rightarrow \ {\color{blue} \underset{ \overset{-}{M}}{ \underbrace}}dx+{\color{blue} \underset{ \overset{-}{N}}{ \underbrace}}dy=0 \ {\color{red}(1)}$$

<P>Q: Find h (Euler intergrate factor) st</P> <P>Eq(1) is exact.</P> <P>Note: (1) already satisfiesexcatness condition 1 on p.6-1:</P> <P> </P>

$${\color{blue}p.5-4} {\color{red}Eq(1):} \ { \overset{-}{M}}_{y}={ \overset{-}{N}}_{x}$$

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$${ \overset{-}{M}}_{y}={(hM)}_{y}={h}_{y}M+h{M}_{y}$$

$${ \overset{-}{N}}_{x}={(hN)}_{x}={h}_{x}N+h{N}_{x}$$

$$\Rightarrow$$

<P>Solve Eq.(1) for h(x,y). Not easy in general.</P> <P>2 Particular cases:</P>

$${\color{blue} \underline{Case \ 1:}} \ If \ ({h}_{y}M)=0, \ (1) \ because$$

$${h}_{x}N+h({N}_{x}-{M}_{y})=0$$

$$\Rightarrow \ \frac{{h}_{x}}{h}=-{\color{red} \underset{known}{ \underbrace}}$$

$$ \underset{logh}{ \underbrace{\int \frac{{h}_{x}}{h}dx}}=-\int \frac{1}{N}({N}_{x}-{M}_{y})dx$$

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$${\underline{Note:}} \ {h}_{y}{\color{blue} \underset{ \neq0}{\underbrace}}=0 \ \Rightarrow \ {h}_{y}=0 $$

$$\Rightarrow$$

$$\Rightarrow$$

If (2) satisfied, then:

$${\color{blue}carl \ Sagan.}$$