### DEPARTMENT OF MATHEMATICS & STATISTICS 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(ex). (3) 4. Find , if the series converges. (6) 5. Test the following series for convergence: (a) ; (b) . (5) 6. Determine whether is absolutely convergent, conditionally convergent or divergent. (7) 7. Find the interval of convergence for the power series (4) 8. To what function does converge when it converges? (5) 9. Find the Taylor series for xex about x = 2. (8) 10. Use a power series to find the solution of the following initial value problem: y'' - x2y' - 2xy = 0, y(0) = 1, y'(0) = 0. (5) 11. Use Gauss-Jordan 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 n × n 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)