Math 427K-H   Homework 6    Due Friday      March 23



1.  Find the solution to the equation


          y" + 2y'  - 8y  = sin t

with y(0) = y'(0) = 0.




2. Find the solution to the equation   y'''  +  8y = 16  with

y(0) = y'(0) = y"(0) = 0.




3.  The notation in matlab for a matrix is A = [ 1  2; 1  3;  2  4] 

where  the  semicolons separate  rows. The matrix A above is  3x2.

A  vector is written v = [1; 3 ; 5]  and a covector is w = [ 1  2   5].


Compute  wA,  wv, vw , (v^T)v ,  w w^T  , A^T A  and A  A^T.  Here we 

are using A^T to indicate the transpose of A.



4. Let  A  [ 1  0  0; 1  2  0 ;  2  3  4]  and B =  [1 4; 2 9].  Compute

A^-1 and B^-1 in any way you like.



5.  Let  y ' = A y  where A = [3 -2; 2, -2].   Find r so that y =

[4 ; 2] exp(rt) solves the vector differential equation.






Required:   Pass in a sheet of paper with the names of the students working
together on a project, a tentative topic, and one reference for the project
which is not the textbook. This should go in the "extra credit" pile.


Extra Credit:  Suppose that N is a strictly upper triangular nxn matrix.

Show that N^n = 0.  Now show that (I - N )^-1 =  I  +  N + N^2 +  N^(n-1).

Suggestion:  work it out for 3X3 matrices first. Note that you don't have

to compute the inverse.  It is only necessary that the given matrix serves

the purpose.