MATH 2503
FINAL EXAMINATION APRIL 1999  TIME: 3 HOURS 
NO CALCULATORS.
SHOW ALL INTERMEDIATE CALCULATIONS.
MARKS  
(5)  1. 
Solve the initial value problem:
Find .  
(5)  2.  Find the general solution to the differential
equation:
y'' + 2y' + y = 4e^{x}.
 
(8)  3.  Use the method of variation of parameters to
solve
y'' + 3y' + 2y = cos(e^{x}).
 
Find , if the series converges.  
(6)  5.  Test the following series for convergence:  
(a)
;  
Determine whether is absolutely convergent, conditionally convergent or divergent.  
(7)  7.  Find the interval of convergence for the power
series
 
To what function does converge when it converges?  
Find the Taylor series for xe^{x} about x = 2.  
(8)  10.  Use a power series to find the solution of the
following initial value problem:
y''  x^{2}y'  2xy = 0, y(0)
= 1, y'(0) = 0.
 
(5)  11.  Use GaussJordan elimination to find all
solutions or show that there are none:
 
(7)  12.  (a)  Find the inverse of the matrix

(b)  Use the inverse found in part (a) to solve the
system:
 
(4)  13.  If A is an matrix with the property A^{1} = 2A determine the value(s) of detA. Give a 2 × 2 example of a matrix with this property.  
(8)  14.  Find the eigenvalues and eigenvectors of the
matrix
 
(80) 