Find the eigenvalues and eigenfunctions of the given boundary
value problem. Assume that all eigenvalues are...
Find the eigenvalues and eigenfunctions of the given boundary
value problem. Assume that all eigenvalues are real. (Let
n represent an arbitrary positive number.)
Find the eigenvalues λn and eigenfunctions
yn(x) for the given boundary-value problem. (Give your
answers in terms of k, making sure that each value of
k corresponds to two unique eigenvalues.)
y'' + λy = 0, y(−π) = 0, y(π) = 0
λ2k − 1 =, k=1,2,3,...
y2k − 1(x) =, k=1,2,3,...
λ2k =, k=1,2,3,...
y2k(x) =, k=1,2,3,...
Find the eigenvalues
λn
and eigenfunctions
yn(x)
for the given boundary-value problem. (Give your answers in
terms of n, making sure that each value of n
corresponds to a unique eigenvalue.)
y'' + λy = 0, y(0) = 0, y(π/6) = 0
λn =
,
n = 1, 2, 3,
yn(x) =
,
n = 1, 2, 3,
Find the eigenvalues
λn
and eigenfunctions
yn(x)
for the given boundary-value problem. (Give your answers in
terms of n, making sure that each value of n
corresponds to a unique eigenvalue.)
x2y'' +
xy' + λy =
0, y(1) =
0, y'(e) = 0
Find the eigenvalues
λn
and eigenfunctions
yn(x)
for the given boundary-value problem. (Give your answers in
terms of n, making sure that each value of n
corresponds to a unique eigenvalue.)
x2y'' + xy' + λy = 0, y'(e−1) =
0, y(1) = 0
λn =
n = 1, 2, 3,
yn(x) =
n = 1, 2, 3,
Find the eigenfunctions for the following boundary value
problem.
x2y?? ? 15xy? + (64 +
?)?y ?=?
0, y(e?1) ?=?
0, ?y(1) ?=? 0.
In the eigenfunction take the arbitrary constant (either
c1 or c2) from the general
solution to be 1
Find the linear space of eigenfunctions for the problem with
periodic boundary conditions
u′′(x) = λu(x)
u(0) = u(2π)
u′(0) = u′(2π)
for (a) λ = −1 (b) λ = 0 (c) λ = 1.
Note that you should look for nontrivial eigenfunctions
Find eigenvalue (?) and eigenfunction and evaluate orthogonality
from the given boundary value problem. ?2?′′ + ??′ + ?? = 0, ?(1) =
0, ?(5) = 0. Hint: Use Cauchy-Euler Equation, (textbook
pp141-143).
(a) Find all positive values of λ for which the following
boundary value problem has a nonzero solution. What are the
corresponding eigenfunctions? X′′ + 4Xʹ + (λ + 4) X = 0, X′(0) = 0
and X′(1) = 0. Hint: the roots of its auxiliary equation are –2 ±
σi, where λ = σ2.
(b) Is λ = 0 an eigenvalue of this boundary value problem? Why
or why not?