[SOLVED] Amath 568 advanced differential equations homework 2

1. Consider the nonhomogeneous problem ⃗x′ = A⃗x + ⃗g(t).
Let ⃗x = M⃗y where the columns of M are the eigenvectors of A.2. Given L = −d
2/dx2 find the eigenfunction expansion solution of
d
2
y
dx2
+ 2y = −10 exp(x)
where y(0) = 0 and y
′
(1) = 0.3. Given L = −d
2/dx2 find the eigenfunction expansion solution of
d
2
y
dx2
+ 2y = −x
where y(0) = 0 and y(1) + y
′
(1) = 0.4. Consider the Sturm-Liouville eigenvalue problem
Lu = −
d
dx 
p(x)
du
dx
+ q(x)u = λρ(x)u
for 0 < x < l with boundary conditions
α1u(0) − β1u
′
(0) = 0
α2u(l) − β2u
′
(l) = 0
and with p(x) > 0, ρ(x) > 0, and q(x) ≥ 0 and with p(x), ρ(x), q(x), and p
′
(x) continuous over 0 < x < l, and the weighted inner product ⟨ϕ, ψ⟩ρ =
R l
0
ρ(x)ϕ(x)ψ
∗
(x)dx.
Show the following:
(a) L is a self-adjoint operator.(b) Eigenfunctions corresponding to different eigenvalues are orthogonal, i.e. ∀n ̸=
m : ⟨un, um⟩ρ = 0.(c) Eigenvalues are real, non-negative, and eigenfunctions may be chosen to be
real valued.(d) Each eigenvalue is simple, i.e. it only has one eigenfunction.