$$Consider the problem:

ax$\text{'}\text{'}+$bx$\text{'}+$cx$=$f$($t$),$x$(0)=0,$x$\text{'}(0)=0,$

for time $0$t$<$$,$where a$,$b$,$c $$are constants and f$($t$)$is a known function. We view this problem as a linear system, where f$($t$)$is

a known input and the solution x$($t$)$is the output. Laplace transforms of the input and output functions satisfy the relation

x$($s$)=$H$($s$)$F$($s$),$where we call

H$($s$)=$x$($s$)$F$($s$)=1$as$2+$bs$+$c

the system transfer function.

Suppose an input f$($t$)=6$t$,$when applied to the linear system above, produces the output

x$($t$)=2($e$^-3$t$-1)+$t$($e$^-3$t$+5),$t$0.$

Find x$($s$)=$L$\{$x$($t$)\}$and F$($s$)=$L$\{$f$($t$)\}.$

x$($s$)=$help $($formulas$)$

F$($s$)=$help $($formulas$)$

Now find the system transfer function, H$($s$).$

H$($s$)=$help $($formulas$)$

What will be the output if a Heaviside unit step input f$($t$)=$u$($t$)$is applied to the system?

New x$($t$)=$help $($formulas$)$