**
MATH 3503**

FINAL EXAMINATION DECEMBER 1994 | TIME: 3 HOURS |

1. | (a) | Find where |

(b) | Use the Laplace Transform to solve | |

where is the function of part (a). | ||

(c) | Solve | |

2. | Evaluate | |

(a) | ||

(b) | ||

(c) | ||

(d) | , using the Convolution Theorem. Do not evaluate the
integral.
| |

3. | Show that is a regular singular point of the
differential equation
x = 0 | |

Find the indicial equation and show that its roots differ by an integer. By considering the smaller root, show that two independent series solutions may be obtained and find the first three non-zero terms in each of the solutions. | ||

4. | (a) | Find the general solution to the system |

and write down a fundamental matrix for the system. | ||

(b) | Use the result of part (a) to find the general solution to the non-homogeneous system | |

5. | (a) | Find the Fourier series for the function |

(b) | Find the complex Fourier series for the function | |

6. | (a) | Find the Fourier cosine series for the function |

(b) | Solve the boundary value problem for | |

where is the function of part (a). |