Question

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

Answer #1

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!!

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)

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.

find the eigenvalues and the eigenfunctions for the equation y''
+ (lambda)y = 0 where y(a) = 0, y(b) = 0 for a<b.

Using separation of variables, write down a complete list of L^2
eigenfunctions and of eigenvalues for the Laplacian on the cylinder
D X [-1, 1], with homogeneous Dirichlet boundary conditions, where
D is the (two-dimensional) disk centered at the origin of radius
2.
b) Use this to solve the heat equation
partial u / partial t = Delta u
on this cylinder with homogeneous Dirichlet boundary conditions,
with initial data u(x, y, z, 0) = z, where z is the...

a)Find the eigenvalues and eigenfunctions of the Sturm-Liouville
system
y"+ lamda y = 0
y(0) = 0
y'(1) = 1
I got y=x and y=sin((sqrt k)x)/((sqrt k) cos(sqrt k))
Please do b
(b) Show that the eigenfunctions Yn and Ym you obtained from the
above
are orthogonal if n not= m.

Find the eigenvalues and eigenfunctions of the Sturm-Liouville
system
y"+ lamda y = 0
y(0) = 0
y'(1) = 1
(b) Show that the eigenfunctions Yn and Ym you obtained from the
above
are orthogonal if n not= m.

find the eigenvalues of the following matrix. then find the
corresponding eigenvector(s) of one ofthose eigenvalues (pick your
favorite).
1 -2 0
-1 1 -1
0 -2 1

Find the characteristic equation of A, the eigenvalues
of A, and a basis for the eigenspace corresponding to each
eigenvalue.
A =
−2
1
6
0
1
1
0
0
9
I found the eigenvalues to be (-2, 1,9).
How do I find the basis for the eigenspace corresponding to each
eigenvalue?
(c) a basis for the eigenspace corresponding to each eigenvalue

For what values of a (if any) does the boundary value
problem
x'' + ax' = 0, x(0) = 0, x(π) = 0
have nontrivial (i.e. nonzero) solutions
Hint: In order to solve, divide the problem into three
cases
1. If a > 0. In this case let a = b^2. 2. If a < 0. In this
case let a = −b^2. 3. If a = 0.

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