So I am in the section of our book about Strum-Liouville problems and all of the previous questions have had the boundary values equal to a constant (such as $y(10) = 0\,\,\,\,or\,\,\ y(L) = 0)$. But this problem has them set equal to other functions so I am at a bit of a loss. Here is my work so far:
The question in the book says: Find the eigenvalues and eigenfunctions for $$y''+λy=0, \,\,\,\,\,\,y(0) = y(1)\,\,\,\,\,\,\,\,y'(0) = y'(1)$$
I am assuming that $λ>0$ and I am replacing $λ$ with $k^2$ and assuming that $k$ is a non-zero constant.
After going through the characteristic equation I find that $r = \pm ki$.
I then apply Euler's formula because of the imaginary roots and get: $$y(x) = A_1\cos(kx)+iA_2\sin(kx)$$
Now this is where I am not sure if I am heading in the right direction. I figure that I should go ahead and plug in the boundary values and see what happens.
So I end up with: $$y(0) = A_1\cos(0)+iA_2\sin(0) = A_1\cos(k)+iA_2\sin(k) = y(1)$$
since $\cos(0)=1$ and $\sin(0)=0$ so I then simplify: $$y(0) = A_1 = A_1\cos(k)+iA_2\sin(k) = y(1)$$
I could then do some algebra and solve for A$_1$ in terms of A$_2$, but I don't know if that's really going to get me anywhere. Am I heading in the right direction so far?