I am given:
y”+λy=0, y(0)=0, (1−λ)y(1)+y′(1)=0
As usual we are looking for not trivial solutions.
Looks like a standard eigenvalue problem and yet I am totally stuck.
The case when lambda = 0 is rather obvious. A=B=0. Not much fun.
But when I start trying for lambda greater or smaller than zero, I get to this:

λ < 0, the solution is of the form:
B((1+ω^2)sinh(ω)+ωcosh(ω))=0 where λ=ω^2 
λ > 0, the solution is of the form:
B((1ω^2)sin(ω)+ωcos(ω))=0 where λ=ω^2
The question states:Find the nontrivial stationary paths,
stating clearly the eigenfunctions y. In the case 1) I cant see any non trivial solutions but… well in the second case I cant see either. I know there are solutions.
Any help would be highly appreciated