**MATH3242/COSC3122**

**TERM
TEST 2**

**Monday, March 22,
1999 14:30 - 15:30**

1. All questions should be attempted.

2. The marks are given at the beginning of each question.

3. The total number of marks for this test is 50.

4. A non-programmable calculator is allowed as an aid.

5. Note that pertinent formulae and theorems are given on page 3.

1. (12 marks) Put the following integrals into the standard form for the type of Gaussian integration indicated.

(a) Gauss-Hermite:

[Note: all of the exponential term should be included in the weight function].

(b) Gauss-Laguerre:

(c) Evaluate the integral in part (b) given the following roots and weights for n = 2:

roots |
weights |

0.5858 |
0.8536 |

3.4142 |
0.1464 |

2. (6 marks) For what values of the parameter q do the following integrals converge?

(a) (b)

3. (11 marks) Evaluate the improper integral in question 2(a) with q = 5/2 by integrating the singular part analytically and using the Elementary Trapezoidal Rule for the remainder.

4. (6 marks) Which of the following initial-value problems satisfy the conditions which guarantee a unique solution? Justify your answer.

(a) y´
= 3xy + x^{2}, 1 < x
< 2, y(1) = 4;

(b) y´
= exp(-2y) + (1 - 2x)^{1/2}, 0 < x < 1, y(0) = -2

5. (3 marks) Transform the second-order initial-value problem

y´´
- 2xy´y + 4xy^{2}
= sin(x), 0 < x < 2, y(0) = 1.2, y´(0)
= 2.2

into a system of first-order differential equations.

You are **NOT **required to find a
solution to this differential equation.

6. (8 marks)

(a) Write out the Taylor method of order two for the initial-value problem

y´
= 2xy + x^{2}, 0 < x < 1, y(0) = 2.0

(b) Find an approximate solution to this system using Euler's method. Take h = 0.5.

7. (4 marks)

(a) Give one advantage and one disadvantage of Gaussian integration methods over Newton-Cotes methods.

(b) What property do the Runge-Kutta-Fehlberg and Bulirsh-Stoer methods have in common?

(c) Write out the general form of the orthogonality property of orthogonal polynomials.