User:EGM6341.s11.TEAM1.WILKS/Mtg5

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010= Mtg 5: Thu, 2 Sep 10

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L2-ODE-VC: linear 2nd order $$ \Rightarrow \ \ $$ need 2 conditions to solve for 2 constants Boundary Value Problem (BVP) Prescribe known values: Initial Value Problem (IVP) Prescribe known values: Solve IVP: 1) ODE [[media: 2010_08_31_15_01_03.djvu | Eq.(3)p.4-3]] for $$ x \in \left [ a, \infty \right [ \ $$, where left [ is defined as closed and right [ defined as open 2) Initial Condition Eq.2 Two Points: A) solution exists and is unique, existence and uniqueness emphasized. Math Proof not done here. Accept as facts.

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B) Superposition of solution B1) Homogeneous Solution $$ y_H(x) \ $$ B2) Particular Solution $$ y_P(x) \ $$ Homogeneous Solution: Particular Solution: Add Eq.(3) and Eq.(4) to get: Linearity of $$ L_2 \Rightarrow L_2(y_H + y_P) \ $$

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Existence and uniqueness of solution $$ y \ $$ in [[media: 2010_09_02_13_55_50.djvu | Eq.(2)p.5-2]], then by [[media: 2010_09_02_13_55_50.djvu | Eq.(5)p.5-2]], we have: $$ y(x)=y_H(x)+y_P(x) \ $$ Question: $$ L_2 \ $$ is second order differential operator; there are 2 constants of integration, thus we need 2 boundary conditions or 2 initial conditions to solve for these 2 constants; where are those constants ? Answer: There are actually 2 homogeneous solutons $$ y_H^1(x) \ $$ and $$ y_H^2(x) \ $$, such that $$ L_2(y_H^1)=L_2(y_H^2)=0 \ $$; in addition, $$ y_H^1 \ $$ and $$ y_H^2 \ $$ are linearly independent. EXAMPLE : $$ \sin x \ $$ and $$ \cos x \ $$ are examples of 2 linearly independent functions. END EXAMPLE $$ y_H \ $$ is a linear combination of $$ y_H^1 \ $$ and $$ y_H^2 \ $$: $$ y_H=cy_H^1+dy_H^2 \ $$, where $$ c,d \in \mathbb R \ $$ 2 parameters to fit 2 boundary conditions or 2 initial conditions $$ y(x)=cy_H^1+dy_H^2+y_P(x) \ $$, where $$ y_H(x)=cy_H^1+dy_H^2 \ $$

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HW Boundary Value Problem: $$ y(a)= \alpha\ \ $$, $$ y(b)= \beta\ \ $$ , find $$ c,d \ $$ in terms of $$ \alpha\ , \beta\ \ $$ END HW EXAMPLE Legendre Differential Equation [[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.31)]]