Find all the eigenvalues and eigenfunctions for the following Boundary value problem:
$y''+ \lambda y = 0$ ; $y(- \pi)=y(\pi)$, $y'(-\pi)=y(\pi)$
I'm just confused at the $\lambda = 0$ case, how was $y(x)=1$ obtained from $y(x)=c_1$. I feel like I am missing something very obvious. Could someone clarify this? Thanks!
The BC read:
$$y(-\pi) =y(\pi)$$ and $$y'(-\pi) = y(\pi)$$
Plugging in the expression for $y$ in the first gives: $c_1 -c_2\pi=c_1 + c_2\pi \Rightarrow c_2 = -c_2 \Rightarrow c_2 =0$
The second results in: $c_2 = c_1 +c_2\pi \Rightarrow c_1 = 0$
Thus, are you sure your BC are right? If the second one reads: $$y'(-\pi) = y'(\pi)$$ you'd get $y(x) = 1$ as an eigenfunction. (note, $y(x) = c_1$ is eigenfunction is equivalent with $y(x) =1$ is an eigenfunction.)