Consider the initial value problem for function yy given by$,$

y$$y$6$y$=2$u$($t$5$$4)$cos$(4($t$5$$4)),$y$(0)=0,$y$(0)=0.$y$$y$6$y$=2$u$($t$5$$4)$cos$(4($t$5$$4)),$y$(0)=0,$y$(0)=0.$

where u$($t$$c$)$u$($t$$c$)$ denotes the step function with step at t$=$ct$=$c$.$

Part $1$: Finding F$($s$)$F$($s$)$

$($a$)$ Find the Laplace Transform of the source function, F$($s$)=$L$[2$u$($t$5$$4)$cos$(4($t$5$$4))]$F$($s$)=$L$[2$u$($t$5$$4)$cos$(4($t$5$$4))].$

F$($s$)=$F$($s$)=$

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Note: We are asking for F$($s$)=$L$[2$u$($t$5$$4)$cos$(4($t$5$$4))]$F$($s$)=$L$[2$u$($t$5$$4)$cos$(4($t$5$$4))],$ not for Y$($s$)=$L$[$y$($t$)]$Y$($s$)=$L$[$y$($t$)].$