User:EGM6321.f09.team1.Zhichao Gong/Mtg14

Mtg 14: Thu, 24 Sep 09

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$${\color{blue}p.13-3} \ {\color{red}(Eq.(1))} \ {\phi}_{y}(k)=\frac{\partial \phi}{\partial y}(k) \ {\color{red}(1)}$$

$$\ 0, \ ..., \ n$$

$${\color{blue}\underline{HW:}} \ {\color{blue}p.13-3} \ Case \ n=1.$$

$$1) \ Find \ {f}_{0} \ in \ terms \ of \ \phi$$

$$2) \ Find \ {f}_{1} \ in \ terms \ of \ \phi \ {\color{blue}{f}_{1}={\phi}_{y}}$$

$$3) \ show \ {f}_{0}-\frac{{f}_{1}}{dx}=0 \Leftrightarrow {\phi}_{xy}={\phi}_{yx} \ {\color{red}(2)}$$

$${\color{blue}\underline{HW:}} \ Case \ n=2.$$

$$1) \ show \ {f}_{1}=\frac{{f}_{2}}{dx}+{\phi}_{y} \ {\color{red}(3)}$$

$$2) \ show \ \frac{d}{dx}({\phi}_{y})={f}_{0} \ {\color{red}(4)}$$

$$3) \ {f}_{0}-\frac{d}{dx}{f}_{1}+\frac{{d}^{2}}{d{x}^{2}}{f}_{2}=0 \ {\color{red}(5)}$$

$$Relate \ {\color{red}(5)} \ to \ {\color{red}Eqs \ (4) \ and \ (5)} \ <\color{blue}p.10-2>$$

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$${\color{blue}\underline{HW:}} \ {\color{blue}Legendre \ L2-ODE-VC}$$

$$F=(1-{x}^{2}){y}^{''}-2x{y}^{'}+n(n+1)y=0 \ {\color{red}(1)}$$

$${\color{blue}Ref:} \ {\color{red} \underset{K}{ \underbrace}} \ (2003) \ p.31$$

$$1) \ Verify \ exactness \ of \ {\color{red}(1)} \ using \ 2 \ method$$

$$1a) \ {\color{blue}p.10-3} \ {\color{red}Eqs. \ (4) \ and \ (5)}$$

$$2a) \ {\color{blue}p.14-1} \ {\color{red}Eqs. \ (5)}$$

$$2) \ If \ not \ exact, \ see \ whether \ it \ can$$

$$be \ made \ exact \ using \ int. \ fact.$$

$$meth. \ w/ \ h(x,y)={x}^{m}{y}^{n}$$

$$Superposition: \ {\color{blue}p.3-3} \ {\color{red}Eq(3)} \ Linearity$$

$${\color{blue}\underline{Read:} \ K.(2003)}$$

$${\color{blue} \ App.3: \ Power \ series}$$

$${\color{blue} \ App.5: \ ODEs \ const.}$$

$${\color{blue}coeff. \ \begin{cases} & int. \ fact. \\ & trial \ soln \  \equiv \ undetermined \ coeff. \end{cases}}$$ page14-3

$${\color{blue}\underline{\color{black}Alternative}} \ defn \ of \ {\color{blue}\underline{\color{black}Linearity:}}$$

L(.) is linear iff ('if and only if' or "equivalent f0")

$${\color{blue}\underline{HW:}} \ show \ {\color{blue}[{\color{red}(1) \ and \ (2)}]} \ \Leftrightarrow \ {\color{blue}[{\color{red}(3)} \ p.3-3]}$$

$${\color{blue}\underline{K. \ p.4}} \ {L}_: \ {C}^{2}(I) \rightarrow \ {C}^{0}(I)$$


 * PEA1.F09.Mtg14.pg3.fig1.svg

$$\ {C}^{0}(I)=space(set)$$

of continous funcs defined on I (superscript "0" means zeroth deriv.)

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C2(I)= space of funcs on I that are cont. , having cont. first and 2nd(superscript"2") deriv.

$$ \begin{cases} & \ sinx, \ cosx, \ exp(x)   \\ & \ {x}^{2}, \ {x}^{3}, \ ... \end{cases}$$

$$\in \ {C}^{\infty}$$

inf. diff.('smooth') duncs


 * PEA1.F09.Mtg14.pg4.fig1.svg