I too much be missing something . . . I found only one solution.
Let be a positive constant.
Consider the equation: .
with boundary conditions: .
For certain discrete values of , this equation can have non-zero solutions.
Find the three smallest values of for which this is the case.
Your preliminary work is correct . . .
So the characteristic equation is: .
The roots are: .
General solution: .
From the given conditions:
Substituting for and taking out
So I thought that for ,
. . I need to make the second group equal to 0 and solve for . Right!
We have: .