User:EGM6321.F10.TEAM1.WILKS/Mtg13

=EGM6321 - Principles of Engineering Analysis 1, Fall 2009= Mtg 13: Tues, 22Sept09

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Euler integrating factor method [[media: Egm6321.f09.mtg12.djvu | Eq.(1) P.12-3 ]] Where: $$ (x^wy^n)=h(x,y) \ $$ HW Find (m,n) such that Eq(1) is exact. END HW Result: A first integration is: Where $$ p=y' \ $$ and $$ k_1 \ $$ and $$ k_2 \ $$ are constants HW Solve Eq(2) for $$ y(x) \ $$ HINT: L1_ODE_VC (integrating factor) END HW A class of exact L2_ODE_VC (how to invent more exact L2_ODE_VC) HW Find mathematical structure of $$ \phi\ \ $$ that yields the above class END HW

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$$ F=\frac{d \phi\ }{dx}= \phi_x(x,y,p)+ \phi_yp+ \phi_pp'=P(x)y''+Q(x)y'+R(x)y \ $$ Where $$ p=y' \ $$ and $$ p'=y'' \ $$ and $$ P(x)= \phi_p \ $$ and $$ Q(x)= \phi_y \ $$ and $$ R(x)y= \phi_x \ $$ Answer: [[media: Egm6321.f09.mtg13.djvu | Eq.(2) P.13-1 ]] : $$ P(x)=x \ $$ and $$ T(x)=2x^{\frac{3}{2}}-1 \ $$ and $$ k=k_1 \ $$ Exact Nn_ODE's, where N means nonlinear and n means nth order Where $$y^{(0)}=y \ $$ and $$y^{(1)}=y' \ $$ and $$y^{(n)}= \frac{d^ny}{dx^n} \ $$

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Condition 1 for exactness: Where $$ \phi_x+ \phi_{y(0)}y^{(1)}+...+ \phi_{y(n-1)}y^{(n)} \ $$ is related term by term to $$ x,y^{(0)},...,y^{n-1} \ $$ Condition 2 of exactness: $$ f_i:= \frac{\partial F}{\partial y^{(i)}} \ $$, where $$ i=1,...,n \ $$ HW Case $$ n=1 \ $$ (N1_ODE) END HW $$ F(x,y,y')=0=\frac{d}{dx} \phi\ (x,y) \ $$ $$ f_0-\frac{df_1}{dx}=0 \Leftrightarrow \phi_{xy}= \phi_{yx} \ $$ HINT: $$ f_1= \phi_y \ $$

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Case $$ n=2 \ $$ (N2_ODE) $$ f(x,y,y',y'')=0=\frac{d \phi\ }{dx}(x,y,y') \ $$ $$ f_0-\frac{df_1}{dx}+\frac{d^2f_2}{dx^2}=0 \ $$