Explore BrainMass
Share

# Eigenfunctions - Expand the function

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

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 March 21, 2019, 11:30 am ad1c9bdddf
https://brainmass.com/math/fourier-analysis/eigenfunctions-expand-the-function-44173

#### Solution Preview

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