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
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 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 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 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 ,
Kayla Marie Saunders
Module Sixteen Lesson Two Assignment
January 12, 2017
The Circulatory System and Respiratory System
While the circulatory system functions to pump blood throughout the body and the
respiratory system aims to carry out the exchange of