User:EGM6321.f09.team1.Zhichao Gong/Mtg22

Mtg 22: Tue, 13 Oct 09

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- p.12-3 - p.20-3: Trial soln for non-homog. L2-ODE-CC Note on m, \alpha, \beta in table: m=0, 1, 2 = number of times that 0 (zero) is root of char. eq. so to make sure that no terms in trial soln (particular soln) is <P>a homog.soln. </P>

$$\alpha \ \in \ R \ is \ a \ root. \ of \ char. \ eq.$$

$$\alpha+i \beta \ \in \ {\color{blue} \underset{ \overset{\uparrow}{set \ of \ complex \ numbers \ (black \ board \ font)}}} \ is \ another \ root \ of \ char. \ eq. \ \beta \in \ R$$

<P><U>HW:</U> K. p.28, pb. 1.3.a.</P>

<P><U>Application:</U></P>

$${y}^{''}-2x{y}^{'}+2y=3$$

Non-homog. L2-ODE-VC

$$Homog. \ soln: \ {u}_{1}(x)=x$$

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(3) p.21-3 $$\Rightarrow \ x{Z}^{'}+2(1-{x}^{2})Z=3$$

Find Z by (4) p.8-2

(4) p.21-3 $$\Rightarrow \ U(x)=\frac{3}{2x}+A \int_{\frac{1}{{s}^{2}}}^{x}exp({s}^{2})ds+B$$

$$y(x)=U(x){\color{blue} \underset{x}{ \underbrace}}$$

<U>HW</U> cf. K. p.28, pb. 1.1 . b

$${\color{blue}b)} \ {\color{blue} \underset{HOW? \ {\color{red}Pretend \ not \ knowing.}}{ \underbrace}}=\frac{sinx}{x} \ homog \ soln \ for$$

$$x{y}^{''}=2{y}^{'}+xy={\color{red} \underset{ \overset{\uparrow}{WORD}}} \ {\color{red}(1)}$$

<P>1) Verify exactness of(1).</P> <P>Int. fact. meth. 2</P> <P>2) Trial solns</P> <P> 2.1) y(x)e<SUP>rx</SUP>, r=const</P> <P>  2.2) y(x)=xe<SUP>rx</SUP>, r=const</P>

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(3) p.21-3 $$\Rightarrow \ x{Z}^{'}+2(1-{x}^{2})Z=3$$

Find Z by (4) p.8-2

(4) p.21-3 $$\Rightarrow \ U(x)=\frac{3}{2x}+A \int_{\frac{1}{{s}^{2}}}^{x}exp({s}^{2})ds+B$$

$$y(x)=U(x){\color{blue} \underset{x}{ \underbrace}}$$

<U>HW</U> cf. K. p.28, pb. 1.1 . b

$${\color{blue}b)} \ {\color{blue} \underset{HOW? \ {\color{red}Pretend \ not \ knowing.}}{ \underbrace}}=\frac{sinx}{x} \ homog \ soln \ for$$

$$x{y}^{''}=2{y}^{'}+xy={\color{red} \underset{ \overset{\uparrow}{WORD}}} \ {\color{red}(1)}$$

<P>1) Verify exactness of(1).</P> <P>Int. fact. meth. 2</P> <P>2) Trial solns</P> <P> 2.1) y(x)e<SUP>rx</SUP>, r=const</P> <P>  2.2) y(x)=xe<SUP>rx</SUP>, r=const</P>

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2.3) y(x)=1/x(e<SUP>rx</SUP>) ,

<P>3) Use undet. fact. meth. </P> <P>to find u<SUB>2</SUB>(x), knowing u<SUB>1</SUB>(x),</P> <P>compare to above results.</P> <P><U>HW:</U> Describe in words step-by-step</P> <P>meth. of attacking similar pbs, given</P> <P>only homog. and non-homog.L2-ODE-VC</P>