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ODE/Sturm-Liouville problem

Meital
in Analysis and Calculus

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ODE/Sturm-Liouville problem.

 

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.]

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