Summary: Math 243A Assignment 7
(1) Let f : Rn+1
be the vector field
f(t, x) =
(1 + t2
(1 + t2 + x 2)1/2
Prove that for every (t0, x0) Rn+1
, the initial value problem
x = f(t, x), x(t0) = x0
has a unique solution defined on the interval - < t < .
(2) Let O Rn
be an open set, and set = R × O. Suppose that
f : Rn
satisfies the hypotheses of the Picard existence
and uniqueness theorem. Suppose also that ¯x O is a stable
equilibrium for f, according to Definition 3.9.2 of the notes.