User:EGM6321.f09.team1.Zhichao Gong/Mtg20

Mtg 20: Thu, 8 Oct 09

page20-1

(3) &amp; (4) p.19-3 : c1(x), c2(x) in (1)p.18-3  

$$y(x)={\color{blue} \underset{{y}^{(x)}_{H}}{ \underbrace}}+{\color{blue} \underset{{y}^{(x)}_{H}}{ \underbrace}}+{\color{blue} \underset{{y}^{(x)}_{P}}{ \underbrace}}$$

Application: $${y}^{''}+y=xsinx=f(x)$$

Homog. soln: $$ \begin{cases} & {u}_{1}(x)=cosx \\ & {u}_{2}(x)=sinx \end{cases}$$

Note:trial soln $${y}_{H}={e}^{x}$$

$${r}^{2}+1=0 \ \Rightarrow \ r=+i, \ -i \ i=\sqrt{-1}$$

de Moivre $${e}^{i{\color{red}x}}=cosx+isinx$$

$$ \underset{-}{W}= \begin{bmatrix} \ cosx & sinx \\ -sinx & cosx \end{bmatrix} \Rightarrow W=det \underset{-}{W}=1$$

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$${d}^{(x)}_{1}=-\int^{}_{x}\frac{(sins)(ssins)}{1}ds$$

$${d}^{(x)}_{2}=-\int^{}_{x}\frac{(coss)(scoss)}{1}ds$$

$$y(x)=Acosx+Bsinx-\frac{1}{4}{x}^{2}cosx+\frac{1}{4}xsinx$$

K. p. 8

<P>Trial solns for non-homog L2-ODE-CC(const coeff)</P> <P>(for part. soln y<SUB>p</SUB>)</P>

$${a}_{2}{y}^{''}+{a}_{1}{y}^{'}+{a}_{0}y=f(x)$$

$${a}_{1}, \ {a}_{2}, \ {a}_{3}=const.$$

$${\color{red}\underline{f(x)}} \ \ {\color{blue}|} \  \ {\color{red}\underline{Trial \ Soln \ y(x)}}$$ $${P}_{n}(x)=\sum_{i=0}^{n}{\color{blue} \underset{ \overset{\uparrow}{known \ coeff}}}{x}^{i} \ \ {\color{blue}|} \  \ y(x)={x}^{m}\sum_{i=0}^{n}{\color{blue} \underset{ \overset{\uparrow}{unknown \ coeff. \ to \ be \ determined}}}{x}^{i}$$

page20-3

$${p}_{n}(x){e}^{{\color{blue} \underset{ \overset{\uparrow}{known}}{{\color{black}\alpha}}}x} \ \  {\color{blue}|} \  \ {x}^{m}(\sum_{n}^{i=0}{c}_{i}{x}^{i}){e}^{\alpha x}$$ $${p}_{n}{e}^{\alpha x} \begin{cases} & sin \beta x \\ & cos \beta x \end{cases} \ \ {\color{blue}|} \ {x}^{m}{e}^{\alpha x} \left[(\sum_{i=0}^{n}{c}_{i}{x}^{i})sin\beta x +(\sum_{i=0}^{n}{d}_{i}{x}^{i}cos \beta x) \right]$$

<P>Trial soln for non-homog. L2-ODE-VC</P> <P>(<U>Note:</U> Trial soln meth. also applicable to Nn-ODE)('Guess')</P>

<U>Application:</U>

$${y}^{''}-\frac{2}{{x}^{2}}y=7{x}^{4}+3{x}^{3}$$

<U>Note:</U> Homog. ODE

$${\color{blue} \underset{Euler \ eq. \ p.15-4}{ \underbrace}}=0$$

Trial soln:

$$y(x)={\color{blue}a}{x}^$$

a, b areunknow coeff.

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$${\color{red} \underset{ \underbrace}}=7{x}^{4}+3{x}^{3} \ {\color{red}(1)}$$

$${\color{blue}1)} \ b=-1 \ \Rightarrow \ {b}^{2}-b-2=0$$