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) ${\displaystyle \frac{\ln 2}{4} + \frac{\ln 3}{9} + \frac{\ln
4}{16} + \frac{\ln 5}{25} + \ldots\;\;}$;
(b) ${\displaystyle \frac{1}{3} + \frac{8}{9} + \frac{27}{27} +
\frac{64}{81} + \frac{125}{243} + \ldots\;\;}$;
(c) ${\displaystyle \frac{2}{1} + \frac{3}{4} + \frac{4}{9} +
\frac{5}{16} + \frac{6}{25} + \ldots\;\;}$.
(18)6. (a) Determine whether the series ${\displaystyle
\sum_{n=1}^{\infty}\;(-1)^{n+1} \frac{\ln(n+1)}{n+1}}$ 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 ${\displaystyle \frac{dy}{dx}}$ and ${\displaystyle \frac{d^2y}{dx^2}}$, given that x = e-t, y = te2t.
(b) Find the total length of the astroid $x = a \cos^3 \theta,\;\;y
= a \sin^3 \theta$.
(c) Evaluate ${\displaystyle \int_C 2x \,ds}$, 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 $\vec{F} = (2xz + \sin y)
\vec{i} + x \cos y \vec{j} + x^2 \vec{k}$ is a conservative field.
(ii) Evaluate ${\displaystyle \int_C \vec{F} \cdot d \vec{r}}$ along the arc
\begin{displaymath}C: \vec{r} = \cos t \vec{i} + \sin t \vec{j} + t \vec{k},\;\;0 \leq
t \leq 2 \pi. \end{displaymath}
(b) Show that div $(r^3 \vec{r}) = 6r^3$ (note: $r = \vert\vec{r}\vert$).
(10)9. (a) Consider ${\displaystyle \int_C (xy + y^2)\,dx +
x^2 dy}$, 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 ${\displaystyle \int_{_{\!\!\mbox{$S$ }}} \! \int \vec{F} \cdot \vec{n}\,dS}$, where $F = 4xz\vec{i} - y^2 \vec{j} + yz \vec{k}$ 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.

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