User:EGM6341.s11.TEAM1.WILKS/Mtg7

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010= Mtg 7: Tue, 7 Sept10

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NOTE Where $$ z \ $$ is a dummy integration variable. Where $$ s \ $$ is a dummy integration variable and $$ y(a) \ $$ is an integration constant. Where $$ k \ $$ is an integration constant. END NOTE [[media: Egm6321.f09.mtg5.djvu| Ref F09 Mtg.5]] for details of Eq.(3) and Eq.(4) Euler integration factor method (continued) From [[media: 2010_09_02_14_58_46.djvu | Eq.(5)p.6-3]]: In general N1_ODE HW  Verify the above statement and give example for L1_ODE END HW Application:

HW Verify Eq.(7) is N1_ODE END HW

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NOTE [[media: 2010_09_02_14_58_46.djvu | Eq.(5)p.6-3]] is in general a N1_ODE, but is not the most general N1_ODE (i.e. it is still a particular N1_ODE) END NOTE EXAMPLE [[media: 2010_09_02_14_58_46.djvu | Eq.(5)p.6-3]] Where $$ \bar M \ (x,y) = M(x,y)+k \ $$ and Eq.(2) has the same structure as [[media: 2010_09_02_14_58_46.djvu| Eq.(5)p.6-3]]. [[media: 2010_09_02_14_58_46.djvu | Eq.(5)p.6-3]] Where $$ \bar M \ (x,y) := M(x,y)-P(x,y) \ $$ and Eq.(3) has the same structure as [[media: 2010_09_02_14_58_46.djvu | Eq.(5)p.6-3]]. see [[media: 2010_09_09_13_54_49.djvu | p.8-2]] and [[media: 2010_09_09_13_54_49.djvu | p.8-3]] END EXAMPLE  Most general N1_ODE: DEFINITION: Exact N1_ODE - Eq(5) is an exact N1_ODE if $$ \exists \ \ $$ a function $$ \phi\ (x,y) \ $$ such that $$ F=\frac{d \phi\ }{dx} \ $$, where $$ \phi\ (x,y) = \ $$ constant k. (continued on p7-3)

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(continued from p7-2) i.e., END DEFINITION From Eq(1) Where $$ \frac{\partial \phi\ (x,y) }{ \partial x} = M(x,y) \ $$ and $$ \frac{\partial \phi\ (x,y) }{\partial y}=N(x,y) \ $$

NOTE If Eq(2) can be found $$ \Rightarrow \ $$ reduce order ($$ y' \ $$ no longer present) in principle, $$ y(x) \ $$ can be obtained by solving Eq(2).$$ k= \ $$ integration constant END NOTE EXAMPLE: Find $$ F(x,y,y') = 0 = \frac{d \phi\ (x,y)}{dx} \ $$ END EXAMPLE: