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),