MATH 2013

FINAL EXAMINATION April 1999 | TIME: 3 HOURS |

MARKS | ||||

(8) | 1. | Solve the following initial value problems: | ||

(a) |
4x dy - y dx = x^{2}dy, y(5) =
5^{1/4}
| |||

(b) |
(x^{3} + y^{3})dx - 3xy^{2}dy =
0, y(2) = 2^{1/3}
| |||

(15) | 2. | Find the general solution for the following equations: | ||

(a) |
y'' - 3y' + 2y = e
^{x} | |||

(b) |
y'' + 9y = 32x cosx
| |||

(c) |
y'' + y' - 2y = 2(1 + x - x^{2}) + 4e^{2x}
| |||

(6) | 3. |
Find the general solution of (use the method of
variation of parameters):
y'' - 2y' = e sin^{x}x.
| ||

(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: | ||

| ; | |||

| ; | |||

| . | |||

| Determine whether the series is absolutely convergent, conditionally convergent or divergent. Give reasons. | |||

| Find the interval of convergence of the series
x + x^{
4} + x^{9} + x^{16} + x^{25} + ... .
| |||

(c) | Give the first four non-zero terms in the expansion of
sin^{-1} x.
| |||

| Find
and
,
given that
x = e^{-t}, y =
te^{2t}.
| |||

| Find the total length of the astroid . | |||

| Evaluate
,
where C consists of
the arc C_{1} of the parabola y = x^{2} from (0,0) to (1,1)
followed by the line segment C_{2} from (1,1) to (0,3).
| |||

| Show that is a conservative field. | |||

Evaluate
along
the arc
| ||||

| Show that div (note: ). | |||

Consider
,
where C is the closed curve of the region bounded by y =
x^{2} and y = x. Use Green's Theorem to set this as a double
integral and then evaluate this double integral.
| ||||

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) |