$(25$ points$)$ Use the Laplace Transform to solve the initial value problem:

$y^{\text{'}\text{'}}2y^{\text{'}}2y=2t,y(0)=2,y^{\text{'}}(0)=-7$

After your complete work and finding the answer, choose the right answer from the following possible choices:

a$.y(t)=t-1e^{-t}(3cos(t)-5sin(t))$

b$.y(t)=1-t{e}^{-t}(5cos(t)-3sin(t))$

c$.y(t)=1t{e}^{-t}(5cos(t)3sin(t))$

d$.$ None of the above