### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 3503

 FINAL EXAMINATION APRIL 1996 TIME: 3 HOURS

NO CALCULATORS PERMITTED
 MARKS 1. Find the Laplace transforms of the following functions: (3) (a) ; (3) (b) ; (4) (c) (4) 2. Find the inverse Laplace transform of (4) 3. Use the convolution product to find the inverse Laplace transform of (7) 4. Use Laplace transforms to solve the differential equation 5. For the differential equation (2) (a) find the roots of the indicial equation; (6) (b) use the Method of Frobenius to find the first 3 non-zero terms in the series solution corresponding to the larger of the 2 roots. 6. Given that the Bessel function of the l kind of order n is (3) (a) show that (4) (b) find a real solution to by using the complex transformation u = ix. (7) 7. Use matrices to solve the initial value problem (7) 8. Find the general solution of the system of differential equations (2) 9. (a) Show that the matrix is a fundamental matrix for the homogeneous system of differential equations (6) (b) Use this fundamental matrix and the method of variation of parameters to find the general solution of the non-homogeneous system (7) 10. Find the Fourier series for the function Sketch the graph of the function to which the series converges over the interval . 11. Consider the initial-boundary value problem for the one-dimensional heat equation with zero boundary conditions (7) (a) Using the method of separation of variables, find an infinite set of non-zero solutions of the form that satisfy the boundary conditions. Show clearly the steps leading to the equations that and must satisfy. (4) (b) From this set of non-zero solutions, find a linear combination which will provide a solution to the initial-boundary value problem if (i) ; (ii) . ___ (80)