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 | ||

x^{2}y'' + (x^{2} - 3x)y' + 4y = 0
| ||||

about x = 0. State the form of a second linearly independent
solution.
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(4) | 6. | Use the substitution
y = x^{-1/2}u(x) to
convert the differential equation
| ||

xy'' + 2y' + xy = 0
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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) |