21-630 Spring 2014
Assignment 2
R. Pego
Problems due Friday Feb. 14:
1.
Suppose f : [0, a] Rn Rn is C 1 and that there is a continuous
function on [0, ) such that |f (t, y)| < (|y|) for all y, and suppose that
1
1
ds = +.
(s)
Prove that for every y0 RN ,
21-630 Spring 2014
Assignment 1
R. Pego
Problems due Wednesday Jan. 29:
1. Suppose f : R R is continuous, y : R R is C 1 and bounded, and
y (t) = f (y(t) for all t R. Suppose y (t0 ) > 0 for some t0 . Prove y is
increasing on R and y = limt y(t) exists wi
21-630 Spring 2014
Assignment 3
R. Pego
Problems due Monday Mar. 3:
1. For u = (u, v), consider the system in the plane:
u (t) = f (u) =
u v u(u2 + v 2 )
u + v v(u2 + v 2 )
Note that this system has the explicit periodic solution u (t) = (u (t), v (t) =
(
21-630 Spring 2014
Assignment 5
R. Pego
Problems due Wednesday April 30:
1. (GH 3.3.1) Show that a smooth system of the form
x =
1 0
x + F (x),
0 1
F (x) = o(|x|),
cannot in general be linearized by a smooth transformation of the form x = h(y) = y + o(|y|
21-630 Spring 2014
Assignment 4
R. Pego
Problems due Wednesday April 16:
1. (Cf. Chicone, exercise 3.41 p282) Stability analysis of Hills equation
leads to the system
y (t) = A(t)y,
A(t) =
0
1
a(t) 0
where a(t) is T -periodic for some T > 0. Let (t) be t