Solving $y'' + \lambda y = 0$, $y(-L) = y(L)$ and $y'(-L) = y'(L)$ I am working on solving the ODE: $y'' + \lambda y = 0$, $y(-L) = y(L)$ and $y'(-L) = y'(L)$.
After applying the first condition, I get: $2c_2  \sin(\sqrt{\lambda} L) = 0$. I avoid letting $c_2 = 0$ to avoid a trivial case. Then $\sqrt{\lambda} = \frac{n \pi}{L}$. Now, I still have to apply the second condition. After doing that I get: $-2\sqrt{\lambda}c_1 \cdot \sin(\sqrt{\lambda}  L) = 0$. If I let $c_1 \neq 0$ then I get $\sqrt{\lambda} = \frac{n\pi}{L}$ again .
Am I supposed to let $c_1 = 0$ or no?
Here is my full work:  

You have solved the problem. Specifically, 
$$y(x) = c_1 \cos(\sqrt{\lambda}x) + c_2 \sin(\sqrt{\lambda}x).$$
If $\sqrt{\lambda} \neq n\pi/L$ then $c_1=c_2=0$. That is, $y(x)=0$. Otherwise, $c_1$ and $c_2$ can take on arbitrary values.
