APM346-2022S > Test 2

Sturm Liouville eigenfunctions

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Weihan Luo:
In the practice term test 2 Variant A Problem 1, I had to solve the the following Sturm Liouville problem:

$$X''+\lambda x = 0$$ with boundary conditions $$X'(0) = X'(4\pi) = 0$$ In the answer key, the eigenfunction corresponding to the eigenvalue $\lambda_{0} = 0$ is $X_0 = \frac{1}{2}$. However, if we substituted $\lambda_0 = 0$ into the ODE, we get:

$$X'' = 0$$ which the solution is simply $$X(x) = \alpha + \beta x$$ Plugging into the boundary conditions and we get that $$\beta = 0$$ so The solution is $$X(x) = \alpha$$ where $\alpha \in \mathbb{R}$. How do we get that $X_0 = \frac{1}{2}$?

Victor Ivrii:
they defined up to a constant