Question

Consider the endpoint problem
y''+ λy = 0; y(−1) = 0, y'(1) = 0.
Find the positive eigenvalues and associated eigenfunctions.

Answer 1

idk what that is sorry

Answer 2

The characteristic equation is \(r^2 +\lambda =0\) hence \(r_{1,2}= \pm\sqrt{-\lambda}\)
hence \(r_{1,2}=\pm i\sqrt{\lambda}\)

Answer 3

We have the solution for the ODE is
\(y=C_1cos(\sqrt{\lambda} x)+C_2 sin(\sqrt{\lambda}x)\)

Answer 4

Replace y(-1) =0 to get the first equation
Take derivative of y and replace y'(1) =0 to get the second equation
Then solve for \(C_1,C_2\)
I think you can handle it, right?