User:EGM6341.s11.TEAM1.WILKS/Mtg11

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010= Mtg 11: Thu, 24 Aug 10

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[[media: 2010_09_14_15_00_52.djvu | HW P.10-4 (continued)]] 2) Assume $$ a_1(x) \ne 0 \forall \ x \ $$, [[media: 2010_09_14_15_00_52.djvu | Eq(8) P.10-3 ]] becomes $$ y'+\frac{a_0(x)}{a_1(x)}y=\frac{b(x)}{a_1(x)} \ $$ Where $$ \frac{a_0(x)}{a_1(x)}= P(x) \ $$ and $$ \frac{b(x)}{a_1(x)} = Q(x) \ $$ from [[media: http://uf.catalog.fcla.edu/uf.jsp?Ntt=king+differential+equations&I=1&Submit=Find&N=20&S=0781249404236215&Ntk=Keyword&V=D&Nty=1#top | K.p.512]] Find expression for $$ y(x) \ $$ in terms of $$ a_0, a_1, b \ $$. 3) $$ a_1(x)=x^2+1 \ $$ $$ b(x)=2x \ $$ $$ a_0(x)=x \ $$ END HW NOTE: [[media: http://uf.catalog.fcla.edu/uf.jsp?Ntt=king+differential+equations&I=1&Submit=Find&N=20&S=0781249404236215&Ntk=Keyword&V=D&Nty=1#top | cf. to K.p.512]] 1) [[media: http://uf.catalog.fcla.edu/uf.jsp?Ntt=king+differential+equations&I=1&Submit=Find&N=20&S=0781249404236215&Ntk=Keyword&V=D&Nty=1#top | K. etal.]] did not derive expression [[media: 2010_09_14_15_00_52.djvu | Eq.(1)p.10-3]] $$ h(x)=e^{\left [ \int_{}^{x} a_0(s)\, ds \right ]} \ $$ "pulling rabbit out of hat" 2) $$ \oint_{}^{x} f(s)\, ds:= \int_{}^{x} f(s)\, ds \ $$ in [[media: http://uf.catalog.fcla.edu/uf.jsp?Ntt=king+differential+equations&I=1&Submit=Find&N=20&S=0781249404236215&Ntk=Keyword&V=D&Nty=1#top | K.2003]] $$ \equiv \int f(x)\, dx$$ without constant

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Lecture: $$ \int_{}^{x} f(s)\, ds = \int f(x)\, dx+k= \oint_{}^{x} f(s)\, ds+k \ $$ [[media: 2010_09_14_15_00_52.djvu | Eq.(6)p.10-3]] :2 constants $$ k_1 \ $$ and $$ k_2 \ $$ [[media: 2010_09_14_15_00_52.djvu | Eq.(1)p.10-3]] : $$ h(x) \rightarrow \ k_1 \ $$ [[media: 2010_09_14_15_00_52.djvu | Eq.(6)p.10-3]] : $$ \int_{}^{x} h(s)b(s) \, ds \rightarrow \ k_2  \ $$ But [[media: 2010_09_14_15_00_52.djvu | Eq.(5)p.10-2]] is L1_ODE_VC HW  $$ \alpha\ \ $$ Show that $$ k_1  \ $$ is not necessary. END HW HW  $$ \beta\ \ $$ Show  [[media: 2010_09_14_15_00_52.djvu | Eq.(6)p.10-3]] agrees with [[media: http://uf.catalog.fcla.edu/uf.jsp?Ntt=king+differential+equations&I=1&Submit=Find&N=20&S=0781249404236215&Ntk=Keyword&V=D&Nty=1#top | K.p.512]], i.e. $$ y(x)=Ay_H(x)+y_P(x) \ $$ END HW HW  $$ \gamma\ \ $$ Find $$ y_H(x) \ $$ independantly, i.e. solve $$ y'+a_0y=0 \ $$ END HW $$ \delta\ \ $$ How about $$ y_P(x) \ $$ ? $$ \Rightarrow \ \ $$ Variation fo parameters (later) END NOTE

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= A class of exact N1-ODE =

Recall [[media: 2010_09_14_15_00_52.djvu | Eq.(7)p.10-1]] (Case 1) One possibility to satisfy this condition: Consider:

Where Eq(6) is a L1-ODE-VC (not necessarily exact, but can be made exact: integrating factor method) APPLICATION Consider $$ a(x)= x^4 \ne \ b(x)= x \Rightarrow \ \bar b \ (x) = \frac{1}{2}x^2 \ $$ $$ k=10 \ $$

F09: Find $$ h(x) \ $$ such that Eq.(7) is exact END APPLICATION

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Question: But [[media: 2010_09_16_13_50_21.djvu | Eq.(6)p.11-3]] is linear! Find N1-ODEs that are exact or can be made exact by integrating factor method.