MATH 520 Homework
Spring 2014
41. Let Lu = (x − 2)u00 + (1 − x)u0 + u on (0, 1).
a) Find the Green’s function for
u0 (0) = 0 u(1) = 0
Lu = f
(Hint: First show that x − 1, ex are linearly independent solutions of Lu = 0.)
b) Find the adjoint operator and boundary conditions.
42. Let φ, ψ be solutions of Lu = a2 (x)u00 + a1 (x)u0 + a0 (x)u = 0 on (a, b) and
W (φ, ψ)(x) = φ(x)ψ 0 (x) − φ0 (x)ψ(x) be the corresponding Wronskian determinant.
a) Show that W is either zero everywhere or zero nowhere. (Suggestion: find a first
order ODE satisfied by W .)
b) If a1 (x) = 0 show that the W is constant.
43. Let Lu = a2 (x)u00 + a1 (x)u0 + a0 (x)u with a02 = a1 , so that L is formally self adjoint.
If B1 u = C1 u(a) + C2 u0 (a), B2 u = C3 u(b) + C4 u0 (b), show that {B1∗ , B2∗ } = {B1 , B2 }.
44. Find the Green’s function for
u00 + 2u0 − 3u = f (x) 0 < x < ∞
u(0) = 0
lim u(x) = 0
x→∞
(Think of the last condition as a ’boundary condition at infinity’.) Using the Green’s
function, find u(2) if f (x) = e−6x .
45. Consider the second order operator
Lu = a2 (x)u00 + a1 (x)u0 + a0 (x)u
a<x<b
with non-separated boundary conditions
B1 u = α11 u(a) + α12 u0 (a) + β11 u(b) + β12 u0 (b) = 0
B2 u = α21 u(a) + α22 u0 (a) + β21 u(b) + β22 u0 (b) = 0
where the vectors (α11 , α12 , β11 , β12 ), (α21 , α22 , β21 , β22 ) are linearly independent. We
again say that two other non-separated boundary conditions B1∗ , B2∗ are adjoint to B1 , B2
with respect to L if J(u, v)|ba = 0 whenever B1 u = B2 u = B1∗ v = B2∗ v = 0.
Find the adjoint operator and boundary conditions in the case that
Lu = u00 + xu0
B1 u = u0 (0) − 2u(1) B2 u = u(0) + u(1)