MATH 3503

FINAL EXAMINATION April 1997 | TIME: 3 HOURS |

## NO CALCULATORS PERMITTED |

MARKS | ||||

1. | Find the Laplace transforms of the following functions: | |||

(5) | (a) | |||

(3) | (b) | . | ||

(6) | 2. | Find the inverse Laplace transform of | ||

(8) | 3. | Use Laplace transforms to solve the differential equation | ||

(8) | 4. | Use Laplace transforms to solve the differential equation | ||

(10) | 5. | Use the Method of Frobenius to find a solution to the differential equation | ||

x^{2}y'' + x(1-x)y' - (1+3x)y = 0
| ||||

about x = 0. State the form of a second linearly independent
solution.
| ||||

(4) | 6. | Use the identities | ||

and | ||||

to find the recurrence formula for Bessel functions. | ||||

(7) | 7. | Use matrix methods to solve the initial value problem | ||

(8) | 8. | Use matrix methods to find the general solution to | ||

(6) | 9. | Find the Fourier series for the function | ||

Sketch the graph of the function to which the series converges over the interval . | ||||

(5) | 10. | Find the Fourier sine series for the function | ||

Sketch the graph of the function to which the series converges over the interval . | ||||

(10) | 11. | Use the method of separation of variables to solve the partial differential equation | ||

subject to | ||||

(80) |