User:EGM6321.F10.TEAM1.WILKS/Mtg19

=EGM6321 - Principles of Engineering Analysis 1, Fall 2009= Mtg 19: Tues, 5Oct09

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HW Legendre differential [[media: Egm6321.f09.mtg14.djvu | Eq.(1) P.14-2 ]] with $$ n=0 \ $$, such that homogeneous solution $$ u_1(x)=1 \ $$. Use reduction of order method 2 (undetermined factor) to find $$ u_2(x) \ $$, second homgenous solution END HW HW [[media: http://books.google.com/books?id=9Cg3HWCnCjAC&printsec=frontcover&dq=differential+equations+billingham&ei=pGR4SpPVLojSMpb07Qw#v=onepage&q=&f=false | K. p28, pb. 1.1.b. ]] END HW Variation of parameters (continued)   P.18-4

Use expression for $$ y' \ $$  Eq.(2) P.18-4  and $$ y'' \ $$   Eq.(3) P.18-4  in non-homogeneous L2_ODE_VC [[media: Egm6321.f09.mtg3.djvu  | Eq.(1) P.3-1 ]]

Where $$ u_1''+a_1u_1'+a_0u_1 \Rightarrow 0 \ $$, because $$ u_1 \ $$ is a homogeneous solution Where $$ u_2''+a_1u_2'+a_0u_2 \Rightarrow 0 \ $$, because $$ u_2 \ $$ is a homogeneous solution

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2 equations Eq.(1) P.18-4  and  Eq.(1) P.19-1  for two unknowns $$ \begin{Bmatrix} c_1' \\ c_2' \end{Bmatrix} \ $$ In matrix form: $$ \begin{bmatrix} u_1 & u_2 \\ u_1' & u_2' \end{bmatrix}  \begin{Bmatrix} c_1' \\ c_2'\end{Bmatrix} = \begin{Bmatrix} 0 \\ f \end{Bmatrix} \ $$ Where $$ \begin{bmatrix} u_1 & u_2 \\ u_1' & u_2' \end{bmatrix} \ $$ is the Wronskian matrix designated as $$ \underline{W} \ $$ The Wronskian, W, is the determinant of $$ \underline{W} \ $$ $$ W= det \underline{W} \ $$ If $$ W \ne \ 0 \ $$, then $$ \underline{W}^{-1} \ $$ exists and $$ \begin{Bmatrix} c_1' \\ c_2'\end{Bmatrix} =  \underline{W} ^{-1}  \begin{Bmatrix} 0 \\ f \end{Bmatrix} \ $$ Theorem: $$\forall \!\,$$ $$ u_1, u_2 \ $$ (function of x) are linearly independant if $$ \underline{W} \ne \ 0  \ $$, where $$ 0= \ $$ zero function.

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Where $$ \begin{Bmatrix} -u_2f \\ u_1f \end{Bmatrix} \ $$ are known Where $$ \int_{}^{x} \frac{u_2(s)f(s)}{W_s}\, ds = d_1(x) \ $$ Where $$ \int_{}^{x} \frac{u_1(s)f(s)}{W_s}\, ds= d_2(x) \ $$