Quiz9 - Quiz 9 w/ Solutions Math 322. Fall, 2009. October,...

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Unformatted text preview: Quiz 9 w/ Solutions Math 322. Fall, 2009. October, 30 2009. NAME: Instructor: Bole Yang, Erica McEvoy Please show ALL of your work. Given the following Sturm-Liouville problem y + y = 0 y (0) = 0 y ( ) = 0 1. Prove that there are no eigenfunctions for < . If < , then the general solution to the ODE is given by y ( x ) = c 1 e kx + c 2 e- kx (1) where k = | | (a real, non-imaginary number). Differentiating gives y ( x ) = c 1 ke kx- c 2 ke- kx (2) Substituting in the boundary conditions y (0) = 0 and y ( ) = 0 gives the following equations 0 = c 1- c 2 (3) 0 = c 1 ke k- c 2 ke- k (4) Solving for c 1 and c 2 gives c 1 = c 2 (from first equation. Substitution into the second equation gives 0 = c 1 ( ke k- 1) (5) The only solution to this equation is c 1 = 0 . It follows that c 2 = 0 , so then the only solution is y ( x ) = 0 , which is NOT an eigenfunction. So no eigenfunctions exist for < . MATH 322. QUIZ 1 MATH 322....
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Quiz9 - Quiz 9 w/ Solutions Math 322. Fall, 2009. October,...

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