**
MATH 3503**

FINAL EXAMINATION APRIL 1996 | TIME: 3 HOURS |

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
is
n | |||

(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) | . | |||

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(80) |