Consider the regular Sturm-Liouville eigenvalue problem \left\{\begin{array}{l}

-\left(x u_{x}\right) x+\frac{u}{x}=\lambda \frac{u}{x} \text { for } 1

u(1)=u(e)=0

\end{array}\right. Determine the eigenvalue problem solved by v(y), where we define v(y)=z(x) \text { with } y=\ln x Using (a), find all eigenvalues {}>1 of problem (1) and show that a cor-responding eigen function d for X, is \phi_{n}(x)=\sqrt{2} \sin (x \pi \ln x) (c) Check directly that the sequence of eigenfunctions {n}>1 of (1) is orthonor-mal with respect to an appropriate inner product.

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