4.2 Example 4（periodic）

[Tianyi Zhang]

$$X^{''} + \lambda X = 0$$
with condition  $$X(0) = X(l), X^{'}(0) = X^{'}(l)$$
how to get the answer $$\lambda_{2n-1} = \lambda_{2n} = (\frac{n\pi}{2l})^{2}$$ and the corresponding eigenfunctions?

[Victor Ivrii]

We did it on lectures: you need to solve constant coefficients ODE and find when and how many non-trivial solutions it has satisfying boundary conditions

http://www.math.toronto.edu/courses/apm346h1/20169/PDE-textbook/Chapter4/S4.2.html#example-4.2.2

Anyone wants to post details here?

PS. It should be $=(\frac{2\pi n }{l})^2$. I will fix misprint tonight.