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Homework Help: ODE/Sturm-Liouville problem.

  1. Dec 7, 2005 #1
    consider the Sturm-Liouville problem:

    y" + [ lambda p (t) - q(t)] y= 0 in (0,1),

    alpha y(0) + beta y'(0) = 0

    gamma y(1) + delta y'(1) = 0,

    Where alpha, beta, gamma, delta are real constants, and p: [0,1] -> R and q:[0,1] -> R are continuous functions with p(t) > 0.

    (a) Suppose alpha*beta doesn't equal 0. Show that if f_n(t) and g_n(t) are eigenfunctions associated with a given eigenvalue lambda_n of the Sturm-Liouville problem, then f_n(t) = c g_n(t), t belongs to [0,1], for some constant c in R.

    (b) Can one remove the restriction alpha*beta doesn't equal to 0 in part (a)? [ Explain and justify your answer.]
  2. jcsd
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