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
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 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