Practice Final, G63.2470, Spring 2009
Olof Widlund April 30, 2009
Motivate your answers and cross out everything that is not part of your answers. 1. Find the solution of x 5x + 4x = 4t2 e2t . 2. Consider the solutions of u (x) + au(x)/x2 = 0, 1 x < . Her
November 1, 2011
ExercisesSet 4, Math 848, Fall 2011
1. Let
1
(x) =
e x2
0
if x > 0
if x 0
Prove that (x) is a C function from R to R.
2. Let 1 (x) = (x) (1 x), where is as in the previous exercise, and
let
x
0 1 (t)dt
(x) = 1
0 1 (t)dt
The function (x)
ExercisesSet 2, Math 848, Fall 2011
1. Consider the system
x = 2x y + z + t2 cos(t)
y = x+y+z
z = x 2y + 3z + 2 t3 exp(t)
Prove that any maximal solution is dened on the whole real line.
2. Let g : R R, be a C 1 function with derivative g (t) for t R.
Ass
ExercisesSet 3, Math 848, Fall 2011
1. Find the general real solution to the dierential equation x = Ax for
each of the following matrices A. Also, draw a rough sketch of the
orbits in each case.
(a)
A=
01
1 0
A=
21
2 3
A=
1
1
1 1
(b)
(c)
(d)
1
20
A = 2 1
Assignment Set 6, G63.2470, Spring 2009
Olof Widlund
April 27, 2009
The following assignments are due no later than May 7, at midnight.
1. Develop a numerical quadrature rule, similar to a Gaussian rule but using
the roots of (1x2 )L (x). Here Ln (x) is t
Assignment Set 5, G63.2470, Spring 2009
Olof Widlund
April 2, 2009
The following assignments are due on April 8, at midnight, but will
be accepted up to one week late.
1. Problem 2 of chapter 7 of Coddington and Levinson.
2. Problem 11 of chapter 7 of Cod
Assignment Set 4, G63.2470, Spring 2009
Olof Widlund
March 14, 2009
The following assignments are due on March 25 at midnight, but
will be accepted up to one week late.
1. Problem 1 in chapter 1 of Coddington and Levinson.
2. Problem 1 in chapter 15 of Co
Assignment Set 3, G63.2470, Spring 2009
Olof Widlund
February 23, 2009
The following assignments are due on March 4 at midnight.
1. Given the Jordan canonical form or a real matrix, show how construct the
real Jordan form. (This canonical form was used bu
September 3, 2011
ExercisesSet 1
1. Suppose T : X Y is a linear map of a Banach space X into Banach
space Y . Let
A = inf cfw_k : | T x | k | x | x X
| Tx |
B = sup
|x|
x=0
C = sup cfw_| T x |
| x |1
Show that A = B = C .
2. Suppose | |1 , | |2 are two n