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Eigenfunctions - Expand the function

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Please show each step of your solution. When you use theorems, definitions, etc., please include in your answer.

Expand the function

f(x) = {x^4, 0 <= x < 2
{0, 2 <= x <= pi

in terms of the eigenfunctions of the given eigenvalue problem. Use computer software, such as the Maple int command, to evaluate the expansion coefficient a_n as a function of n.

(a) y" + lambda y = 0, y'(0) = 0, y'(pi) = 0
(b) y" + lambda y = 0, y'(0) = 0, y(pi) = 0
(b) y" + lambda y = 0, y(0) = 0, y'(pi) = 0

Please see the attached.

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We start with solving the Sturm-Liouville (S-L) problem (b), whose general solution is:

See attached

The boundary conditions lead to:

See attached

Since A couldn’t be zero, we get an eigenvalues problem, that is there are certain values of (λ) which satisfy (4):

See attached

These are the eigenvalues of our S-L problem. The ...

Solution Summary

This shows how to expand a given function in terms of eigenfunctions of a given eigenvalue problem. B is solved in this solution, which is step by step and detailed, including diagrams.