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APM346-2016F => APM346--Lectures => Chapter 4 => Topic started by: Tianyi Zhang on November 02, 2016, 02:10:55 PM

Title: 4.2 Example 4(periodic)
Post by: Tianyi Zhang on November 02, 2016, 02:10:55 PM
$$X^{''} + \lambda X = 0$$
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?
Title: Re: 4.2 Example 4(periodic)
Post by: Victor Ivrii on November 02, 2016, 04:58:40 PM
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 (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.