### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 2503

 FINAL EXAMINATION December 1997 TIME: 3 HOURS

### NO CALCULATORS

 MARKS (8) 1. Solve the following initial value problems: (a) ; (b) . (13) 2. Find the general solution to the following equations: (a) y'' - 2y' - 3y = 2e3x; (b) ; (c) y'' + 2y' = x + 2. (7) 3. Use variation of parameters to find the general solution of (7) 4. Use power series to solve the following differential equation. Find the recurrence relation and terms up to and including those involving x5 in the general solution of y'' + 2xy' + 2y = 0. (16) 5. Test the following series for convergence or divergence: (a) (b) (c) (d) (17) 6. (a) Determine whether the series is absolutely convergent, conditionally convergent or divergent. Give reasons. (b) Find the interval of convergence of the series (c) Give the first four non-zero terms in the expansion of (10) 7. Use Gauss-Jordan elimination method to solve (11) 8. (a) Find the inverse of the matrix and use this to find the solution of the system (b) Use Cramer's rule to solve the system for x. (9) 9. (a) Show that the vector is an eigenvector of (b) (i) Find the characteristic equation of (ii) Show is an eigenvalue of A and find the eigenvectors corresponding to it. (100)