Determine the eigenvalues and eigenfunctions of the following
operator (assume σ(x) ≡ 1): L(u) = u''−2u x ∈ (−1,1) with periodic
boundary conditions u(−1) = u(1), u'(−1) = u'(1). Box your final
answer
Determine the eigenvalues and the corresponding normalized
eigenfunctions of the following Sturm–Liouville problem: y''(x) +
λy(x) = 0, x ∈ [0;L], y(0) = 0, y(L) = 0,
Find the eigenvalues and eigenfunctions of the given boundary
value problem. Assume that all eigenvalues are real. (Let
n represent an arbitrary positive number.)
y''+λy=
0,
y(0)= 0,
y'(π)= 0
Show that Hermitian operators have real eigenvalues. Show that
eigenvectors of a
Hermitian operator with unique eigenvalues are orthogonal. Use
Dirac notation
for this problem.
Q3. Consider the matrix A .
Use R statistical software to determine the eigenvalues and
normalized eigenvectors of A, trace of
A, determinant of A, and inverse
of A. Also determine the eigenvalues and
normalized eigenvectors of
A-1. Your answer
should include your R code (annotated with comments) and a
hand-written or typed summary of the answers from the R output.
Find the eigenvalues and eigenfunctions of the Sturm-Liouville
system
y"+ lamda y = 0 o<x<1
y(0) = 0
y'(1) = 0
(b) Show that the eigenfunctions Yn and Ym you obtained from the
above
are orthogonal if n not= m.