MATH 2503

FINAL EXAMINATION December 1997 | TIME: 3 HOURS |

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 = 2e^{3x};
| |||

(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 x^{5} in the general
solution of
| ||

y'' + 2xy' + 2y = 0.
| ||||

(16) | 5. | Test the following series for convergence or divergence: | ||

(a) |
||||

(c) |
||||

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.
| ||||

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) |