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Quiz9 - Quiz 9 w Solutions Math 322 Fall 2009 October 30...

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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 + λ y = 0 y (0) = 0 y ( π ) = 0 1. Prove that there are no eigenfunctions for λ < 0 . If λ < 0 , then the general solution to the ODE is given by y ( x ) = c 1 e kx + c 2 e - kx (1) where k = | λ | (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 ( π ) = 0 gives the following equations 0 = c 1 - c 2 (3) 0 = c 1 ke k π - c 2 ke - k π (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 π - 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 λ < 0 .

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M ATH 322. Q UIZ 1 ———————————————————————————— 2 2. Find all eigenfunctions and corresponding eigenvalues for λ 0 . When λ > 0 , then the general solution is y ( x ) = c 1 cos ( λ x ) + c 2 sin ( λ x ) (6) where y ( x ) = - c 1 λ sin ( λ x ) + c 2 λ cos ( λ x ) (7) Subbing in the boundary conditions gives
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