Question

Differential Equations Eigenvalue Problem:
Find the positive Eigenvalues for \[y ^{''} +\lambda y = 0\]
where
\[ y ^{'}(-\pi)=y^{'}(\pi)=0\]

So far here is what I have done:
\[\lambda> 0\]
used the characteristic equ. took derivative use endpoint values and found

\[y^{'}(-\pi) = A \sqrt{\lambda}\sin(\sqrt{\lambda} x) + B \cos(\sqrt{\lambda} x)\]
\[y^{'}(\pi) = -A \sqrt{\lambda}\sin(\sqrt{\lambda} x) + B \cos(\sqrt{\lambda} x)\]

then found that
\[0= 2Bcos(\sqrt{\lambda}\pi) \]
so when
\[b \neq0 ; \lambda = (2n-1)^2/4\]
\[n \in \mathbb{N}\]

But I am having trouble findi

Answer 1

But I am having trouble finding the additional eigenvalues when B=0

Answer 2

here is the part of the solution i do not understand