User:EGM6321.f09.team1.Zhichao Gong/Mtg16

Mtg 16: Tue, 29 Sep 09

page16-1

Application: Homog. L2-ODE-VC

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

$${\color{blue} \underset{Transf. \ of \ var.}{ \underline {Meth1:}}} \ Let \ {\color{red} \underset{ \overset{\uparrow}{current \ var.}}}={e}^ \ {\color{red}(2)}$$

$${(.)}^{'}:=\frac{d}{dx}(.){\color{blue}={(.)}_{x}} \ {\color{green} Need \ symbol \ to \ designate \ \frac{d}{dt}(.)}$$

$$\overset{0}{(.)}:=\frac{d}{dt}(.){\color{blue}{(.)}_{t}}$$

$${\color{red}(1) \ \Rightarrow} \ {x}^{2}{y}_{xx}-2x{y}_{x}+2y=0 \ {\color{red}(3)}$$

$$Note: \ y(x)=y(x(t)) \ {\color{red}(4)}$$

$${\color{blue} \underset{{y}_{t}}{ \underbrace}}={\color{blue} \underset{{y}_{x}}{ \underbrace}}{\color{black} \underset{{e}^{t}}{ \underbrace}} \ \Rightarrow$$

page16-2

$$\frac{dy}{dx}=\frac{dy}{dt}{\color{blue} \underset{{(\frac{dx}{dt})}^{-1}={e}^{-t}}{ \underbrace}}=(\frac{dt}{dx} \frac{d}{dt})(y)$$

$$\Rightarrow$$

$$\frac{{d}^{2}y}{d{x}^{2}}=\frac{d}{dx}(\frac{d}{dx}y)={\color{red} \underset{ \underbrace}}$$

$$\Rightarrow$$

page16-3

Eq(3) p.16-1 $${y}_{tt}-3{y}_{t}+2y=0 \ {\color{red}(8)}$$

Homog. L2-ODE-CC CC means Const. Coeff.

Meth2: under coeff or

$${\color{red} \underline trial \ soln} \ {\color{blue}y={e}^{rt}} \ {\color{red}(9)}$$

Put(9) into (8) $$\Rightarrow$$ charac. eq. for r (=root)

$${r}^{2}-3r+2=0 \ \Rightarrow \ \begin{cases} & \ {r}_{1}=1 \\ & \ {r}_{2}=2 \end{cases}$$

$$y={\color{blue} \underset{lin. \ combo. \ of \ 2 \ trial \ soln.}{ \underbrace{{\color{black}{C}_{1}{e}^{{r}_{1}t}+{C}_{2}{r}^{2}}t}}}$$

$$={C}_{1}{({\color{blue} \underset{x}{ \underbrace}})}^{{r}_{1}}+{C}_{2}{({\color{blue} \underset{x}{ \underbrace}})}^{{r}_{2}}$$

page16-4

HW: Try solve Eq(1)p.16-1 directly using meth. of trial soln y=erx for b.c. y(1)=3 and y(2)=4 comp. soln w/ Eq(10) p.16-3 <P>Use matlab to plot Soln.</P>

$${\color{blue} \underline Appl:} \ {x}^{3}{y}^{'}-{x}^{2}{y}^{}-2x{y}^{'}-4y=0$$

$$Trial \ Soln: \ y={x}^{r}$$

$$Char. \ eq: \ ({r}^{2}+1)(r-4)=0$$

$$\Rightarrow$$

$$ {r}_{1}=4, \ \begin{cases} & {r}_{2}= +i, \ i=\sqrt{-1} \\ & {r}_{3}= -i \end{cases}$$

$$\ y(x)={C}_{1}{x}^{4}+{C}_{2}cos(logx)+{C}_{3}sin(logx)$$