User:EGM6321.F10.TEAM1.WILKS/Mtg25

=EGM6321 - Principles of Engineering Analysis 1, Fall 2009= [tp://upload.wikimedia.org/wikiversity/en/e/e5/Egm6321.f09.mtg25.djvu Mtg 25:] Tues, 20Oct09

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Pre-recorded Thursday, 15 Oct09 [[media: Egm6321.f09.mtg23.djvu |  P.23-1 ]] : Motivate L2_ODE_VC Review: Class exact L2_ODE_VC [[media: Egm6321.f09.mtg13.djvu | P.13-2 ]] : $$ \phi\ (x,y,p)=P(x)p+T(x)y+k \ $$, where  $$ p=y' \ $$ Let: $$ P(x)=x^2 \ $$ and $$ T(x)=x^4 \ $$, where $$ k=0 \ $$ $$ \phi\ = x^2p+x^4y=C \ $$ where $$ C= \ $$ constant $$ \phi_x =2xp+4x^3y \ $$ $$ \phi_y =x^4 \ $$ $$ \phi_p =x^2 \ $$ $$ \frac{d \phi\ }{dx}= \left [ 2xp+4x^3y \right ] + \left [ x^4 \right ]p + \left [ x^2 \right ]p'=0 \ $$ Where $$ p'=y'' \ $$ Exact L2_ODE_VC $$ x^2y''+x(2+x^3)y'+4x^3y=0 \ $$

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Generate non-exact L2_ODE_VC that can be made exact by intergral factor method: $$ x^a \left [ x^{2-a}y''+x^{1-a}(2+x^3)y'+4x^{3-a}y \right ]=0 \ $$ Where $$ x^a \ $$ is the integrating factor $$ h(x,y)=x^my^n\ $$ and $$ \left [ x^{2-a}y''+x^{1-a}(2+x^3)y'+4x^{3-a}y \right ]=0 \ $$ is L2_ODE_VC, not necessarily exact "$$ a \ $$" can be any number, $$ a=1, \frac{3}{2},5... \ $$