AM3815-Assignment4-Solns1

# AM3815-Assignment4-Solns1 - AM 3815 Solutions to Assignment...

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Unformatted text preview: AM 3815 - Solutions to Assignment 4 1. We separate the variables in the usual way to get- T T =- X 00 X = Î» . We first consider the BVP X 00 =- Î» X , X (- Ï€ ) = X ( Ï€ ), X (- Ï€ ) = X ( Ï€ ). We know the eigenvalues of this problem are real, and we consider three possible cases. Case 1: If Î» > 0, then X = A cos âˆš Î» x + B sin âˆš Î» x . The X (- Ï€ ) = X ( Ï€ ) BCs imply A cos(- âˆš Î»Ï€ ) + B sin(- âˆš Î»Ï€ ) = A cos âˆš Î»Ï€ + B sin âˆš Î»Ï€ â‡” A cos âˆš Î»Ï€- B sin âˆš Î»Ï€ = A cos âˆš Î»Ï€ + B sin âˆš Î»Ï€ â‡” 2 B sin âˆš Î»Ï€ = which holds for either B = 0 or Î» = n 2 , where n â‰¥ 1 is an integer (recall that we have assumed Î» > 0). The X (- Ï€ ) = X ( Ï€ ) BCs imply- A âˆš Î» sin(- âˆš Î»Ï€ ) + B âˆš Î» cos(- âˆš Î»Ï€ ) =- A âˆš Î» sin âˆš Î»Ï€ + B âˆš Î» cos âˆš Î»Ï€ â‡” A âˆš Î» sin âˆš Î»Ï€ + B âˆš Î» cos âˆš Î»Ï€ =- A âˆš Î» sin âˆš Î»Ï€ + B âˆš Î» cos âˆš Î»Ï€ â‡” 2 A âˆš Î» sin âˆš Î»Ï€ = which holds for either A = 0 or Î» = n 2 , where n â‰¥ 1 is an integer. Note that if Î» 6 = n 2 , then A = B = 0 and we would have only a trivial solution to the eigenvalue problem. For given n â‰¥ 1, then, the eigenfunction corresponding to Î» n = n 2 is X n ( x ) = A n cos nx + B n sin nx ....
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## This note was uploaded on 01/17/2012 for the course AM 3815 taught by Professor Llll during the Spring '11 term at UWO.

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AM3815-Assignment4-Solns1 - AM 3815 Solutions to Assignment...

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