User:EGM6321.F10.TEAM1.WILKS/Mtg17

=EGM6321 - Principles of Engineering Analysis 1, Fall 2009= Mtg 17: Thur, 10Oct09

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Linearity $$ \Rightarrow \ \ $$ superposition $$ y=y_H+y_P \ $$ Homogeneous Solution $$ y_H \ $$ - Euler Equations - Trial solution (undefined coefficient) - Reduction of order method 2: Undetermined factor Homogeneous L2_ODE_VC: cf. [[media: Egm6321.f09.mtg3.djvu | Eq.(1) P.3-1 ]] Where $$ a_0(x) \ $$ can be substituted for $$ a_1(x) \ $$ or $$ a_0y \ $$ in Eq(1) Given one homogeneous solution $$ u_1(x) \ $$ known Find second homogeneous solution $$ u_2(x) \ $$ such that Where$$ k_1,k_2 \ $$ are constants Assume full homogeneous solution Where$$ U(x) \ $$ is an unknown to be determined Where$$ u_1(x) \ $$ is known

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"Full" = includes $$ u_2(x) \ $$ Add the following: $$ a_0(x) \left [ y=Uu_1 \right ] \ $$ and $$ a_1(x) \left [ y'=Uu_1'+U'u_1 \right ] $$ and $$ \left [ y=Uu_1+2U'u_1'+U''u_1 \right ] $$ To get $$ a_0y+a_1y'+y=U \left [ a_0u_1+a_1u_1'+u_1 \right ] + U'\left [ a_1u_1+2u_1' \right ]+U''u_1=0 \ $$ by Eq(1) p17-1 Reduce to $$ u_1''+a_1u_1'+a_0u_1=0 \ $$ Since $$ u_1 \ $$ is a homogeneous solution $$ \Rightarrow \ 0=U'(a_1u_1+2u_1')+U''u_1 \ $$, NOTE missing dependent variable U in front of $$ U' \ $$ term Let $$ Z:=U' \Rightarrow \ \ $$ homogeneous L1_ODE_VC for Z

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Solve for Z, HW - integration factorial method END HW - Direct integration (because Eq(1) is homogeneous) $$ \Rightarrow \ \frac{Z'}{Z}+(a_1+\frac{2u_1'}{u_1})=0 \ $$ Where $$ a_1, \frac{2u_1'}{u_1} \ $$ are known Integrate $$ \log \left | Z \right \vert +2 \log \left | u_1 \right \vert+ \int_{}^{x} a_1(s)\, ds=k $$, where k is a constant where $$ c=e^k \ $$ $$ \Rightarrow \ U(x)=\int_{}^{x} \frac{c}{(u_1(t))^2}e^ \, dt+\tilde{c} \ $$ where $$ \bar a \ _1(t)=\int_{}^{t} a_1(s)\, ds \ $$

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Homogeneous solution where $$ \tilde{c}=k_1 \ $$ and $$ c=k_2 \ $$ HW obtain [[media: Egm6321.f09.mtg17.djvu | Eq.(2) P.17-3 ]] $$ Z(x) \ $$ using the integrating factor method END HW