Show that the function is an eigenfunction of the equation I'm not sure how to use the bbcode so I've taken a screenshot instead: 

Came up on a past exam paper that I'm working towards and I'm not sure how to answer it.
I assumed that EQN . EIGENFUNCTION = EIGENVECTOR . EIGENFUNCTION 
(from ef definition)
But it doesn't cancel out to a constant for eigenvector value.
Appreciate any help!
Endnote: The question continues: 

And I'm not 100% sure about that either. 
Again, thank you!
 A: $$(\sin\lambda_nx)'=\lambda_n\cos\lambda_nx$$
$$(\sin\lambda_nx)'=-\lambda_n^2\sin\lambda_nx$$
So putting $\,\psi(x):=\sin\lambda_nx\,$ , we easily find the above are solutions to the given differential equation, and in order to have $\,\psi(1)=0\,$ we must choose $\,\lambda_n=k_n\pi\,\,,\,\,k_n\in\Bbb Z\,$
A: The eigenfunctions result from the boundary conditions.  The generic solution to the differential equation is
$$ \psi(x) = A \cos{\lambda x} + B \sin{\lambda x} $$
We then apply the boundary condition $\psi(0) = 0$ and get
$$A \cos{0} + B \sin{0} = A = 0$$
The boundary condition $\psi(1) = 0$ implies that
$$B \sin{\lambda} = 0$$
We assume that $B \ne 0$.  This latter equation puts a condition on the values of $\lambda$:
$$\sin{\lambda} = 0 \implies \lambda = n \pi \; \forall n \in \mathbb{Z}$$
We then may define $\lambda_n = n \pi$ as an eigenvalue of the differential operator defined above (the equation plus the boundary conditions).  The general solution to this equation is a linear combination of eigenfunctions, that is, $\psi_n(x) = \cos{\lambda_n x}$.
By the way, maybe I am missing something, but (c) makes no sense to me, as $1-x$ does not satisfy the boundary conditions satisfied by the eigenfunctions $\psi_n(x)$.  
