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: !enter image description here 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: !enter image description here And I'm not 100% sure about that either. Again, thank you!

$$(\sin\lambda_nx)'=\lambda_n\cos\lambda_nx$$

$$(\sin\lambda_nx)'=-\lambda_n^2\sin\lambda_nx$$

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 $\,\lambda_n=k_n\pi\,\,,\,\,k_n\in\Bbb Z\,$