Consider the Sturm-Liouville problem
X′′(x) + λX(x) = 0 subject toX′(0) = 0, X(l) = 0.
Are the boundary conditions symmetric?
Do these boundary conditions yield negative eigenvalues?
Determine the eigenvalues and eigenfunctions, Xn(x). (It is
enough in some cases to provide the equation that determines the
eigenvalues rather than an explicit formula.)
Are the eigenfunctions orthogonal?
Determine the eigenvalues and the corresponding normalized
eigenfunctions of the following Sturm–Liouville problem: y''(x) +
λy(x) = 0, x ∈ [0;L], y(0) = 0, y(L) = 0,
Find the eigenvalues and eigenfunctions of the Sturm-Liouville
system
y"+ lamda y = 0 o<x<1
y(0) = 0
y'(1) = 0
(b) Show that the eigenfunctions Yn and Ym you obtained from the
above
are orthogonal if n not= m.
We consider the Boundary Value Problem :
u'(x)+u(x)=f(x), 0<x<1
u(0)-eu(1)=a
,a is real number kai f is continue in [0,1].
1. Find a a necessary and sufficient condition ,that Boundary
value problem is solvabled.
2. Solve the Boundary value problem with a=0.
Consider a semi-infinite square well: U(x)=0 for 0 ≤ x ≤ L,
U(x)=U0 for x > L, and U(x) is infinity otherwise.
Determine the wavefunction for E < Uo , as far as possible,
and
obtain the transcendental equation for the allowable energies E.
Find the necessary condition(s) on E for the solution to exist.