Question One $[25]$

$1\mathrm{.}1.$ The motion of a particle is described by the differential equation

$\frac{d^{2}x}{d{t}^2}+3\frac{dx}{dt}+2x=3e^{-2t}$

subject to $x(0)=1$ and $x^{\text{'}}(0)=0.$ Use the method of undetermined coefficients to determine the displacement $x(t)$ and discuss the progress of the motion as $t$

$1\mathrm{.}2.$ Use the power series generated about $x=0$ to determine the recurrence relation leading to the solution of the differential equation and determine the solution up to three $(3)$ non$-$zero terms.

$x{y}^{\text{'}\text{'}}+x{y}^{\text{'}}-2y=0$

Question Two $[22]$

$2\mathrm{.}1.$ Establish the existence of extremum points of $f(x$;$y$;$z)=x^3+2xy-y^2+1,$ and categorize each of the points.

$2\mathrm{.}2.$ Consider a solid formed by the region bounded by $y=x$ and $y=x^2$ and with density $(x$;$y)=2xk\frac{g}{m^3},$ determine the moment of inertia of the solid about the X $-$axis and Y $-$axis.

Question Three $[21]$

$3\mathrm{.}1.$ Determine whether the vector field $F(x$;$y$;$z)=(x{y}^2)i+z^{2}j+2yzk$ is conservative or not.

$3\mathrm{.}2.$ Determine the curl of the vector field $F(x$;$y$;$z)=(x{y}^{2}z)i+xzj+3yzk$ at the point $(-2$;$3$;$-1).$

$3\mathrm{.}3.$ Evaluate the line integral ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F.dr$ where $F(x$;$y$;$z)=(xy)i+x{z}^{2}j+2yzk$ is a vector field and $C$ is a curve described by $r(t)=ti-t^{2}j+t^{2}k,0t1.$

Question Four $[22]$

$4\mathrm{.}1.$ Sketch the region bounded by the curves $y=x,y=x^2$ and $x0$ and then use Green's Theorem to evaluate the integral ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}(x-x{y}^2)dx-(y-2y{x}^2)dy$ where $C$ is the boundary of the region bounded by the curves.

$(9)$