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I have a Sturm-Liouville problem as follows:

$$u'' + \lambda u = 0 , 0<x<1$$ $$u'(0) = u(1) = 0. $$

It is asked to estimate the first two eigenvalues via Rayleigh-ritz method and compare to the exact values. Then, to choose polynomial trial functions which resemble what our first two eigenfunctions should like.

Now, I showed that the eigenvalues cannot be negative or $0$. If $\lambda >0$ we will have a general solution $$u(x)=A \cos(\mu x) + B \sin(\mu x)$$ where $\mu^2 = \sqrt \lambda$. Appyling boundary conditions, we find that the eigenvalues must be of the form $$\lambda _n = n\pi$$ and corresponding eigenfunctions are given by $$\phi_n = A \cos(n \pi x)$$ for $ n \in \mathbb{Z}.$

Now, how can I proceed?

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    $\begingroup$ Your eigenvalues are incorrect. The right answer is $\lambda_n = (2n+1)\frac{\pi}{2}$ and $\phi_n = \cos((2n+1)\frac{\pi}{2}x)$ $\endgroup$ – Dylan May 23 at 8:09

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