User:EGM6321.F10.TEAM1.WILKS/Mtg21

=EGM6321 - Principles of Engineering Analysis 1, Fall 2009= Mtg 21: Thurs, 8Oct09

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P.20-4 (continued) $$ \Rightarrow ax^{-1} \ $$ is a homgenous solution $$ \forall \!\, a\ $$ $$ u_1(x)=c_1x^{-1} \ $$ 2) $$ b=2 \Rightarrow ax^2 \ $$ is another homogeneous solution since $$ b^2-b-2=0 \ $$ $$ u_2(x)=c_2x^2 \ $$ (Verify $$ u_1 \ $$ and $$ u_2 \ $$ are linearly independant components of $$ W \ $$ 3) $$ b=6, a=\frac{1}{4} \Rightarrow \ $$ left hand side of Eq(1) p20-4 $$ =7x^4 \ $$, where $$ =7x^4 \ $$  is the 1st term on the right hand side for $$ b=6, a=\frac{1}{4} \Rightarrow  \frac{1}{4}x^6 \ $$  4) $$ b=5, a=\frac{1}{6} \Rightarrow \ $$ left hand side of Eq(1) p20-4  $$ =3x^3 \ $$ , where $$ =3x^3 \ $$  is the 2nd term on the right hand side for $$ b=5, a=\frac{1}{6} \Rightarrow \frac{1}{6}x^5 \ $$

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Llinearity of ordinary differential equation $$ \Rightarrow \ $$ superposition $$ y_P(x)=\frac{1}{4}x^6+\frac{1}{6}x^5 \ $$ $$ y(x)=c_1x^{-1}+c_2x^2+y_P(x) \ $$, where  $$  c_1x^{-1}+c_2x^2=y_H(x) \ $$ Alternative method to obtain full solution for non-homogeneous L2_ODE_VC knowing only one homogeneous solution (e.g. obtained by trial solution) (bypassing reduction of order method2-undertermined factor for $$ u_2 \ $$ and variation of parameter method) [[media: Egm6321.f09.mtg3.djvu | Eq.(1) P.3-1 ]] =    $$ y''+a_1(x)y'+a_0(x)y=f(x) \ $$ Assume having found $$ u_1(x) \ $$, a homogeneous solution: $$ u_1''+a_1(x)u_1'+a_0(x)u_1=0 \ $$ Consider: $$ y(x)=U(x)u_1(x) \ $$, where $$ U(x) \ $$ is an undetermined factor

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Follow the same argument as on [[media: Egm6321.f09.mtg17.djvu | P.17-2 ]] to obtain: NOTE: this equation is missing the dependant variable $$ U \ $$ in front of $$ U' \ $$ term due to reduction of order method $$ \phi\ \ $$ where $$ u_1(x) \ $$ and $$ \left [ a_1(x)u_1(x)+2u_1'(x) \right ] \ $$ are known Non-homogeneous L1_ODE_VC solution for $$ Z(x) \ $$ : [[media: Egm6321.f09.mtg8.djvu |  Eq.(4) P.8-2 ]] ref: [[media: http://books.google.com/books?id=9Cg3HWCnCjAC&printsec=frontcover&dq=differential+equations+billingham&ei=pGR4SpPVLojSMpb07Qw#v=onepage&q=&f=false | K p.28, problem 1.1ab ]] a) $$ u_1(x)=e^x \ $$, $$ (x-1)y''-xy'+y=0 \ $$ Trial solution $$ y(x)=e^{rx} \ $$ , where $$ r= \ $$ constant

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Find $$ r_1,r_2 \ $$ How many valid homogeneous solutions to $$ u_1=e^{r_1x} \ $$, find $$ u_2 \ $$ using undetermined factor method