User:EGM6321.f09.team1.Zhichao Gong/Mtg4

Mtg 4: Tue, 1 Sep 09

page4-1

$$Redu{\color{blue}\underline} \ of \ {\color{blue}\underline} \ \phi \ :$$

$$M{\color{blue}\underline} \ able.$$

$${\color{red}\underline}$$

$$G({y}^{n}, \ {y}^{n-1}{\color{red}order \ n}, \ ... \, \ {y}^{''}, \ {y}^{'}, \ y{\color{blue}(dependent \ varible)}, \ x{\color{blue}(independent \ varible)}) \ = \ 0 \ {\color{red}(1)}$$

$$Missing \ y \ in \ G(.) \ :$$

$$G({y}^{n}, \ {y}^{n-1}, \ ... \, \ {y}^{''}, \ {y}^{'}, \ x) \ = \ 0 $$

$$Reduce \ order \ by \ $$

$$\Rightarrow$$

$$If \ we \ can \ solve \ (3) \ for \ p(x) \ then \ (2) \ \Rightarrow$$

page4-2

$${\color{blue}\underline{Application}}$$

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

$$G({y}^{}, \ {y}^{'}, \ y \ x) \ = \ {y}^{}+{y}^{'}-x \ = \ 0 $$

$$let \ p(x)={y}^{'}(x) \ \Rightarrow \ {p}^{'}+p=x \ {\color{blue}linear \ first \ order \ ODE}$$

Use method ofintergate factor :

$$p(x)= \underset{ \overset{ \uparrow}}{A} {e}^{-x}+x-1 \ {\color{red}(2)}$$

$$y(x)=\int_{}^{x}p(z)dz=-\underset{ \overset{ \uparrow}}{A}{e}^{-x}+\frac{{x}^{2}}{2}-x+\underset{ \overset{ \uparrow}}{B}$$

HW: Derive Eq(2) using intergate factor.

Intergrate factor method

$$1st \ order \ ODEs \ (can \ be \ nonlinear)$$

$$\Rightarrow$$

$$M(x,y)+N(x,y){y}^{'}=0$$

M, N nonlinear wrt (x,y)

page4-3

$${\color{blue}\underline{Application}}$$

$$M(x,y)=2{x}^{2}+\sqrt{y}$$

$$N(x,y)={x}^{5}{y}^{3}$$

$${\color{blue} \underset{M(x,y)}{ \underbrace}}+{\color{blue} \underset{N(x,y)}{ \underbrace}}{\color{black}{y}^{'}=0} \ {\color{red}(1)}$$

HW: show that the 1st order ODE above is nonlinear