### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 2013

 FINAL EXAMINATION April 1999 TIME: 3 HOURS
NO CALCULATORS

 MARKS (8) 1. Solve the following initial value problems: (a) 4x dy - y dx = x2dy, y(5) = 51/4 (b) (x3 + y3)dx - 3xy2dy = 0, y(2) = 21/3 (15) 2. Find the general solution for the following equations: (a) y'' - 3y' + 2y = ex (b) y'' + 9y = 32x cosx (c) y'' + y' - 2y = 2(1 + x - x2) + 4e2x (6) 3. Find the general solution of (use the method of variation of parameters): y'' - 2y' = ex sinx. (6) 4. Find the general solution of (use the method of power series): y'' + xy' + y = 0. (12) 5. Test the following series for convergence or divergence: (a) ; (b) ; (c) . (18) 6. (a) Determine whether the series is absolutely convergent, conditionally convergent or divergent. Give reasons. (b) Find the interval of convergence of the series x + x 4 + x9 + x16 + x25 + ... . (c) Give the first four non-zero terms in the expansion of sin-1 x. (15) 7. (a) Find and , given that x = e-t, y = te2t. (b) Find the total length of the astroid . (c) Evaluate , where C consists of the arc C1 of the parabola y = x2 from (0,0) to (1,1) followed by the line segment C2 from (1,1) to (0,3). (10) 8. (a) (i) Show that is a conservative field. (ii) Evaluate along the arc (b) Show that div (note: ). (10) 9. (a) Consider , where C is the closed curve of the region bounded by y = x2 and y = x. Use Green's Theorem to set this as a double integral and then evaluate this double integral. (b) Consider , where and S is the surface of the cube bounded by x = 0,  x = 1,  y = 0,  y = 1,  z = 0,  z = 1. Use divergence theorem to set this as a triple integral and then evaluate this triple integral. (100)