Sturm-Liouville Problem, boundary condition

[kidsasd987]

given the generalized SL conditions

Let's say psi_m and psi_n are eigenfunctions of the given y.
Its Wronskian is 0 because otherwise the boundary condition doesn't make sense much.

However, I wonder if it is possible to have,
S={ x | W[psi_m(x) , psi_n(x)] =/= 0 }
otherwise W[psi_m(x) , psi_n(x)] = 0
then alpha and alpha' are 0 at such points, but still satisfies SL conditions.
Would it matter because at those points we can have any psi_m(x) and psi_n(x)?

If not could anyone tell me why?

 

[kidsasd987]

ah, forgot to mention that, alpha' = alpha_2. whereas alpha = alpha_1

I could not find the equation picture of SL conditions of wronskian using psi_m, psi_n and alpha and alpha'.