Find the Laplace transforms of the given function.

$($a$),f(t)=t^{5}3t^3-4$

$($b$)f(t)=2sin4t-4cos2t$

$(1$ mark$)$

$($c$)f(t)=32tcos4t$

$(1$ mark$)$

$(2$ marks$)$

Find the inverse transforms of the given function.

$($d$)F(s)=\frac{1}{s}-\frac{2a}{s^2}\frac{12b}{(s^{2}9)}$

$(2$ marks$)$

$($c$),F(s)=\frac{3s1}{(s-1)(s^{2}1)}$

$(4$ marks$)$

The function $y(t)$ is subject to the initial condition $y(0)=0$ and $y^{\text{'}}(0)=0.$

$2y^{\text{'}\text{'}}8y=3sin2t$

$($f$)$ Solve the given differential equation.

$(10$ marks$)$