8. Problems 31
6. Prove that
d
dx1
ln |x1| = p.v.
1
x1
.
7. Consider a system of ordinary differential equations with
C∞
coef-
ficients,
du
dx
= A(x)u,
where u(x) = (u1(x),...,um(x)) and A(x) is an m × m
C∞
matrix. Using Proposition 7.5 and the existence of a
C∞
funda-
mental matrix of solutions (i.e., an m × m matrix Φ(x) such that
dΦ(x)
dx
= A(x)Φ(x) and Φ(0) = I, where I is the identity matrix),
prove that any distribution solution of
du
dx
= A(x)u is a regular
functional corresponding to a
C∞
function, i.e., any distribution
solution is a classical solution.
8. Denote θ(x1,x2) = θ(x1)θ(x2), where θ(t) = 1 for t 0 and θ(t) =
0 for t 0. Prove that
∂2
∂x1∂x2
θ(x1,x2) = δ,
where δ is the delta-function in
R2.
9. A distribution Ey (y is a parameter) is called a fundamental solu-
tion to the ordinary differential equation Lu = f if
LEy = δ(x y).
a) Find the general form of fundamental solution to the second
order ordinary differential equation
p0(x)u (x) + p1(x)u (x) + p2(x)u = f,
pk(x)
C∞(R1),
k = 0, 1, 2, p0(x) = 0.
b) Find the fundamental solution to
n
k=0
pk(x)u(k)
= f, pk(x)
C∞,
k = 0,...,n, p0(x) = 0,
such that Ey = 0 for x y.
c) Find the fundamental solution to u +4u = f such that Ey = 0
for x y.
10. Let u(x, t) be a smooth function in
R2
outside of a curve x = x(t).
Suppose that there exist limits
u+(t) = lim
(x,t)→(x(t),t)
u(x, t), x x(t),
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