User:EGM6321.f09.team1.Zhichao Gong/Mtg12

Mtg 12: Thu, 17 Sep 09

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p.11-3

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

$${\color{blue}\underline{HW:}} \ use \ {\phi}_{xy}={\phi}_{yx} \ to \ obtain \ {\color{blue}p.10-2} \ {\color{red}Eq(4)}$$

1st rel. in exactness condition 2 for N2. ODEs

Application: ZGenerate exact N2. ODEs (reverse engineering)

$${\color{blue}\underline{Method:}} \ {\color{blue}1)} \ Select \ \phi \ (x, \ y, \ {\color{blue} \underset{ \overset{||}{{y}^{'}}}})$$

$$F(x, \ y, \ {y}^{'}, \ {y}^{''})=\frac{d}{dx}\phi (x, \ y, \ {y}^{'})$$

$$Consider: \ \phi(x, \ y, \ p)=4{x}^{5}{p}^{2}+2{x}^{2}y$$

$${\phi}_{x}=20{x}^{4}{p}^{2}+4xy \ {\phi}_{p}=8{x}^{5}p$$

$${\phi}_{y}=2{x}^{2}$$

$$(8{x}^{5}{y}^{'}){y}^{''}+2{x}^{2}{y}^{'}+20{x}^{4}{{y}^{'}}^{2}+4xy=0$$

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Show above N2. ODE exact.

$${\color{blue}\underline} \ Condition \ 1:$$

$$f=8{x}^{5}p, \ g=2{x}^{2}p+20{x}^{4}{p}^{2}+4xy$$

$${\color{blue}\underline} \ Condition \ 2:$$

p.10-2 Eqs (4) and (5)HW.

$$A \ first \ int. \ : \ f={\phi}_{p}$$

$$\Rightarrow \phi=h(x,y)+{\color{blue} \underset{b}{ \underbrace{\color{black}\int_{}^{}{\color{blue} \underset{8{x}^{5}p}{ \underbrace}}dp}}}$$

$$g={\phi}_{x}+{\phi}_{y}p \Rightarrow$$

$${\color{red}\underline{\color{black}2{x}^{2}p}}+{\color{blue} \underset{get \ rid \ of}{ \underbrace}}+4xy=({h}_{x}+{\color{blue} \underset{get \ rid \ of}{ \underbrace}})+({h}_{y}+0){\color{red}\underline{\color{black}p}}$$

$$Assume {h}_{y}=2{x}^{2} \Rightarrow \ h=2{x}^{2}y+k(x)$$

$${h}_{x}=4xy=4xy+{k}_{x} \ \Rightarrow \ {k}_{x}=0$$

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$$\Rightarrow \ k(x)=k \ Constant$$

$$h(x,y)=2{x}^{2}y+k$$

$$\phi(x, \ y, \ p)=2{x}^{2}y+4{x}^{5}{p}^{2}+k$$

$${\color{blue}\underline{\color{black}Non-exact \ L2-ODE-VC:}}$$

$${\color{blue}\underline{\color{black}Application:}} \ \sqrt{x}{y}^{''}+2x{y}^{'}+3y=0 \ {\color{red}(1)}$$

HW: Verify exactness of above ODE.

Recipe: For special power(No sinx, cosx ...) form

$${\color{blue}L1-ODE-VC:} \ {\color{red}(Actually, \ N1-ODE-VC \ for \ b,d \neq 0)}$$

$${x}^{a}{y}^{b}(py+qx{y}^{'})+{x}^{c}{y}^{d}(ry+sx{y}^{'})$$

$$where \ (a, \ b, \ c, \ d), \ (p, \ q, \ r, \ s)\in \ R$$

$${\color{blue}L2-ODE-VC:} $$

$$\ \alpha{x}^{r}{y}^{''}+\beta{x}^{s}{y}^{'}+\gamma{x}^{t}y$$

$$where \ (\alpha, \ \beta, \ \gamma),(r, \ s, \ t) \in \ R$$

$$Consider \ int. \ feet. \ h(x,y)={x}^{'}{y}^{}$$

$$where \ (m, \ n)\in \ R \ to \ be \ determined$$

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Back to applp,12-3 Eq(1), final (m,n) st:  

$$({x}^{m}{y}^{n})[\sqrt{x}{y}^{''}+2x{y}^{'}+3y]=0 \ is \ exact.$$