User:EGM6321.F10.TEAM1.WILKS/Mtg9

=EGM6321 - Principles of Engineering Analysis 1, Fall 2009= Mtg 9: Thur, 10Sept09

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Note: Symbol notations: $$ \Delta\ $$ defined as equal by definition Non-symmetric notation $$ := $$ means equal by definition as well, better notation than $$ \Delta\ $$ Goal: Derive mathematical structure of a class of exact N1_ODEs Exact L1_ODE_VC: Application: Just invent any $$ a(x),b(x) \ $$ Let $$ a(x)=x^4 \ $$ $$ b(x)=x \Rightarrow \ \bar b \ (x)=\frac{1}{2}x^2 \ $$ $$ k=10 \ $$

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HW Show that Eq(3) L1_ODE_VC is "exact". See Note [[media: Egm6321.f09.mtg10.djvu | P.10-1 ]] END HW Question: How about N1_ODEs? Eq.(2)P.6-4 N1 means Nonlinear, 1st Order Where $$ \int_{}^{x} b(s) = \bar b \ (x) \ $$ [[media: Egm6321.f09.mtg6.djvu | Eq.(2) P.6-4]]: $$ \frac{1}{N}(N_x-M_y)=\frac{1}{ \bar b \ (x) c(y)} \left [ b(x)c(y)-a(x)c(y) \right ]=-f(x) \ $$

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[[media: Egm6321.f09.mtg9.djvu | Eq.(4) P.9-2 ]] $$ M(x,y)=a(x)\int_{}^{y} c(s) \ $$ Where $$ \int_{}^{y} c(s)= \bar c \ (y) \ $$ From [[media: Egm6321.f09.mtg9.djvu | Eq.(6) P.9-3 ]], [[media: Egm6321.f09.mtg9.djvu  | Eq.(4) P.9-2 ]] and [[media: Egm6321.f09.mtg4.djvu  | Eq.(3) P.4-2 ]] obtain: Application: Consider the following $$ a(x)=5x^3+2 \ $$ $$ b(x)=x^2 \Rightarrow \ \bar b \ (x)=\frac{1}{3}x^3 \ $$ $$ c(y)=y^4 \Rightarrow \ \bar c \ (y)=\frac{1}{5}y^5 \ $$ HW Show Eq(8) is "exact" N1_ODE. Note [[media: Egm6321.f09.mtg10.djvu | P.10-1 ]] END HW

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L2_ODE_VC with missing dependant variable $$ P(x)y''+Q(x)y'+R(x)y=S(x) \ $$ Where $$ R(x) \rightarrow \ 0 \ $$ due to missing dependant variable y. [[media: Egm6321.f09.mtg2.djvu | Eq.(1)P.2-3]] $$ p:=y' \rightarrow \ P(x)p'+Q(x)p=S(x) \ $$ is a L1_ODE_VC Solution: [[media: Egm6321.f09.mtg8.djvu | Eq.(4)P.8-2]] Exact N2_ODEs: General N2_ODEs: $$ F(x,y,y',y'')=0 \ $$ Application: $$ (x^3+2x^5y^2)y''+(x^{\frac{3}{2}}+10) \sqrt{y} y'+y^{100}=0 \ $$ $$ F=0 \ $$ is exact means $$ \exists \phi\ (x,y,y') \ $$ such that $$ F(x,y,y',y'')=\frac{d}{dx} \phi\ (x,y,y') \ $$