Exercise $4(20$ points$)$

Use Laplace transform to solve the given initial$-$value problem

$y^{\text{'}\text{'}}-6y^{\text{'}}9y=t,y(0)=0,y^{\text{'}}(0)=1$

Possible Answers

$($a$.)y(t)=\frac{2}{27}-\frac{1}{9}t\frac{2}{27}{e}^{3t}-\frac{10}{9}t{e}^{3t}$

$($b$.)y(t)=\frac{2}{27}\frac{1}{9}t-\frac{2}{27}{e}^{3t}\frac{10}{9}t{e}^{3t}$

$($c$.)y(t)=\frac{2}{27}\frac{1}{9}t-\frac{2}{27}{e}^{3t}\frac{10}{9}t{e}^{3t}$

$($d$.)y(t)=\frac{2}{27}-\frac{1}{9}t\frac{2}{27}{e}^{-3t}-\frac{10}{9}t{e}^{-3t}$

$($e$.)$ None of the above