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 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 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
For the wave equation, utt = c2uxx, with the following boundary
and initial conditions,
u(x, 0) = 0
ut(x, 0) = 0.1x(π − x)
u(0,t) = u(π,t) = 0
(a) Solve the problem using the separation of variables.
(b) Solve the problem using D’Alembert’s solution. Hint: I would
suggest doing an odd expansion of ut(x,0) first; the final solution
should be exactly like the one in (a).
Consider the following wave equation for u(t, x) with boundary
and initial conditions defined for 0 ≤ x ≤ 2 and t ≥ 0.
∂ 2u ∂t2 = 0.01 ∂ 2u ∂x2 (0 ≤ x ≤ 2, t ≥ 0) (1)
∂u ∂x(t, 0) = 0 and ∂u ∂x(t, 2) = 0 (2)
u(0, x) = f(x) = x if 0 ≤ x ≤ 1 1 if 1 ≤ x ≤ 2.
(a) Compute the coefficients a0, a1, a2, ....
Consider the ODE u" + lambda u=0 with the boundary conditions
u'(0)=u'(M)=0, where M is a fixed positive constant. So u=0 is a
solution for every lambda,
Determine the eigen values of the differential operators: that
is
a: find all lambda such that the above ODE with boundary
conditions has non trivial sol.
b. And, what are the non trivial eigenvalues you obtain for each
eigenvalue