The discussion revolves around finding the eigenvalues for the differential equation $$y''+\lambda y=0$$ with specific boundary conditions. Participants explore the nature of the eigenvalues, questioning whether they are complex or real, and delve into the implications of the characteristic equation and the cosine function.
Participants express differing views on whether the eigenvalues are complex or real, with some arguing that $\lambda$ must be complex while others suggest it can still be real. The discussion remains unresolved regarding the definitive nature of the eigenvalues.
There are unresolved assumptions about the definitions of $\lambda$ and the implications of the boundary conditions. The discussion also highlights the complexity introduced by working with complex numbers and the multivalued nature of certain functions.
I like Serena said:That sounds very plausible. (Mmm)
Actually, $y=0$ is always a solution, regardless of the value of $\lambda$, so $\lambda=0$ is not really the corresponding eigenvalue.
So it makes sense to leave that one out.
Furthermore, the eigenfunctions for $-\ln^2(2\pm\sqrt 3)$ are identical.
That is because $$\frac 1{2-\sqrt 3} = 2+\sqrt 3$$.
So we might consider them as the same.
I like Serena said:Have you tried to calculate the Wronskian for 2 eigenfunctions of which 1 has an imaginary eigenvalue? (Wondering)
mathmari said:So should I try it for $u(x)=\sin{(x(-i \ln{(2+\sqrt{3})}+2 \pi n))}$ and $v(x)=v^*(x)=(2+\sqrt{3})^x-(2+\sqrt{3})^{-x}$ ?
I like Serena said:Yes... (Sweating)