$-3-$

MAT$3700$

May$/$June $2023$

QUESTION $1$

Solve the following first order differential equations:

$1\mathrm{.}1(x^2-y^2)dy=xydx$

$1\mathrm{.}2\frac{dy}{dx}+ytanx=y^{3}se{c}^{4}x$

QUESTION $2$

Find the general solution for the following differential equation using the method of undetermined coefficients $\frac{d^{2}y}{dx}+3\frac{dy}{dx}+2y=3sinx.$

$[10]$

QUESTION $3$

Find the general solutions of the following differential equations using D$-$operator methods:

$3\mathrm{.}1(D^2+D-2)y=x^2+e^{2x}$

$3\mathrm{.}2(D^2-4D+4)y=e^{2x}(x^2+1)$

$[17]$

QUESTION $4$

Solve only for $y$ in the following set of simultaneous differential equations by using

D$-$operator methods:

$(2D-1)x+(D+1)y=5sint$

$(3D-1)x+(2D+1)y=e^t$

QUESTION $5$

Determine:

$5\mathrm{.}1L\{H(t-3)\}$

$5\mathrm{.}2$ Find the inverse Laplace transform of $L^{-1}\{\frac{5s+2}{(s+1)(s+2)}\}.$

$[6]$

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QUESTION $6$

$-4-$

MAT$3700$

May$/$June $2023$

A vehicle rests on a spring$-$shock absorber system on each of four wheels. The system yields the model $y^{\text{'}\text{'}}(t)+2y^{\text{'}}(t)+17y(t)=e^{-t}cos4t,$ where $e^{-t}cos4t$ represents the force resulting from a bumpy road. If $y(0)=0$ and $y^{\text{'}}(0)=0,$ use Laplace transforms to solve the differential equation.

$[8]$

QUESTION $7$

The equation of motion of a body performing damped forced oscillations is:

$\frac{d^{2}x}{d{t}^2}+5\frac{dx}{dt}+6x=cost,$ with initial conditions $x(0)=\frac{1}{10}$ and $x^{\text{'}}(0)=0$