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. Cons
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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,
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
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)
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 ru
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 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 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
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