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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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.
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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:
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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 ...
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