ODE/Sturm-Liouville problem.

[ODEMath]

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.]

 

[Aaron Bex]

Answer:
(a) Let y_1 and y_2 be two eigenfunctions of the Sturm-Liouville problem with the same eigenvalue lambda_n. Then, according to the Sturm-Liouville theory, we have

y_1(t) = c y_2(t),

where c is a constant.

To prove this, let us consider the following Wronskian of y_1 and y_2:

W(y_1,y_2) = y_1(0)y_2'(0) - y_1'(0)y_2(0).

Since y_1 and y_2 are both solutions of the Sturm-Liouville problem, they satisfy the boundary conditions at x=0 and x=1. This implies that,

alpha y_1(0) + beta y_1'(0) = 0

gamma y_2(1) + delta y_2'(1) = 0.

Substituting these in the Wronskian, we get,

W(y_1,y_2) = alpha gamma y_1(0)y_2