DEPARTMENT OF MATHEMATICS & STATISTICS 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 x2y'' + 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)