### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 2503

 FINAL EXAMINATION April 1997 TIME: 3 HOURS

### NO CALCULATORS

MARKS
(12) 1. Solve the following differential equations :
(a)
(b)
(c)
(8)2. Find the general solution for the following differential equations:
(a) y'' + 2y' + 2y = 0
(b) y'' + y' = 2x + 1
(6)3. Solve the following initial value problem:
(16)4. (a) Solve, using the method of variation of parameters .
(b) Use power series to solve y'' + xy' + y = 0.
(12)5. Test the following series for convergence or divergence:
(a) (b) (c)
(15)6. (a) Test whether, the following series is absolutely, conditionally convergent or divergent:
(b) Find the interval of convergence of the series
(c) Find the Maclauren series for .
(5)7. Use Gauss-Jordan elimination method to find all the solutions, if any, of the following system
 x + 3y + z = 2 x - 2y - 2z = 3 2x + y - z = 5 3x - y - 3z = 8
(8)8. (a) Find the inverse of the matrix
(b) Use (a) to solve
 x + 2y + 3z = 0 x + y + 2z = 1 y + 2z = 2
(8)9. (a) Evaluate the following determinant:
(b) Use Cramer's rule to find the value of z in the following system:
 x - 2y + z = 0 y - 2z = 0 x + 3y - z = 1
(10)10. Let .
(a) Find the characteristic equation of A.
(b) Show 4 is an eigenvalue of A.
(c) Find an eigenvector of A corresponding to the eigenvalue 4.

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