### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 3503

FINAL EXAMINATION
April 1998
TIME: 3 HOURS

### NO CALCULATORS PERMITTED

 MARKS (3) 1. Use the definition of the Laplace transform to show that (5) 2. Find the Laplace transform of (5) 3. Find the inverse Laplace transform of (8) 4. Use Laplace transforms to solve the differential equation (8) 5. Use the Method of Frobenius to find a solution to the differential equation x2y'' + (x2 - 3x)y' + 4y = 0 about x = 0. State the form of a second linearly independent solution. (4) 6. Use the substitution y = x-1/2u(x) to convert the differential equation xy'' + 2y' + xy = 0 to a Bessel equation. Use this to solve the differential equation. (7) 7. Use matrix methods to solve the system given that the characteristic equation is . (7) 8. Use matrix methods to solve the system (5) 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 on . (8) 11. Use the method of separation of variables to solve the partial differential equation subject to (65)