ODE/PDE- eighenvalues question

[Roni1985]

Homework Statement

X''+$$\lambda$$*X=0 0<x<2pi

X(0)=X(2pi) and X'(0)=X'(2pi)

Homework Equations

The Attempt at a Solution

My only problem is the case when $$\lambda$$=0
in such case the general solution is X(x)= C1+C2*x

Now, after applying the BCs, this is what I get:

C1=C2*2pi
and

C2=C2

now, what should C2 be ? zero or any value ?
if C2 is zero, we have the trivial solution only. However, if C2 is any number, we have a nontrivial solution.

how can I solve this question?


Thanks.

 

[Dick]

After applying the first BC, I get C1=C1+C2*2*pi, not C1=C2*2*pi.

 

[fzero]

C1=C2*2pi

This should be

$$C_1 = C_1 + 2\pi C_2.$$

How does the solution with $$\lambda\neq 0$$ behave in the limit as $$\lambda \rightarrow 0$$? Is this consistent with your solution for $$\lambda = 0$$?