Find the values of λ (eigenvalues) for which the given problem
has a nontrivial solution. Also...
Find the values of λ (eigenvalues) for which the given problem
has a nontrivial solution. Also determine the corresponding
nontrivial solutions (eigenfunctions).
(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?
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
2. (a) Find the values of a and b such that the eigenvalues of A
= |a 1 are 2 and -5. (b) Find the values of a, b and c such that
the eigenvalues of A = | 0 1 0 | 0 0 1 | a b c are 3, -2, and
5.
b 1|
(a) Let λ be a real number. Compute A − λI.
(b) Find the eigenvalues of A, that is, find the values of λ for
which the matrix A − λI is not invertible. (Hint: There should be
exactly 2. Label the larger one λ1 and the smaller λ2.)
(c) Compute the matrices A − λ1I and A − λ2I.
(d) Find the eigenspace associated with λ1, that is the set of
all solutions v = v1 v2 to (A...
Problem #2:
Use separation of variables with λ = 36 to find a
product solution to the following partial differential
equation,
y
∂2u
∂x2
+
∂u
∂y
= 0
that also satisfies the conditions u(0, 0) = 9 and
ux(0, 0) = 4.
For which values of λ does the system of equations
(λ − 2)x + y = 0
x + (λ − 2)y = 0
have nontrivial solutions? (That is, solutions other than x = y
= 0.) For each such λ find a nontrivial solution.
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,