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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.

    © BrainMass Inc. brainmass.com April 3, 2020, 2:43 pm ad1c9bdddf
    https://brainmass.com/math/fourier-analysis/eigenfunctions-expand-the-function-44173

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    Solution:

    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.

    $2.19

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