MATH 2503

FINAL EXAMINATION April 1998 | TIME: 3 HOURS |

MARKS | |||

(8) | 1. | Solve the following initial value problems: | |

(a) | ; | ||

(b) | . | ||

(12) | 2. | Find the general solution to the following equations: | |

(a) |
y'' - 2y' + y = x^{2} + 1;
| ||

(b) |
y'' - 3y' + 2y = e^{x};
| ||

(c) | . | ||

(8) | 3. | Use variation of parameters to find the general solution of | |

(8) | 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.
| |||

(8) | 5. | Find the sum of the following series: | |

(a) | |||

(12) | 6. | Test the following series for convergence or divergence: | |

(a) | |||

(12) | 7. | (a) | Find the interval of convergence of the series |

(b) | Find the Taylor series for
f(x) = e^{2x} about a = 1. Write
the general term.
| ||

(8) | 8. | Use Gauss-Jordan elimination method to solve | |

(14) | 9. | (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 y.
| |||

(10) | 10. | (a) | Find the characteristic equation of |

(b) | Show
is an eigenvalue of A.
| ||

(c) | Find the eigenvectors of A corresponding to the smallest
eigenvalue of A.
| ||

(100) |