Eigenfunction boundary conditions order

[ognik]

Hi, just want to confirm that with the eigenfunction boundary condition $ p(x) v^*(x)u'(x)|_{x=a} = 0 $, the order of (solutions) v, u doesn't matter? I ask because a problem like this had one solution = a constant, so making that the u solution makes $ p(x) v^*(x)u'(x) = 0 $ no matter the limits...

Also checking that p(x) is what I think of as $ P(x) = \frac{p_1(x)}{p_0(x)} $ ? (for ODE's)

 

[jvicens]

Hello,

Yes, the order of solutions v and u does not matter in this case. The eigenfunction boundary condition only requires that the product of the solutions v and u satisfies the given condition, regardless of their order.

As for your second question, yes, p(x) is often represented as P(x) in ODEs, where P(x) is the ratio of the coefficients p1(x) and p0(x). This is a common notation used in ODEs to simplify the representation of the problem.

I hope this helps clarify your doubts. Let me know if you have any further questions.