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.
Determine the eigenvalues and eigenfunctions of the following
operator (assume σ(x) ≡ 1): L(u) = u''−2u x ∈ (−1,1) with periodic
boundary conditions u(−1) = u(1), u'(−1) = u'(1). Box your final
answer
Consider the ODE y''(x) = λy(x) for some real constant λ.
Determine ALL values of λ for which there exists solutions
satisfying the boundary conditions y(0) = y(10) = 0. For each such
λ, give all possible solutions. Are they unique?
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
Determine whether or not the following languages are regular. If
the language is regular then give an NFA or regular expression for
the language. Otherwise, use the pumping lemma for regular
languages or closure properties to prove the language is not
regular.
1) L = { 0 n1 k : k ≤ n ≤ 2k}
2) L = { 0 n1 k : n > 0, k > 0 } È { 1 k0 n : k > 0, n...