Question

Consider the boundary value problem below (assume λ > 0): y ′′ + λy = 0 y(0) = 0 y ′ (π) = 0 Find the eigenvalues and the associated eigenfunctions for this problem. Show all work.

Answer #1

Find all eigenvalues and corresponding eigenfunctions for the
following boundary value problem (x^2)y'' + λy = 0, (1 < x <
2), y(1) = 0 = y(2) and in particular the three cases μ < 1/2, μ
= 1/2, and μ > 1/2 associated with the sign and vanishing of the
discriminant of the characteristic equation

Given: The following boundary value problem:
y"+ lamda*y = 0;
0 < x < 2; y(0) =
0; y’(2) = 0
Find corresponding eigenvalues, (lamda)n
and normalized eigenfunctions yn
Expand the function f(x) = x, in terms of the eigen functions
obtained in (i)

find the eigenvalues and eigenfunctions for the given
boundary-value problem.
y'' + (lambda)y = 0, y(-pi)=0, y(pi)=0
Please explain where alpha = (2n+1)/2 comes from in the
lambda>0 case. Thank you!!

Consider the boundary value problem y''(t) + y(t) = f(t) , y(0)
= 0 and y(π/2) = 0.
Find the solution to the boundary value problem

A nontrivial solution of the boundary value problem y′′ + 9y =
0; y′(0) = 0, y′(π) = 0 is:

Solve the given boundary-value problem.
y'' − 2y' + 2y = 2x − 2,
y(0) = 0, y(π) = π

find the boundary- value problem
y’’ + 2y = 4x, y(0) = 0, y(1) + y’(1) = 0

MATLAB CODE REQUIRED
Consider the problem of estimating y(0.5) for the boundary-value
problem
y''+ y' = y + 2, y(0) = 0, y'(1) = 2. Find the solution using 2
approaches:
(b) Use finite difference with n =
10.Please provide Matlab code. What is y(0.5) value with
Matlab?
(c) Using bvp4c. Please provide Matlab
code. What is y(0.5) value with Matlab?

Sturm-Liouville problem:
y′′+λy= 0, y′(0) = 0, y(1) +y′(1) = 0.
Determine the four smallest eigenvalues and corresponding
eigenfuntions.
For lambda = 0, < 0, > 0

solve the boundary value problem:
y''(x)+y(x)=e^x for 0<x<pi with y(0)=0 and
y(pi)+y'(pi)=0.
please show all steps.

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