# Sturm-liouville and Hermit's differential equation

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1. (a) Find the eigenvalues and eigenfunctions of the boundary-value problem.

x2y'' + xy' + λ y = 0, y(1) = 0, y(5) = 0.

(b) Put the differential equation in self -adjoint form.

(c) Give an orthogonality relation.

2. Hermite's differential equation

y'' -2xy' + 2ny = , n = 0, 1, 2,...

has polynomial solutions Hn(x). Put the equation in self-adjoint and give an orthogonality relation.

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#### Solution Summary

This is an example of a Sturm Liouville problem involving eigenvalues and eigenfunctions, Hermit's differential equation, and an orthogonality relation.

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