Problem $51$: $(20$ points$)$

A linear time$-$invariant system with input f$($t$)$ and output y$($t$)$ is represented by the ordinary$-$differential equation

d$2$y

dt$2+12$ dy

dt $+20$ y$($t$)=40$ f$($t$).$

Suppose that the initial conditions are y$(0)=2,$y$(0)=0$ and that the input to the system is

f$($t$)=(1+2$e$2$t$)$u$($t$).$

$1.(7$ points$)$ Using the method of Laplace transforms, solve for the zero$-$input response yzi $($t$).$

$2.(11$ points$)$ Using the method of Laplace transforms, solve for the zero$-$state response yzs$($t$).$

$3.(2$ point$)$ Determine the total response y$($t$)$ of the system.

Verify your results by reviewing the solutions to Problem Set $3,$ Parts $13\mathrm{.}4$ and $13\mathrm{.}5.$